Gamma distribution

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Gamma distribution - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"81a02315-9431-48c8-8f07-c42f065da46b","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Gamma_distribution","wgTitle":"Gamma distribution","wgCurRevisionId":1258598224,"wgRevisionId":1258598224,"wgArticleId":207079,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","CS1 maint: multiple names: authors list","CS1 maint: numeric names: authors list","Articles with short description","Short description matches Wikidata","All articles with unsourced statements","Articles with unsourced statements from September 2012","Articles with unsourced statements from May 2019","Continuous distributions","Factorial and binomial topics","Conjugate prior distributions","Exponential family distributions", "Infinitely divisible probability distributions","Survival analysis","Gamma and related functions"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Gamma_distribution","wgRelevantArticleId":207079,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":70000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled" :false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q117806","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=[ "ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/1200px-Gamma_distribution_pdf.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="900"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/800px-Gamma_distribution_pdf.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="600"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/640px-Gamma_distribution_pdf.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="480"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Gamma distribution - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Gamma_distribution"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Gamma_distribution&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Gamma_distribution"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Gamma_distribution rootpage-Gamma_distribution skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Gamma+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Gamma+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Gamma+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Gamma+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> 1 Definitions </div> </a> <button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definitions subsection </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Characterization_using_shape_α_and_rate_β" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_using_shape_α_and_rate_β"> <div class="vector-toc-text"> 1.1 Characterization using shape α and rate β </div> </a> <ul id="toc-Characterization_using_shape_α_and_rate_β-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization_using_shape_k_and_scale_θ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_using_shape_k_and_scale_θ"> <div class="vector-toc-text"> 1.2 Characterization using shape k and scale θ </div> </a> <ul id="toc-Characterization_using_shape_k_and_scale_θ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 2 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Mean_and_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mean_and_variance"> <div class="vector-toc-text"> 2.1 Mean and variance </div> </a> <ul id="toc-Mean_and_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skewness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skewness"> <div class="vector-toc-text"> 2.2 Skewness </div> </a> <ul id="toc-Skewness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_moments"> <div class="vector-toc-text"> 2.3 Higher moments </div> </a> <ul id="toc-Higher_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Median_approximations_and_bounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Median_approximations_and_bounds"> <div class="vector-toc-text"> 2.4 Median approximations and bounds </div> </a> <ul id="toc-Median_approximations_and_bounds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Summation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Summation"> <div class="vector-toc-text"> 2.5 Summation </div> </a> <ul id="toc-Summation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling"> <div class="vector-toc-text"> 2.6 Scaling </div> </a> <ul id="toc-Scaling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponential_family" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponential_family"> <div class="vector-toc-text"> 2.7 Exponential family </div> </a> <ul id="toc-Exponential_family-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithmic_expectation_and_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_expectation_and_variance"> <div class="vector-toc-text"> 2.8 Logarithmic expectation and variance </div> </a> <ul id="toc-Logarithmic_expectation_and_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Information_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_entropy"> <div class="vector-toc-text"> 2.9 Information entropy </div> </a> <ul id="toc-Information_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kullback–Leibler_divergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kullback–Leibler_divergence"> <div class="vector-toc-text"> 2.10 Kullback–Leibler divergence </div> </a> <ul id="toc-Kullback–Leibler_divergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laplace_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace_transform"> <div class="vector-toc-text"> 2.11 Laplace transform </div> </a> <ul id="toc-Laplace_transform-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> 3 Related distributions </div> </a> <button aria-controls="toc-Related_distributions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related distributions subsection </button> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> 3.1 General </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compound_gamma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compound_gamma"> <div class="vector-toc-text"> 3.2 Compound gamma </div> </a> <ul id="toc-Compound_gamma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weibull_and_stable_count" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weibull_and_stable_count"> <div class="vector-toc-text"> 3.3 Weibull and stable count </div> </a> <ul id="toc-Weibull_and_stable_count-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Statistical_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Statistical_inference"> <div class="vector-toc-text"> 4 Statistical inference </div> </a> <button aria-controls="toc-Statistical_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Statistical inference subsection </button> <ul id="toc-Statistical_inference-sublist" class="vector-toc-list"> <li id="toc-Parameter_estimation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parameter_estimation"> <div class="vector-toc-text"> 4.1 Parameter estimation </div> </a> <ul id="toc-Parameter_estimation-sublist" class="vector-toc-list"> <li id="toc-Maximum_likelihood_estimation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Maximum_likelihood_estimation"> <div class="vector-toc-text"> 4.1.1 Maximum likelihood estimation </div> </a> <ul id="toc-Maximum_likelihood_estimation-sublist" class="vector-toc-list"> <li id="toc-Caveat_for_small_shape_parameter" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Caveat_for_small_shape_parameter"> <div class="vector-toc-text"> 4.1.1.1 Caveat for small shape parameter </div> </a> <ul id="toc-Caveat_for_small_shape_parameter-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Closed-form_estimators" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Closed-form_estimators"> <div class="vector-toc-text"> 4.1.2 Closed-form estimators </div> </a> <ul id="toc-Closed-form_estimators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bayesian_minimum_mean_squared_error" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bayesian_minimum_mean_squared_error"> <div class="vector-toc-text"> 4.1.3 Bayesian minimum mean squared error </div> </a> <ul id="toc-Bayesian_minimum_mean_squared_error-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bayesian_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bayesian_inference"> <div class="vector-toc-text"> 4.2 Bayesian inference </div> </a> <ul id="toc-Bayesian_inference-sublist" class="vector-toc-list"> <li id="toc-Conjugate_prior" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conjugate_prior"> <div class="vector-toc-text"> 4.2.1 Conjugate prior </div> </a> <ul id="toc-Conjugate_prior-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Occurrence_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Occurrence_and_applications"> <div class="vector-toc-text"> 5 Occurrence and applications </div> </a> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_variate_generation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Random_variate_generation"> <div class="vector-toc-text"> 6 Random variate generation </div> </a> <ul id="toc-Random_variate_generation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 7 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 8 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Gamma distribution</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 29 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-29" aria-hidden="true" > 29 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%BA%D8%A7%D9%85%D8%A7" title="توزيع غاما – Arabic" lang="ar" hreflang="ar" data-title="توزيع غاما" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qamma_paylanmas%C4%B1" title="Qamma paylanması – Azerbaijani" lang="az" hreflang="az" data-title="Qamma paylanması" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%AE%E0%A6%BE_%E0%A6%AC%E0%A6%A3%E0%A7%8D%E0%A6%9F%E0%A6%A8" title="গামা বণ্টন – Bangla" lang="bn" hreflang="bn" data-title="গামা বণ্টন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B0-%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Гама-размеркаванне – Belarusian" lang="be" hreflang="be" data-title="Гама-размеркаванне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_gamma" title="Distribució gamma – Catalan" lang="ca" hreflang="ca" data-title="Distribució gamma" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rozd%C4%9Blen%C3%AD_gama" title="Rozdělení gama – Czech" lang="cs" hreflang="cs" data-title="Rozdělení gama" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gammaverteilung" title="Gammaverteilung – German" lang="de" hreflang="de" data-title="Gammaverteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_gamma" title="Distribución gamma – Spanish" lang="es" hreflang="es" data-title="Distribución gamma" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%DA%AF%D8%A7%D9%85%D8%A7" title="توزیع گاما – Persian" lang="fa" hreflang="fa" data-title="توزیع گاما" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_Gamma" title="Loi Gamma – French" lang="fr" hreflang="fr" data-title="Loi Gamma" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_gamma" title="Distribución gamma – Galician" lang="gl" hreflang="gl" data-title="Distribución gamma" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%90%EB%A7%88_%EB%B6%84%ED%8F%AC" title="감마 분포 – Korean" lang="ko" hreflang="ko" data-title="감마 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_Gamma" title="Distribuzione Gamma – Italian" lang="it" hreflang="it" data-title="Distribuzione Gamma" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%92%D7%9E%D7%90" title="התפלגות גמא – Hebrew" lang="he" hreflang="he" data-title="התפלגות גמא" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gamma-eloszl%C3%A1s" title="Gamma-eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Gamma-eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gamma-verdeling" title="Gamma-verdeling – Dutch" lang="nl" hreflang="nl" data-title="Gamma-verdeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AC%E3%83%B3%E3%83%9E%E5%88%86%E5%B8%83" title="ガンマ分布 – Japanese" lang="ja" hreflang="ja" data-title="ガンマ分布" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_gamma" title="Rozkład gamma – Polish" lang="pl" hreflang="pl" data-title="Rozkład gamma" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_gama" title="Distribuição gama – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição gama" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%BC%D0%B0-%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Гамма-распределение – Russian" lang="ru" hreflang="ru" data-title="Гамма-распределение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Gama_rozdelenie" title="Gama rozdelenie – Slovak" lang="sk" hreflang="sk" data-title="Gama rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Porazdelitev_gama" title="Porazdelitev gama – Slovenian" lang="sl" hreflang="sl" data-title="Porazdelitev gama" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_gamma" title="Sebaran gamma – Sundanese" lang="su" hreflang="su" data-title="Sebaran gamma" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gamma-jakauma" title="Gamma-jakauma – Finnish" lang="fi" hreflang="fi" data-title="Gamma-jakauma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gammaf%C3%B6rdelning" title="Gammafördelning – Swedish" lang="sv" hreflang="sv" data-title="Gammafördelning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gamma_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Gamma dağılımı – Turkish" lang="tr" hreflang="tr" data-title="Gamma dağılımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%BC%D0%B0-%D1%80%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB" title="Гамма-розподіл – Ukrainian" lang="uk" hreflang="uk" data-title="Гамма-розподіл" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BC%BD%E7%91%AA%E5%88%86%E4%BD%88" title="伽瑪分佈 – Cantonese" lang="yue" hreflang="yue" data-title="伽瑪分佈" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BC%BD%E7%8E%9B%E5%88%86%E5%B8%83" title="伽玛分布 – Chinese" lang="zh" hreflang="zh" data-title="伽玛分布" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q117806#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Gamma_distribution" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Gamma_distribution" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Gamma_distribution">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Gamma_distribution&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Gamma_distribution&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Gamma_distribution">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Gamma_distribution&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Gamma_distribution&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Gamma_distribution" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Gamma_distribution" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Gamma_distribution&oldid=1258598224" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Gamma_distribution&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Gamma_distribution&id=1258598224&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGamma_distribution">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGamma_distribution">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Gamma_distribution&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Gamma_distribution&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Gamma_distribution" hreflang="en">Wikimedia Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Statistics/Distributions/Gamma" hreflang="en">Wikibooks</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q117806" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1247679731">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Gamma</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability density function</div><a href="/wiki/File:Gamma_distribution_pdf.svg" class="mw-file-description" title="Probability density plots of gamma distributions"><img alt="Probability density plots of gamma distributions" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/325px-Gamma_distribution_pdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/488px-Gamma_distribution_pdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/650px-Gamma_distribution_pdf.svg.png 2x" data-file-width="800" data-file-height="600" /></a></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><a href="/wiki/File:Gamma_distribution_cdf.svg" class="mw-file-description" title="Cumulative distribution plots of gamma distributions"><img alt="Cumulative distribution plots of gamma distributions" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/325px-Gamma_distribution_cdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/488px-Gamma_distribution_cdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Gamma_distribution_cdf.svg/650px-Gamma_distribution_cdf.svg.png 2x" data-file-width="800" data-file-height="600" /></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td class="infobox-data infobox-data-a"> <ul><li>k > 0 <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li> <li>θ > 0 <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li></ul></td><td class="infobox-data infobox-data-b"> <div><ul><li>α > 0 <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li><li>β > 0 <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb9a21cd92ae991a12ba8aaa186dd60922862c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.5ex; height:2.843ex;" alt="{\displaystyle x\in (0,\infty )}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb9a21cd92ae991a12ba8aaa186dd60922862c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.5ex; height:2.843ex;" alt="{\displaystyle x\in (0,\infty )}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959c7540287d0d1be72c2e39826eaf2060ce907e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.651ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3373bd46a7261833556cec0a5c95f51ab4b0e74d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.293ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206171e583bb2aa2e14a65530b7cd9e5596d922e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.124ex; height:6.009ex;" alt="{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd8662ff6fdff9e9f96399ac86737471db7d976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.819ex; height:6.009ex;" alt="{\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128f9dbb916e39f620213e206592bf14f13554d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.302ex; height:2.176ex;" alt="{\displaystyle k\theta }"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9485a5ec6938007ea1b5f5f09946f39a14a42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.324ex; height:5.176ex;" alt="{\displaystyle {\frac {\alpha }{\beta }}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td class="infobox-data infobox-data-a"> No simple closed form</td><td class="infobox-data infobox-data-b"> No simple closed form</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9e723ed5d8686d56eda7a66c3f1a1fe3db880f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.533ex; height:2.843ex;" alt="{\displaystyle (k-1)\theta {\text{ for }}k\geq 1}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{\text{ for }}k<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>k</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{\text{ for }}k<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83ac368070086266d1302f673abb085ec278d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.582ex; height:2.176ex;" alt="{\displaystyle 0{\text{ for }}k<1}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>β</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>α</mi> <mo>≥</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>, </mtext> </mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>α</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0209d0c976e5707524172b0d4071a3a73ad6f12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.108ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\theta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\theta ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9fb845e6076205ac8deabbbc533acedc15103d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.676ex;" alt="{\displaystyle k\theta ^{2}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84135b4731462ff6c9a9c23a434a06e9305ee85e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:3.227ex; height:5.343ex;" alt="{\displaystyle {\frac {\alpha }{\beta ^{2}}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\sqrt {k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mi>k</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45eff79aec609af84bf67d9272aebc1fa8681994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.983ex; height:6.176ex;" alt="{\displaystyle {\frac {2}{\sqrt {k}}}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\sqrt {\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mi>α</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8a164ebce39b73870af9d23275a9a76c15e6eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:4.26ex; height:6.176ex;" alt="{\displaystyle {\frac {2}{\sqrt {\alpha }}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Excess kurtosis</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea39cf9fc4c3ff5dc63035a85205b437db7e426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.047ex; height:5.343ex;" alt="{\displaystyle {\frac {6}{k}}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mi>α</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{\alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ce5c506f119628c417e693b0368db51ee874ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.324ex; height:5.176ex;" alt="{\displaystyle {\frac {6}{\alpha }}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badcd9de56aee1547b1743cd802484471cf3555c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.86ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a73a699028dd881daebebc59d36f894e74f187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.655ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>t</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>θ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48369eff20942cb3a2e4cfbef6c31d75c2fca94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.993ex; height:5.343ex;" alt="{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>β</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>t</mi> <mo><</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae293b4a1441aae3cafacb8a4ce61d23bfa48cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.372ex; height:6.509ex;" alt="{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\theta it)^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mi>i</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\theta it)^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5768a597d0f48f216a027bd3e92a8b3e67e0371b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.912ex; height:3.176ex;" alt="{\displaystyle (1-\theta it)^{-k}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>t</mi> </mrow> <mi>β</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9b1c060b013f2c7f698d58711983d9ddf0b18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.465ex; height:6.509ex;" alt="{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi>k</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ca1fa0ecbcf09ce7fc348b6c3363faac0408e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.412ex; height:6.509ex;" alt="{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi>α</mi> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6881c4895c0e9bbc951e99f996f94e2a3690aae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.05ex; height:6.509ex;" alt="{\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></th><td class="infobox-data infobox-data-a"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>V</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79060985aa8683bbf0b380d57ca56522822342ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.895ex; height:6.676ex;" alt="{\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de7bf8b64325f4129e05929e2385f3ca37bb88bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.731ex; height:6.509ex;" alt="{\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }"></td><td class="infobox-data infobox-data-b"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>V</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87074b8ec525badd064920b64dcff7be1c51ceaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.526ex; height:6.676ex;" alt="{\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\frac {E[X]}{V[X]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>V</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {E[X]}{V[X]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187bc571898043026331662ae41bb70d4104d429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.328ex; height:6.509ex;" alt="{\displaystyle \beta ={\frac {E[X]}{V[X]}}}"></td></tr></tbody></table> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the gamma distribution is a versatile two-<a href="/wiki/Statistical_parameter" title="Statistical parameter">parameter</a> family of continuous <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>.<a href="#cite_note-1">[1]</a> The <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>, and <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> are special cases of the gamma distribution.<a href="#cite_note-2">[2]</a> There are two equivalent parameterizations in common use: <ol><li>With a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> k and a <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> θ</li> <li>With a shape parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492bbf1283920d57a534b63fca94a2e2ff4d1946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.797ex; height:2.176ex;" alt="{\displaystyle \alpha =k}"> and a <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate parameter</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =1/\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664250d7209fe5e66b2de5a0c74c7a28ef17b395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \beta =1/\theta }">⁠</li></ol> In each of these forms, both parameters are positive real numbers. The distribution has important applications in various fields, including <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, life testing.<a href="#cite_note-3">[3]</a> In econometrics, the (k, θ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a> for integer k values. Bayesian statistics prefer the (α, β) parameterization, utilizing the gamma distribution as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations. The probability density and cumulative distribution functions of the gamma distribution vary based on the chosen parameterization, both offering insights into the behavior of gamma-distributed random variables. The gamma distribution is integral to modeling a range of phenomena due to its flexible shape, which can capture various statistical distributions, including the exponential and chi-squared distributions under specific conditions. Its mathematical properties, such as mean, variance, skewness, and higher moments, provide a toolset for statistical analysis and inference. Practical applications of the distribution span several disciplines, underscoring its importance in theoretical and applied statistics.<a href="#cite_note-4">[4]</a> The gamma distribution is the <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy probability distribution</a> (both with respect to a uniform base measure and a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/x}"> base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln X] = ψ(k) + ln θ = ψ(α) − ln β is fixed (ψ is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a>).<a href="#cite_note-5">[5]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=1" title="Edit section: Definitions">edit</a>]</div> The parameterization with k and θ appears to be more common in <a href="/wiki/Econometrics" title="Econometrics">econometrics</a> and other applied fields, where the gamma distribution is frequently used to model waiting times. For instance, in <a href="/wiki/Accelerated_life_testing" title="Accelerated life testing">life testing</a>, the waiting time until death is a <a href="/wiki/Random_variable" title="Random variable">random variable</a> that is frequently modeled with a gamma distribution. See Hogg and Craig<a href="#cite_note-6">[6]</a> for an explicit motivation. The parameterization with α and β is more common in <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, where the gamma distribution is used as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> distribution for various types of inverse scale (rate) parameters, such as the λ of an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> or a <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a><a href="#cite_note-7">[7]</a> – or for that matter, the β of the gamma distribution itself. The closely related <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse-gamma distribution</a> is used as a conjugate prior for scale parameters, such as the <a href="/wiki/Variance" title="Variance">variance</a> of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. If k is a positive <a href="/wiki/Integer" title="Integer">integer</a>, then the distribution represents an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>; i.e., the sum of k independent <a href="/wiki/Exponentially_distributed" class="mw-redirect" title="Exponentially distributed">exponentially distributed</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a>, each of which has a mean of θ. <div class="mw-heading mw-heading3"><h3 id="Characterization_using_shape_α_and_rate_β">Characterization using shape α and rate β</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=2" title="Edit section: Characterization using shape α and rate β">edit</a>]</div> The gamma distribution can be parameterized in terms of a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> α = k and an inverse scale parameter β = 1/θ, called a <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate parameter</a>. A random variable X that is gamma-distributed with shape α and rate β is denoted <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>Gamma</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e9787d3e7e96c76c9b0e247fecbf8d3b25335d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.976ex; height:2.843ex;" alt="{\displaystyle X\sim \Gamma (\alpha ,\beta )\equiv \operatorname {Gamma} (\alpha ,\beta )}"> The corresponding probability density function in the shape-rate parameterization is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf760a328d5b468fea5f9f1d47cca54b558b6da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.747ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta x}\beta ^{\alpha }}{\Gamma (\alpha )}}\quad {\text{ for }}x>0\quad \alpha ,\beta >0,\\[6pt]\end{aligned}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (\alpha )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffad1626eab58fe191891edd57e7d27fa9a2def8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.75ex; height:2.843ex;" alt="{\displaystyle \Gamma (\alpha )}"> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. For all positive integers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\alpha )=(\alpha -1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (\alpha )=(\alpha -1)!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df21f16281b82ecb7a2dc5132fc54e88741b6855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.795ex; height:2.843ex;" alt="{\displaystyle \Gamma (\alpha )=(\alpha -1)!}">. The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is the regularized gamma function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6e29c7cfefa404f14fea26c52ea1471116570c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.963ex; height:6.509ex;" alt="{\displaystyle F(x;\alpha ,\beta )=\int _{0}^{x}f(u;\alpha ,\beta )\,du={\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (\alpha ,\beta x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (\alpha ,\beta x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0387febf90a45b820b486bf53b5a89a6d14fa664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.255ex; height:2.843ex;" alt="{\displaystyle \gamma (\alpha ,\beta x)}"> is the lower <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>. If α is a positive <a href="/wiki/Integer" title="Integer">integer</a> (i.e., the distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>), the cumulative distribution function has the following series expansion:<a href="#cite_note-Papoulis-8">[8]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>x</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56150efa42bf73a899c72690ba1004ffcfa397e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.652ex; height:7.343ex;" alt="{\displaystyle F(x;\alpha ,\beta )=1-\sum _{i=0}^{\alpha -1}{\frac {(\beta x)^{i}}{i!}}e^{-\beta x}=e^{-\beta x}\sum _{i=\alpha }^{\infty }{\frac {(\beta x)^{i}}{i!}}.}"> <div class="mw-heading mw-heading3"><h3 id="Characterization_using_shape_k_and_scale_θ">Characterization using shape k and scale θ</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=3" title="Edit section: Characterization using shape k and scale θ">edit</a>]</div> A random variable X that is gamma-distributed with shape k and scale θ is denoted by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>Gamma</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b0320535d26cf0c3b96e123ab0a046b6db3b1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.941ex; height:2.843ex;" alt="{\displaystyle X\sim \Gamma (k,\theta )\equiv \operatorname {Gamma} (k,\theta )}"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma-PDF-3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/320px-Gamma-PDF-3D.png" decoding="async" width="320" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/480px-Gamma-PDF-3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Gamma-PDF-3D.png/640px-Gamma-PDF-3D.png 2x" data-file-width="2040" data-file-height="1653" /></a><figcaption>Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5, and 6. One can see each θ layer by itself here <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-k.png">[2]</a> as well as by k <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-Theta.png">[3]</a> and x. <a class="external autonumber" href="https://commons.wikimedia.org/wiki/File:Gamma-PDF-3D-by-x.png">[4]</a>.</figcaption></figure> The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> using the shape-scale parametrization is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf176962d326ad7af8186d5f4cd3f3e7fae4852" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.381ex; height:6.509ex;" alt="{\displaystyle f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.}"> Here Γ(k) is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> evaluated at k. The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is the regularized gamma function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10c41c86ac78205b3f7cd5a1ba84052714ff662" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.119ex; height:8.343ex;" alt="{\displaystyle F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bab1607e067f732c5fc272613cb87e793c79af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.836ex; height:5.009ex;" alt="{\displaystyle \gamma \left(k,{\frac {x}{\theta }}\right)}"> is the lower <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>. It can also be expressed as follows, if k is a positive <a href="/wiki/Integer" title="Integer">integer</a> (i.e., the distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a>):<a href="#cite_note-Papoulis-8">[8]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b50954cd5d3328b10d9922b97b893c25cd76a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.275ex; height:7.343ex;" alt="{\displaystyle F(x;k,\theta )=1-\sum _{i=0}^{k-1}{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}e^{-x/\theta }=e^{-x/\theta }\sum _{i=k}^{\infty }{\frac {1}{i!}}\left({\frac {x}{\theta }}\right)^{i}.}"> Both parametrizations are common because either can be more convenient depending on the situation. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=4" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Mean_and_variance">Mean and variance</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=5" title="Edit section: Mean and variance">edit</a>]</div> The mean of gamma distribution is given by the product of its shape and scale parameters: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k\theta =\alpha /\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>k</mi> <mi>θ</mi> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =k\theta =\alpha /\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a58ddcce52712f203ad4142baba4a474b8bef8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.882ex; height:2.843ex;" alt="{\displaystyle \mu =k\theta =\alpha /\beta }"> The variance is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>k</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88cc5e24c9d0eb7176ce48ec7069281aea69a2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.979ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=k\theta ^{2}=\alpha /\beta ^{2}}"> The square root of the inverse shape parameter gives the <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.5</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52db1d7831315dcadbff298e415c9fc48de55076" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.663ex; height:3.343ex;" alt="{\displaystyle \sigma /\mu =k^{-0.5}=1/{\sqrt {\alpha }}}"> <div class="mw-heading mw-heading3"><h3 id="Skewness">Skewness</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=6" title="Edit section: Skewness">edit</a>]</div> The <a href="/wiki/Skewness" title="Skewness">skewness</a> of the gamma distribution only depends on its shape parameter, k, and it is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/{\sqrt {k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2/{\sqrt {k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7958ea261b36951e15f1c9363cd01a384a8217d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.119ex; height:3.176ex;" alt="{\displaystyle 2/{\sqrt {k}}.}"> <div class="mw-heading mw-heading3"><h3 id="Higher_moments">Higher moments</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=7" title="Edit section: Higher moments">edit</a>]</div> The n-th <a href="/wiki/Raw_moment" class="mw-redirect" title="Raw moment">raw moment</a> is given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b2480d1a1107cd132f94b5280db23ddde59033" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.709ex; height:6.843ex;" alt="{\displaystyle \mathrm {E} [X^{n}]=\theta ^{n}{\frac {\Gamma (k+n)}{\Gamma (k)}}=\theta ^{n}\prod _{i=1}^{n}(k+i-1)\;{\text{ for }}n=1,2,\ldots .}"> <div class="mw-heading mw-heading3"><h3 id="Median_approximations_and_bounds">Median approximations and bounds</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=8" title="Edit section: Median approximations and bounds">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/320px-Gamma_distribution_median_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/480px-Gamma_distribution_median_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gamma_distribution_median_bounds.png/640px-Gamma_distribution_median_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption>Bounds and asymptotic approximations to the median of the gamma distribution. The cyan-colored region indicates the large gap between published lower and upper bounds before 2021.</figcaption></figure> Unlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation. The median for this distribution is the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932c212790dbeb09a8b9244e3eeb9b8fbb31b309" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.766ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}\int _{0}^{\nu }x^{k-1}e^{-x/\theta }dx={\frac {1}{2}}.}"> A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the gamma distribution was handled first by Chen and Rubin, who proved that (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f287a8873fea8959af7d41cd891678d8875d3a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =1}">) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo><</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50148b370f2224a88b71230d9b89d393ce2a35ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.358ex; height:5.176ex;" alt="{\displaystyle k-{\frac {1}{3}}<\nu (k)<k,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (k)=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (k)=k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dd26b6f7f7f39a301f14b3c05dff07882ba906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.732ex; height:2.843ex;" alt="{\displaystyle \mu (k)=k}"> is the mean and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7023bd901ef6b557d850ffa2089200e625d370e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.253ex; height:2.843ex;" alt="{\displaystyle \nu (k)}"> is the median of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Gamma}}(k,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Gamma}}(k,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0648b913658b37bf6c4db6034c038b8e385e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.238ex; height:2.843ex;" alt="{\displaystyle {\text{Gamma}}(k,1)}"> distribution.<a href="#cite_note-9">[9]</a> For other values of the scale parameter, the mean scales to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>k</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =k\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e18aba75babe49bc0b1eb2a0e35726e395b93b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.802ex; height:2.676ex;" alt="{\displaystyle \mu =k\theta }">, and the median bounds and approximations would be similarly scaled by θ. K. P. Choi found the first five terms in a <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a> asymptotic approximation of the median by comparing the median to <a href="/wiki/Ramanujan_theta_function" title="Ramanujan theta function">Ramanujan's <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> function</a>.<a href="#cite_note-10">[10]</a> Berg and Pedersen found more terms:<a href="#cite_note-Pedersen,_Henrik_L.-2006-11">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mrow> <mn>405</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>184</mn> <mrow> <mn>25515</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2248</mn> <mrow> <mn>3444525</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19006408</mn> <mrow> <mn>15345358875</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06425d870fa9b7a504eaaef775e51f7937ac4b24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:83.426ex; height:6.176ex;" alt="{\displaystyle \nu (k)=k-{\frac {1}{3}}+{\frac {8}{405k}}+{\frac {184}{25515k^{2}}}+{\frac {2248}{3444525k^{3}}}-{\frac {19006408}{15345358875k^{4}}}-O\left({\frac {1}{k^{5}}}\right)+\cdots }"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_Lyon_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/320px-Gamma_distribution_median_Lyon_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/480px-Gamma_distribution_median_Lyon_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Gamma_distribution_median_Lyon_bounds.png/640px-Gamma_distribution_median_Lyon_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption> Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de282e9589256ce4d8b25101b4a64b840fa4c820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.129ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx 2^{-1/k}(A+k)}">, and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. The cyan shaded region is the remaining gap between upper and lower bounds (or conjectured bounds), including these new bounds and the bounds in the previous figure.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gamma_distribution_median_loglog_bounds.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/320px-Gamma_distribution_median_loglog_bounds.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/480px-Gamma_distribution_median_loglog_bounds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Gamma_distribution_median_loglog_bounds.png/640px-Gamma_distribution_median_loglog_bounds.png 2x" data-file-width="2333" data-file-height="1750" /></a><figcaption><a href="/wiki/Log%E2%80%93log_plot" title="Log–log plot">Log–log plot</a> of upper (solid) and lower (dashed) bounds to the median of a gamma distribution and the gaps between them. The green, yellow, and cyan regions represent the gap before the Lyon 2021 paper. The green and yellow narrow that gap with the lower bounds that Lyon proved. Lyon's bounds proved in 2023 further narrow the yellow. Mostly within the yellow, closed-form rational-function-interpolated conjectured bounds are plotted along with the numerically calculated median (dotted) value. Tighter interpolated bounds exist but are not plotted, as they would not be resolved at this scale.</figcaption></figure> Partial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated. Berg and Pedersen also proved many properties of the median, showing that it is a convex function of k,<a href="#cite_note-Berg-12">[12]</a> and that the asymptotic behavior near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8e0dbcff88172cb178dc51b01046ea6a4a0c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.012ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx e^{-\gamma }2^{-1/k}}"> (where γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>), and that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"> the median is bounded by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mo><</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f7ba6d1b9eabdd7621fc99f96065e5b1f96dda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.962ex; height:3.343ex;" alt="{\displaystyle k2^{-1/k}<\nu (k)<ke^{-1/3k}}">.<a href="#cite_note-Pedersen,_Henrik_L.-2006-11">[11]</a> A closer linear upper bound, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"> only, was provided in 2021 by Gaunt and Merkle,<a href="#cite_note-13">[13]</a> relying on the Berg and Pedersen result that the slope of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7023bd901ef6b557d850ffa2089200e625d370e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.253ex; height:2.843ex;" alt="{\displaystyle \nu (k)}"> is everywhere less than 1: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\leq k-1+\log 2~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\leq k-1+\log 2~~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b15d0fc8940dbe55f21e5697550d78722c4983" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.088ex; height:2.843ex;" alt="{\displaystyle \nu (k)\leq k-1+\log 2~~}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"> (with equality at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}">) which can be extended to a bound for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"> by taking the max with the chord shown in the figure, since the median was proved convex.<a href="#cite_note-Berg-12">[12]</a> An approximation to the median that is asymptotically accurate at high k and reasonable down to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aea83a57db153f846b8b7b1a8306e59c4e43ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.281ex; height:2.176ex;" alt="{\displaystyle k=0.5}"> or a bit lower follows from the <a href="/wiki/Wilson%E2%80%93Hilferty_transformation" class="mw-redirect" title="Wilson–Hilferty transformation">Wilson–Hilferty transformation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>9</mn> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4061f7fcbab5e475380fda6e2dfcf445d100e0dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.25ex; height:6.509ex;" alt="{\displaystyle \nu (k)=k\left(1-{\frac {1}{9k}}\right)^{3}}"> which goes negative for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<1/9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<1/9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68a68b15e862c49b0f09a218ff1a94b061271e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.797ex; height:2.843ex;" alt="{\displaystyle k<1/9}">. In 2021, Lyon proposed several approximations of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f59966fa882b4fdd8a8abbfe3882d7b0e5cbedda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.893ex; height:3.343ex;" alt="{\displaystyle \nu (k)\approx 2^{-1/k}(A+Bk)}">. He conjectured values of A and B for which this approximation is an asymptotically tight upper or lower bound for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}">.<a href="#cite_note-Lyon-2021a-14">[14]</a> In particular, he proposed these closed-form bounds, which he proved in 2023:<a href="#cite_note-Lyon-2021b-15">[15]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2591377ce5a7b571e43130a7aa0ae5e591c8bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.98ex; height:5.176ex;" alt="{\displaystyle \nu _{L\infty }(k)=2^{-1/k}(\log 2-{\frac {1}{3}}+k)\quad }"> is a lower bound, asymptotically tight as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\to \infty }"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed520e224ba985e574fb77b71ee5bb6883512f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.604ex; height:3.343ex;" alt="{\displaystyle \nu _{U}(k)=2^{-1/k}(e^{-\gamma }+k)\quad }"> is an upper bound, asymptotically tight as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcc764df099ea8bc091518756ccc1221f6e37ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.988ex; height:2.176ex;" alt="{\displaystyle k\to 0}"> Lyon also showed (informally in 2021, rigorously in 2023) two other lower bounds that are not <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expressions</a>, including this one involving the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>, based on solving the integral expression substituting 1 for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b201e900a30da19a1f1e4bdddcc70fe7e502be4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.535ex; height:2.509ex;" alt="{\displaystyle e^{-x}}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>></mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> </msup> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec07f2440f178b43a965d030b6d17daadbacfc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.418ex; height:6.843ex;" alt="{\displaystyle \nu (k)>\left({\frac {2}{\Gamma (k+1)}}\right)^{-1/k}\quad }"> (approaching equality as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcc764df099ea8bc091518756ccc1221f6e37ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.988ex; height:2.176ex;" alt="{\displaystyle k\to 0}">) and the tangent line at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"> where the derivative was found to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0.9680448</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad24afe34a23eb1560b491689eec1e7dea040c63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.959ex; height:3.009ex;" alt="{\displaystyle \nu ^{\prime }(1)\approx 0.9680448}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4d3c22d938a07214c417f6975f7cbd17fbf1a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.655ex; height:3.009ex;" alt="{\displaystyle \nu (k)\geq \nu (1)+(k-1)\nu ^{\prime }(1)\quad }"> (with equality at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}">) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\geq \log 2+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>2</mn> <mi>Ei</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\geq \log 2+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcb4e9ccbc2695aea8316f010e96d42d178e3d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.674ex; height:2.843ex;" alt="{\displaystyle \nu (k)\geq \log 2+(k-1)(\gamma -2\operatorname {Ei} (-\log 2)-\log \log 2)}"> where Ei is the <a href="/wiki/Exponential_integral" title="Exponential integral">exponential integral</a>.<a href="#cite_note-Lyon-2021a-14">[14]</a><a href="#cite_note-Lyon-2021b-15">[15]</a> Additionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (1)=\log 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (1)=\log 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48b9898b0be0be38bbdfc4049164982bce09ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.824ex; height:2.843ex;" alt="{\displaystyle \nu (1)=\log 2}">) and has a maximum relative error less than 0.6%. Interpolated approximations and bounds are all of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8435e11856bb84972296cd69082bd5ac49609938" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.333ex; height:2.843ex;" alt="{\displaystyle \nu (k)\approx {\tilde {g}}(k)\nu _{L\infty }(k)+(1-{\tilde {g}}(k))\nu _{U}(k)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf709e979316ee494a3f076f7e1d97be44a3f8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\displaystyle {\tilde {g}}}"> is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b991fb29325fbaa24480e075e54871f5128a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle g(k)}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ea545145faf7173d894259d15c1b2ed59cf70d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.737ex; height:6.509ex;" alt="{\displaystyle g(k)={\frac {\nu _{U}(k)-\nu (k)}{\nu _{U}(k)-\nu _{L\infty }(k)}}}"> For the simplest interpolating function considered, a first-order rational function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e65ab5d33035c86360b97f1dde6799ddeec9d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.344ex; height:5.843ex;" alt="{\displaystyle {\tilde {g}}_{1}(k)={\frac {k}{b_{0}+k}}}"> the tightest lower bound has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>405</mn> </mfrac> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>≈</mo> <mn>0.143472</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4d17f6dff2d8b4188f7d396887129affd9edb8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:47.587ex; height:8.843ex;" alt="{\displaystyle b_{0}={\frac {{\frac {8}{405}}+e^{-\gamma }\log 2-{\frac {\log ^{2}2}{2}}}{e^{-\gamma }-\log 2+{\frac {1}{3}}}}-\log 2\approx 0.143472}"> and the tightest upper bound has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.374654</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8762f4ac41b74d0ac8c557125bda65e5431ee804" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:33.216ex; height:8.343ex;" alt="{\displaystyle b_{0}={\frac {e^{-\gamma }-\log 2+{\frac {1}{3}}}{1-{\frac {e^{-\gamma }\pi ^{2}}{12}}}}\approx 0.374654}"> The interpolated bounds are plotted (mostly inside the yellow region) in the <a href="/wiki/Log%E2%80%93log_plot" title="Log–log plot">log–log plot</a> shown. Even tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.<a href="#cite_note-Lyon-2021a-14">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Summation">Summation</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=9" title="Edit section: Summation">edit</a>]</div> If Xi has a Gamma(ki, θ) distribution for i = 1, 2, ..., N (i.e., all distributions have the same scale parameter θ), then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d35fd0b7cd574ea4f7a49eb96f107f6ffe5edb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.531ex; height:7.509ex;" alt="{\displaystyle \sum _{i=1}^{N}X_{i}\sim \mathrm {Gamma} \left(\sum _{i=1}^{N}k_{i},\theta \right)}"> provided all Xi are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. For the cases where the Xi are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> but have different scale parameters, see Mathai <a href="#cite_note-16">[16]</a> or Moschopoulos.<a href="#cite_note-17">[17]</a> The gamma distribution exhibits <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinite divisibility</a>. <div class="mw-heading mw-heading3"><h3 id="Scaling">Scaling</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=10" title="Edit section: Scaling">edit</a>]</div> If <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391514411e8eaf5fdc115ca83083262b1fc48424" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.891ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Gamma} (k,\theta ),}"> then, for any c > 0, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615789eea5ea246ac96af8627865abbad8f1b404" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.292ex; height:2.843ex;" alt="{\displaystyle cX\sim \mathrm {Gamma} (k,c\,\theta ),}"> by moment generating functions, or equivalently, if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3247407b9d18e6a48db04762641c9887835677c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.149ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Gamma} \left(\alpha ,\beta \right)}"> (shape-rate parameterization) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>c</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d2967d0b2f02976cb6572d8be0bb3281711d7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.638ex; height:6.176ex;" alt="{\displaystyle cX\sim \mathrm {Gamma} \left(\alpha ,{\frac {\beta }{c}}\right),}"> Indeed, we know that if X is an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential r.v.</a> with rate λ, then cX is an exponential r.v. with rate λ/c; the same thing is valid with Gamma variates (and this can be checked using the <a href="/wiki/Moment-generating_function" title="Moment-generating function">moment-generating function</a>, see, e.g.,<a rel="nofollow" class="external text" href="http://www.stat.washington.edu/thompson/S341_10/Notes/week4.pdf">these notes</a>, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale). <div class="mw-heading mw-heading3"><h3 id="Exponential_family">Exponential family</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=11" title="Edit section: Exponential family">edit</a>]</div> The gamma distribution is a two-parameter <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> with <a href="/wiki/Natural_parameters" class="mw-redirect" title="Natural parameters">natural parameters</a> k − 1 and −1/θ (equivalently, α − 1 and −β), and <a href="/wiki/Natural_statistics" class="mw-redirect" title="Natural statistics">natural statistics</a> X and ln X. If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a <a href="/wiki/Natural_exponential_family" title="Natural exponential family">natural exponential family</a>. <div class="mw-heading mw-heading3"><h3 id="Logarithmic_expectation_and_variance">Logarithmic expectation and variance</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=12" title="Edit section: Logarithmic expectation and variance">edit</a>]</div> One can show that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\ln \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\ln \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e31141e3a3f0f1aa1ec728a7276269b702836c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.59ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\ln \beta }"> or equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\ln X]=\psi (k)+\ln \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\ln X]=\psi (k)+\ln \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac33a1a15f14d72a259a487b16152938fd409770" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.072ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\ln X]=\psi (k)+\ln \theta }"> where ψ is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a>. Likewise, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} [\ln X]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} [\ln X]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c82551037f1e1392e0d1dc34c8192aec01fe09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.11ex; height:3.343ex;" alt="{\displaystyle \operatorname {var} [\ln X]=\psi ^{(1)}(\alpha )=\psi ^{(1)}(k)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{(1)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f824d0b8ec54f716ece0f1d8acf4898cab2f7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.847ex; height:3.176ex;" alt="{\displaystyle \psi ^{(1)}}"> is the <a href="/wiki/Trigamma_function" title="Trigamma function">trigamma function</a>. This can be derived using the <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> formula for the <a href="/wiki/Exponential_family#Moment_generating_function_of_the_sufficient_statistic" title="Exponential family">moment generating function of the sufficient statistic</a>, because one of the sufficient statistics of the gamma distribution is ln x. <div class="mw-heading mw-heading3"><h3 id="Information_entropy">Information entropy</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=13" title="Edit section: Information entropy">edit</a>]</div> The <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln p(X)]\\[4pt]&=\operatorname {E} [-\alpha \ln \beta +\ln \Gamma (\alpha )-(\alpha -1)\ln X+\beta X]\\[4pt]&=\alpha -\ln \beta +\ln \Gamma (\alpha )+(1-\alpha )\psi (\alpha ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">H</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>α</mi> <mi>ln</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> <mo>+</mo> <mi>β</mi> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln p(X)]\\[4pt]&=\operatorname {E} [-\alpha \ln \beta +\ln \Gamma (\alpha )-(\alpha -1)\ln X+\beta X]\\[4pt]&=\alpha -\ln \beta +\ln \Gamma (\alpha )+(1-\alpha )\psi (\alpha ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61733af14d6071115cd2f558874cdac5180f6dfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:50.503ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {H} (X)&=\operatorname {E} [-\ln p(X)]\\[4pt]&=\operatorname {E} [-\alpha \ln \beta +\ln \Gamma (\alpha )-(\alpha -1)\ln X+\beta X]\\[4pt]&=\alpha -\ln \beta +\ln \Gamma (\alpha )+(1-\alpha )\psi (\alpha ).\end{aligned}}}"> In the k, θ parameterization, the <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a> is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} (X)=k+\ln \theta +\ln \Gamma (k)+(1-k)\psi (k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">H</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} (X)=k+\ln \theta +\ln \Gamma (k)+(1-k)\psi (k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c832644ca07117f067d7b85bff6540faf23e08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.784ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} (X)=k+\ln \theta +\ln \Gamma (k)+(1-k)\psi (k).}"> <div class="mw-heading mw-heading3"><h3 id="Kullback–Leibler_divergence">Kullback–Leibler divergence</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=14" title="Edit section: Kullback–Leibler divergence">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma-KL-3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/320px-Gamma-KL-3D.png" decoding="async" width="320" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/480px-Gamma-KL-3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Gamma-KL-3D.png/640px-Gamma-KL-3D.png 2x" data-file-width="2048" data-file-height="1339" /></a><figcaption>Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. Here β = β0 + 1 which are set to 1, 2, 3, 4, 5, and 6. The typical asymmetry for the KL divergence is clearly visible.</figcaption></figure> The <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> (KL-divergence), of Gamma(αp, βp) ("true" distribution) from Gamma(αq, βq) ("approximating" distribution) is given by<a href="#cite_note-18">[18]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5567126eea1d9b2264fbc9b966d3621a739c5de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:62.413ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})={}&(\alpha _{p}-\alpha _{q})\psi (\alpha _{p})-\log \Gamma (\alpha _{p})+\log \Gamma (\alpha _{q})\\&{}+\alpha _{q}(\log \beta _{p}-\log \beta _{q})+\alpha _{p}{\frac {\beta _{q}-\beta _{p}}{\beta _{p}}}.\end{aligned}}}"> Written using the k, θ parameterization, the KL-divergence of Gamma(kp, θp) from Gamma(kq, θq) is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bae9104706395e7b3d9da9f41c6b7a69b8d9423" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:60.028ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(k_{p},\theta _{p};k_{q},\theta _{q})={}&(k_{p}-k_{q})\psi (k_{p})-\log \Gamma (k_{p})+\log \Gamma (k_{q})\\&{}+k_{q}(\log \theta _{q}-\log \theta _{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Laplace_transform">Laplace transform</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=15" title="Edit section: Laplace transform">edit</a>]</div> The <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of the gamma PDF is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>θ</mi> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>β</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d5c9dac2cf6d0ae13d59b77058972caa492522" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.037ex; height:6.176ex;" alt="{\displaystyle F(s)=(1+\theta s)^{-k}={\frac {\beta ^{\alpha }}{(s+\beta )^{\alpha }}}.}"> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=16" title="Edit section: Related distributions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=17" title="Edit section: General">edit</a>]</div> <ul><li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}"> be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> independent and identically distributed random variables following an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with rate parameter λ, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1973eff94821789d2a7ff09c2f5dc31be87f88d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.466ex; height:5.509ex;" alt="{\displaystyle \sum _{i}X_{i}}"> ~ Gamma(n, λ) where n is the shape parameter and λ is the rate, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n*\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>Gamma</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>∗</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n*\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c7f94ec07eab6afc5d1110e916c1c73c980f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.955ex; height:3.343ex;" alt="{\textstyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}\sim \operatorname {Gamma} (n,n*\lambda )}">.</li> <li>If X ~ Gamma(1, λ) (in the shape–rate parametrization), then X has an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with rate parameter λ. In the shape-scale parametrization, X ~ Gamma(1, λ) has an exponential distribution with rate parameter 1/λ.</li> <li>If X ~ Gamma(ν/2, 2) (in the shape–scale parametrization), then X is identical to χ2(ν), the <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with ν degrees of freedom. Conversely, if Q ~ χ2(ν) and c is a positive constant, then cQ ~ Gamma(ν/2, 2c).</li> <li>If θ = 1/k, one obtains the <a href="/wiki/Schulz-Zimm_distribution" class="mw-redirect" title="Schulz-Zimm distribution">Schulz-Zimm distribution</a>, which is most prominently used to model polymer chain lengths.</li> <li>If k is an <a href="/wiki/Integer" title="Integer">integer</a>, the gamma distribution is an <a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a> and is the probability distribution of the waiting time until the k-th "arrival" in a one-dimensional <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> with intensity 1/θ. If</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>Y</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>θ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea8fc216dc5113f4d745dcee1fd769c4608dff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.598ex; height:5.009ex;" alt="{\displaystyle X\sim \Gamma (k\in \mathbf {Z} ,\theta ),\qquad Y\sim \operatorname {Pois} \left({\frac {x}{\theta }}\right),}"></dd></dl></dd> <dd>then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X>x)=P(Y<k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo><</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X>x)=P(Y<k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd3aee7ab286228c1c22b1ad3371d3f9aa2538a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.346ex; height:2.843ex;" alt="{\displaystyle P(X>x)=P(Y<k).}"></dd></dl></dd></dl> <ul><li>If X has a <a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a> with parameter a, then</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519b56baa947c79685e8946bb422b39654551779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.924ex; height:6.176ex;" alt="{\displaystyle X^{2}\sim \Gamma \left({\frac {3}{2}},2a^{2}\right).}"></dd></dl></dd></dl> <ul><li>If X ~ Gamma(k, θ), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b112df3c2dc3c9c78d5b897a9d195737fd12371a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.339ex; height:2.509ex;" alt="{\textstyle \log X}"> follows an exponential-gamma (abbreviated exp-gamma) distribution.<a href="#cite_note-19">[19]</a> It is sometimes referred to as the log-gamma distribution.<a href="#cite_note-20">[20]</a> Formulas for its mean and variance are in the section <a href="#Logarithmic_expectation_and_variance">#Logarithmic expectation and variance</a>.</li> <li>If X ~ Gamma(k, θ), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>X</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {X}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a98f7181b035dbddcd947c5f1408b9c5ed23cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.916ex; height:3.009ex;" alt="{\displaystyle {\sqrt {X}}}"> follows a <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a> with parameters p = 2, d = 2k, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1d6f6db29b41fef2dcdc6f1cfee82644296c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.355ex; height:3.009ex;" alt="{\displaystyle a={\sqrt {\theta }}}"> [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>].</li> <li>More generally, if X ~ Gamma(k,θ), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff55e24541193805d60671f025c948707ce9933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.985ex; height:2.343ex;" alt="{\displaystyle X^{q}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482e0a33d9e8fd6307b5f68a5182c2d0d14efc9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>0}"> follows a <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a> with parameters p = 1/q, d = k/q, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\theta ^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\theta ^{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142cecd27f4bc31247ef7624e1ab3c509f343039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.407ex; height:2.343ex;" alt="{\displaystyle a=\theta ^{q}}">.</li> <li>If X ~ Gamma(k, θ) with shape k and scale θ, then 1/X ~ Inv-Gamma(k, θ−1) (see <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse-gamma distribution</a> for derivation).</li> <li>Parametrization 1: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f4bce9ca7bb5b08324fd5622ccee7d524aed9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})\,}"> are independent, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8dddf330ecfa29ac3839ed463ee66efe4e3f93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.369ex; height:5.843ex;" alt="{\displaystyle {\frac {\alpha _{2}\theta _{2}X_{1}}{\alpha _{1}\theta _{1}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}">, or equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18ad8b86fd514386d69012b06ed5a2724f3141eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.072ex; height:6.176ex;" alt="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\theta _{1}}{\theta _{2}}}\right)}"></li> <li>Parametrization 2: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b3bb7f3563f635649d7dc74ad13f248e80bd53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.775ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}"> are independent, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d4abcc11916af8cb166426737f02fc6feed39e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.595ex; height:5.843ex;" alt="{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}">, or equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8feeac0cc0fe85a8a361119b77b78cf628096f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.297ex; height:6.176ex;" alt="{\displaystyle {\frac {X_{1}}{X_{2}}}\sim \beta '\left(\alpha _{1},\alpha _{2},1,{\frac {\beta _{2}}{\beta _{1}}}\right)}"></li> <li>If X ~ Gamma(α, θ) and Y ~ Gamma(β, θ) are independently distributed, then X/(X + Y) has a <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> with parameters α and β, and X/(X + Y) is independent of X + Y, which is Gamma(α + β, θ)-distributed.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}\sim {\text{Beta}}(\alpha ,n\beta )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Beta</mtext> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>n</mi> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}\sim {\text{Beta}}(\alpha ,n\beta )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddde13b1fdd627576cf721b3323511bda989ac9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.431ex; height:2.843ex;" alt="{\displaystyle X_{n}\sim {\text{Beta}}(\alpha ,n\beta )\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{n}=nX_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{n}=nX_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7da58296a3cdc7a6c3108ed0b779ed734c5dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.205ex; height:2.509ex;" alt="{\displaystyle Y_{n}=nX_{n}}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19a1b3bf39298aacb7e2daeab9320130a986fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle Y_{n}}"> converges in distribution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Gamma}}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Gamma}}(\alpha ,\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67f310d4169a13d4a8ba188dc39bc638f81dfd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.684ex; height:2.843ex;" alt="{\displaystyle {\text{Gamma}}(\alpha ,\beta )}"> defined under parametrization 2.</li> <li>If Xi ~ Gamma(αi, 1) are independently distributed, then the vector (X1/S, ..., Xn/S), where S = X1 + ... + Xn, follows a <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a> with parameters α1, ..., αn.</li> <li>For large k the gamma distribution converges to <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean μ = kθ and variance σ2 = kθ2.</li> <li>The gamma distribution is the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for the precision of the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with known <a href="/wiki/Mean" title="Mean">mean</a>.</li> <li>The <a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">matrix gamma distribution</a> and the <a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart distribution</a> are multivariate generalizations of the gamma distribution (samples are positive-definite matrices rather than positive real numbers).</li> <li>The gamma distribution is a special case of the <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a>, the <a href="/wiki/Generalized_integer_gamma_distribution" title="Generalized integer gamma distribution">generalized integer gamma distribution</a>, and the <a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">generalized inverse Gaussian distribution</a>.</li> <li>Among the discrete distributions, the <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a> is sometimes considered the discrete analog of the gamma distribution.</li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie distributions</a> – the gamma distribution is a member of the family of Tweedie <a href="/wiki/Exponential_dispersion_model" title="Exponential dispersion model">exponential dispersion models</a>.</li> <li>Modified <a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal distribution</a> – the Gamma distribution is a member of the family of <a href="/wiki/Modified_half-normal_distribution" title="Modified half-normal distribution">Modified half-normal distribution</a>.<a href="#cite_note-21">[21]</a> The corresponding density is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>β</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>γ</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <msqrt> <mi>β</mi> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d857ffeb61fc56ae24b19d6a4dbe58e0b8b9f4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:42.281ex; height:10.509ex;" alt="{\displaystyle f(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mo>;</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f318f1c6f5b6c50886d35fe09b9205c3e66784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.494ex; height:7.509ex;" alt="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}"> denotes the <a href="/wiki/Fox%E2%80%93Wright_Psi_function" class="mw-redirect" title="Fox–Wright Psi function">Fox–Wright Psi function</a>.</li> <li>For the shape-scale parameterization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x|\theta \sim \Gamma (k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x|\theta \sim \Gamma (k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0acf17a39a34862746426e99e25aa8ee005b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:2.843ex;" alt="{\displaystyle x|\theta \sim \Gamma (k,\theta )}">, if the scale parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \sim IG(b,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>∼</mo> <mi>I</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \sim IG(b,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7c9a793a31d07edd3f6d59d75202965b08b09d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.191ex; height:2.843ex;" alt="{\displaystyle \theta \sim IG(b,1)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IG}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1cbed103b7527b8761b26fb2b729ad65844d7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.998ex; height:2.176ex;" alt="{\displaystyle IG}"> denotes the <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse-gamma distribution</a>, then the marginal distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim \beta '(k,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim \beta '(k,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a350b021b5230bc13e56f87a39b1c64f816c04c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:3.009ex;" alt="{\displaystyle x\sim \beta '(k,b)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.022ex; height:2.843ex;" alt="{\displaystyle \beta '}"> denotes the <a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime distribution</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Compound_gamma">Compound gamma</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=18" title="Edit section: Compound gamma">edit</a>]</div> If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The <a href="/wiki/Compound_distribution" class="mw-redirect" title="Compound distribution">compound distribution</a>, which results from integrating out the inverse scale, has a closed-form solution known as the <a href="/wiki/Compound_gamma_distribution" class="mw-redirect" title="Compound gamma distribution">compound gamma distribution</a>.<a href="#cite_note-Dubey-1970-22">[22]</a> If, instead, the shape parameter is known but the mean is unknown, with the prior of the mean being given by another gamma distribution, then it results in <a href="/wiki/K-distribution" title="K-distribution">K-distribution</a>. <div class="mw-heading mw-heading3"><h3 id="Weibull_and_stable_count">Weibull and stable count</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=19" title="Edit section: Weibull and stable count">edit</a>]</div> The gamma distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k)\,(k>1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>k</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;k)\,(k>1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e5dddb7d9a9add51167f6c7786a39a798895f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.331ex; height:2.843ex;" alt="{\displaystyle f(x;k)\,(k>1)}"> can be expressed as the product distribution of a <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a> and a variant form of the <a href="/wiki/Stable_count_distribution" title="Stable count distribution">stable count distribution</a>. Its shape parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> can be regarded as the inverse of Lévy's stability parameter in the stable count distribution: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>u</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>k</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fdb1229d8546f9459b5bfdae27e98ee789a192" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.21ex; height:6.176ex;" alt="{\displaystyle f(x;k)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k}\left({\frac {x}{u}}\right)\left[ku^{k-1}\,{\mathfrak {N}}_{\frac {1}{k}}\left(u^{k}\right)\right]\,du,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16503339cce78afe1b7b86dcf6d064fb7f34b979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}"> is a standard stable count distribution of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1/k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe1c09916f6155dccb18cd323dcce7170ad5fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.122ex; height:2.843ex;" alt="{\displaystyle \alpha =1/k}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{k}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb78f76255ce2e2a500863818ce36691169fb9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.421ex; height:2.843ex;" alt="{\displaystyle W_{k}(x)}"> is a standard Weibull distribution of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">. <div class="mw-heading mw-heading2"><h2 id="Statistical_inference">Statistical inference</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=20" title="Edit section: Statistical inference">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Parameter_estimation">Parameter estimation</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=21" title="Edit section: Parameter estimation">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Maximum_likelihood_estimation">Maximum likelihood estimation</h4>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=22" title="Edit section: Maximum likelihood estimation">edit</a>]</div> The likelihood function for N <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> observations (x1, ..., xN) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e59f233200284ea82290f9fe3d1055c1f8bc02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.77ex; height:7.343ex;" alt="{\displaystyle L(k,\theta )=\prod _{i=1}^{N}f(x_{i};k,\theta )}"> from which we calculate the log-likelihood function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln x_{i}-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln \theta -N\ln \Gamma (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>θ</mi> </mfrac> </mrow> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>N</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln x_{i}-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln \theta -N\ln \Gamma (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66eaf70de1c8f25f6531bc30e60177c104dba1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.38ex; height:7.343ex;" alt="{\displaystyle \ell (k,\theta )=(k-1)\sum _{i=1}^{N}\ln x_{i}-\sum _{i=1}^{N}{\frac {x_{i}}{\theta }}-Nk\ln \theta -N\ln \Gamma (k)}"> Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimator of the θ parameter, which equals the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"> divided by the shape parameter k: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b434bb6cc9b63502d363cd5396852735fb532cbf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.088ex; height:7.343ex;" alt="{\displaystyle {\hat {\theta }}={\frac {1}{kN}}\sum _{i=1}^{N}x_{i}={\frac {\bar {x}}{k}}}"> Substituting this into the log-likelihood function gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln x_{i}-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln \Gamma (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>N</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln x_{i}-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln \Gamma (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2d3ca919f335d783d5ad3236b2411caa8d78c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.572ex; height:7.343ex;" alt="{\displaystyle \ell (k)=(k-1)\sum _{i=1}^{N}\ln x_{i}-Nk-Nk\ln \left({\frac {\sum x_{i}}{kN}}\right)-N\ln \Gamma (k)}"> We need at least two samples: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a366b1cf2a65d58e6be08fede1a447e0771926b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.325ex; height:2.343ex;" alt="{\displaystyle N\geq 2}">, because for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85982022b9eb1f295b44de55023687a490db0a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=1}">, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f050e906764d1b0f675a509a5a34a818f656aee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.99ex; height:2.843ex;" alt="{\displaystyle \ell (k)}"> increases without bounds as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf168f27ae911d0e1f71ce7e2c8fc9d1a3adeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\to \infty }">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}">, it can be verified that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f050e906764d1b0f675a509a5a34a818f656aee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.99ex; height:2.843ex;" alt="{\displaystyle \ell (k)}"> is strictly <a href="/wiki/Concave_function" title="Concave function">concave</a>, by using <a href="/wiki/Polygamma_function#Inequalities" title="Polygamma function">inequality properties of the polygamma function</a>. Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln k-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln k-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12fbe70b36b48cf309a3ddd33c2bda2288dc38c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.48ex; height:7.509ex;" alt="{\displaystyle \ln k-\psi (k)=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}"> where ψ is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\ln x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\ln x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/297c72b01e6457eae87c4b56f8bd06e8c4c3b61f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.771ex; height:3.009ex;" alt="{\displaystyle {\overline {\ln x}}}"> is the sample mean of ln x. There is no closed-form solution for k. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>. An initial value of k can be found either using the <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a>, or using the approximation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln k-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln k-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8de4ca637e88113ae889e550bce8282c9654514" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.244ex; height:6.176ex;" alt="{\displaystyle \ln k-\psi (k)\approx {\frac {1}{2k}}\left(1+{\frac {1}{6k+1}}\right)}"> If we let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d5e4e535c59fed0ec5f4f0c71c0abee169540f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.659ex; height:7.509ex;" alt="{\displaystyle s=\ln \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}=\ln {\bar {x}}-{\overline {\ln x}}}"> then k is approximately <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <mi>s</mi> </msqrt> </mrow> </mrow> <mrow> <mn>12</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb19579e029ef52f2677c6aaa01d443a93dc4a85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.616ex; height:6.176ex;" alt="{\displaystyle k\approx {\frac {3-s+{\sqrt {(s-3)^{2}+24s}}}{12s}}}"> which is within 1.5% of the correct value.<a href="#cite_note-23">[23]</a> An explicit form for the Newton–Raphson update of this initial guess is:<a href="#cite_note-24">[24]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leftarrow k-{\frac {\ln k-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">←</mo> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\leftarrow k-{\frac {\ln k-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b13a1ddfbf19881de52a0f0e647b9d7c3dbf6c77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.203ex; height:7.343ex;" alt="{\displaystyle k\leftarrow k-{\frac {\ln k-\psi (k)-s}{{\frac {1}{k}}-\psi ^{\prime }(k)}}.}"> At the maximum-likelihood estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {k}},{\hat {\theta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {k}},{\hat {\theta }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7912fe7770f431c4e61a0e4595918a0848e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.41ex; height:3.343ex;" alt="{\displaystyle ({\hat {k}},{\hat {\theta }})}">, the expected values for x and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}"> agree with the empirical averages: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln {\hat {\theta }}&={\overline {\ln x}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> <mtd> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln {\hat {\theta }}&={\overline {\ln x}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77120c377a372035d5ef92b6e1be681791494fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.358ex; height:3.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln {\hat {\theta }}&={\overline {\ln x}}.\end{aligned}}}"> <div class="mw-heading mw-heading5"><h5 id="Caveat_for_small_shape_parameter">Caveat for small shape parameter</h5>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=23" title="Edit section: Caveat for small shape parameter">edit</a>]</div> For data, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe46c866379cec130e31f750cf08fa3ce5edfaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.393ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{N})}">, that is represented in a <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> format that underflows to 0 for values smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }">, the logarithms that are needed for the maximum-likelihood estimate will cause failure if there are any underflows. If we assume the data was generated by a gamma distribution with cdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52dc64c0a15754ca6e07f1772061da406e13308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.249ex; height:2.843ex;" alt="{\displaystyle F(x;k,\theta )}">, then the probability that there is at least one underflow is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>underflow</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de6fb21a1e6a5e2df503c4a69f91db17c876706" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.184ex; height:3.176ex;" alt="{\displaystyle P({\text{underflow}})=1-(1-F(\varepsilon ;k,\theta ))^{N}}"> This probability will approach 1 for small k and large N. For example, at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=10^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=10^{-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55da0cb5f3e282ffc0d2c211ad2e7c2e24d978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.967ex; height:2.676ex;" alt="{\displaystyle k=10^{-2}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=10^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=10^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92c893655aade068aa13c22966cb222d4bbac837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.541ex; height:2.676ex;" alt="{\displaystyle N=10^{4}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =2.25\times 10^{-308}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>2.25</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>308</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =2.25\times 10^{-308}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753624cf1c18b917bc51c09f9bc7261d1b1fbbe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.458ex; height:2.676ex;" alt="{\displaystyle \varepsilon =2.25\times 10^{-308}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{underflow}})\approx 0.9998}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>underflow</mtext> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0.9998</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\text{underflow}})\approx 0.9998}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4145cd4e3821fab09005f66981c9745abd2ed752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.133ex; height:2.843ex;" alt="{\displaystyle P({\text{underflow}})\approx 0.9998}">. A workaround is to instead have the data in logarithmic format. In order to test an implementation of a maximum-likelihood estimator that takes logarithmic data as input, it is useful to be able to generate non-underflowing logarithms of random gamma variates, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b098ac734780e67edb80b5a1039aea80f81595a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k<1}">. Following the implementation in <code>scipy.stats.loggamma</code>, this can be done as follows:<a href="#cite_note-Marsaglia-2000-25">[25]</a> sample <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7e4951dd26ab192dfe5719aac257ae932942a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.041ex; height:2.843ex;" alt="{\displaystyle Y\sim {\text{Gamma}}(k+1,\theta )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\sim {\text{Uniform}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Uniform</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\sim {\text{Uniform}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2d79a68d9de22a25a154cd29effa606ea01a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.285ex; height:2.176ex;" alt="{\displaystyle U\sim {\text{Uniform}}}"> independently. Then the required logarithmic sample is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\ln(Y)+\ln(U)/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\ln(Y)+\ln(U)/k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8e3d978cd6908a457738820b19f2bb85e42929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.046ex; height:2.843ex;" alt="{\displaystyle Z=\ln(Y)+\ln(U)/k}">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad42d845708982e924f81ef9a668a8c1464ba237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.307ex; height:2.843ex;" alt="{\displaystyle \exp(Z)\sim {\text{Gamma}}(k,\theta )}">. <div class="mw-heading mw-heading4"><h4 id="Closed-form_estimators">Closed-form estimators</h4>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=24" title="Edit section: Closed-form estimators">edit</a>]</div> There exist consistent closed-form estimators of k and θ that are derived from the likelihood of the <a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized gamma distribution</a>.<a href="#cite_note-26">[26]</a> The estimate for the shape k is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>N</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c91a0a26cb3f2fa4bc828a0c30cf81bd5fede8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.605ex; height:7.509ex;" alt="{\displaystyle {\hat {k}}={\frac {N\sum _{i=1}^{N}x_{i}}{N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}}}}"> and the estimate for the scale θ is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6558c0ea90b9602485c6ba273feadb033fb4121" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.998ex; height:7.509ex;" alt="{\displaystyle {\hat {\theta }}={\frac {1}{N^{2}}}\left(N\sum _{i=1}^{N}x_{i}\ln x_{i}-\sum _{i=1}^{N}x_{i}\sum _{i=1}^{N}\ln x_{i}\right)}"> Using the sample mean of x, the sample mean of ln x, and the sample mean of the product x·ln x simplifies the expressions to: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f9ada852433174f5c66beb07973e6809458a4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.158ex; height:3.343ex;" alt="{\displaystyle {\hat {k}}={\bar {x}}/{\hat {\theta }}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1993cfa1e3a5633d096f41763a1fb4a2dd588b3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.53ex; height:3.176ex;" alt="{\displaystyle {\hat {\theta }}={\overline {x\ln {x}}}-{\bar {x}}{\overline {\ln {x}}}.}"> If the rate parameterization is used, the estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66836da08bc157ab0a5edf7974395e117eaee26c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.231ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}=1/{\hat {\theta }}}">. These estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators. They have however similar efficiency as the maximum likelihood estimators. Although these estimators are consistent, they have a small bias. A bias-corrected variant of the estimator for the scale θ is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e944316bfe95be86b91886fd1ed2c430835ccd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.713ex; height:5.343ex;" alt="{\displaystyle {\tilde {\theta }}={\frac {N}{N-1}}{\hat {\theta }}}"> A bias correction for the shape parameter k is given as<a href="#cite_note-27">[27]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a767a0e0a4372c8ae599487d3bcb2af62413fa8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.413ex; height:7.509ex;" alt="{\displaystyle {\tilde {k}}={\hat {k}}-{\frac {1}{N}}\left(3{\hat {k}}-{\frac {2}{3}}\left({\frac {\hat {k}}{1+{\hat {k}}}}\right)-{\frac {4}{5}}{\frac {\hat {k}}{(1+{\hat {k}})^{2}}}\right)}"> <div class="mw-heading mw-heading4"><h4 id="Bayesian_minimum_mean_squared_error">Bayesian minimum mean squared error</h4>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=25" title="Edit section: Bayesian minimum mean squared error">edit</a>]</div> With known k and unknown θ, the posterior density function for theta (using the standard scale-invariant <a href="/wiki/Prior_probability" title="Prior probability">prior</a> for θ) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>k</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>θ</mi> </mfrac> </mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88b49f3b39c1e9dfa4e3f11c9576df51513012d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.838ex; height:7.343ex;" alt="{\displaystyle P(\theta \mid k,x_{1},\dots ,x_{N})\propto {\frac {1}{\theta }}\prod _{i=1}^{N}f(x_{i};k,\theta )}"> Denoting <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>k</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffae7a36c6134471ab6b856eb7354a68f2ee8130" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.041ex; height:7.343ex;" alt="{\displaystyle y\equiv \sum _{i=1}^{N}x_{i},\qquad P(\theta \mid k,x_{1},\dots ,x_{N})=C(x_{i})\theta ^{-Nk-1}e^{-y/\theta }}"> Integration with respect to θ can be carried out using a change of variables, revealing that 1/θ is gamma-distributed with parameters α = Nk, β = y. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mi>y</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fbb69f9a62c02e04870f6859733934f0d6c087" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.166ex; width:69.255ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }\theta ^{-Nk-1+m}e^{-y/\theta }\,d\theta =\int _{0}^{\infty }x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma (Nk-m)\!}"> The moments can be computed by taking the ratio (m by m = 0) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ae01ae77aa6c640cbaa1bb2a8863454827916a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.069ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} [x^{m}]={\frac {\Gamma (Nk-m)}{\Gamma (Nk)}}y^{m}}"> which shows that the mean ± standard deviation estimate of the posterior distribution for θ is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mi>k</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de03eb3d598f8c2c6481e9c44542904ee3858e9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.989ex; height:7.676ex;" alt="{\displaystyle {\frac {y}{Nk-1}}\pm {\sqrt {\frac {y^{2}}{(Nk-1)^{2}(Nk-2)}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Bayesian_inference">Bayesian inference</h3>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=26" title="Edit section: Bayesian inference">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Conjugate_prior">Conjugate prior</h4>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=27" title="Edit section: Conjugate prior">edit</a>]</div> In <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, the gamma distribution is the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> to many likelihood distributions: the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a>, <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a>, <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a> (with known mean), <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a>, gamma with known shape σ, <a href="/wiki/Inverse_gamma" class="mw-redirect" title="Inverse gamma">inverse gamma</a> with known shape parameter, and <a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> with known scale parameter. The gamma distribution's <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> is:<a href="#cite_note-Fink-28">[28]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo>∣</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>q</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d513935bdc7db0e23f116aab5397368604572ccd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:33.269ex; height:7.009ex;" alt="{\displaystyle p(k,\theta \mid p,q,r,s)={\frac {1}{Z}}{\frac {p^{k-1}e^{-\theta ^{-1}q}}{\Gamma (k)^{r}\theta ^{ks}}},}"> where Z is the normalizing constant with no closed-form solution. The posterior distribution can be found by updating the parameters as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>p</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <msub> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>q</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>q</mi> <mo>+</mo> <msub> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>r</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo>+</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>s</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo>+</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404516eb571949b6c74db5cc102e9e5341a9de21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:16.932ex; height:14.009ex;" alt="{\displaystyle {\begin{aligned}p'&=p\prod \nolimits _{i}x_{i},\\q'&=q+\sum \nolimits _{i}x_{i},\\r'&=r+n,\\s'&=s+n,\end{aligned}}}"> where n is the number of observations, and xi is the i-th observation. <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=28" title="Edit section: Occurrence and applications">edit</a>]</div> Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate β. Then the waiting time for the n-th event to occur is the gamma distribution with integer shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9961376aecd3518f5e3b88e2b02034dc36ab8e4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.981ex; height:1.676ex;" alt="{\displaystyle \alpha =n}">. This construction of the gamma distribution allows it to model a wide variety of phenomena where several sub-events, each taking time with exponential distribution, must happen in sequence for a major event to occur.<a href="#cite_note-29">[29]</a> Examples include the waiting time of <a href="/wiki/Cell_division" title="Cell division">cell-division events</a>,<a href="#cite_note-30">[30]</a> number of compensatory mutations for a given mutation,<a href="#cite_note-31">[31]</a> waiting time until a repair is necessary for a hydraulic system,<a href="#cite_note-32">[32]</a> and so on. In biophysics, the dwell time between steps of a molecular motor like <a href="/wiki/ATP_synthase" title="ATP synthase">ATP synthase</a> is nearly exponential at constant ATP concentration, revealing that each step of the motor takes a single ATP hydrolysis. If there were n ATP hydrolysis events, then it would be a gamma distribution with degree n.<a href="#cite_note-33">[33]</a> The gamma distribution has been used to model the size of <a href="/wiki/Insurance_policy" title="Insurance policy">insurance claims</a><a href="#cite_note-34">[34]</a> and rainfalls.<a href="#cite_note-35">[35]</a> This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a <a href="/wiki/Gamma_process" title="Gamma process">gamma process</a> – much like the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> generates a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>. The gamma distribution is also used to model errors in multi-level <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regression</a> models because a <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture</a> of <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distributions</a> with gamma-distributed rates has a known closed form distribution, called <a href="/wiki/Negative_binomial" class="mw-redirect" title="Negative binomial">negative binomial</a>. In wireless communication, the gamma distribution is used to model the <a href="/wiki/Multi-path_fading" class="mw-redirect" title="Multi-path fading">multi-path fading</a> of signal power;[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] see also <a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh distribution</a> and <a href="/wiki/Rician_distribution" class="mw-redirect" title="Rician distribution">Rician distribution</a>. In <a href="/wiki/Oncology" title="Oncology">oncology</a>, the age distribution of <a href="/wiki/Cancer" title="Cancer">cancer</a> <a href="/wiki/Disease_incidence" class="mw-redirect" title="Disease incidence">incidence</a> often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of <a href="/wiki/Carcinogenesis" title="Carcinogenesis">driver events</a> and the time interval between them.<a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> In <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>, the gamma distribution is often used to describe the distribution of <a href="/wiki/Temporal_coding" class="mw-redirect" title="Temporal coding">inter-spike intervals</a>.<a href="#cite_note-Robson-38">[38]</a><a href="#cite_note-Wright,_2015-39">[39]</a> In <a href="/wiki/Bacterial_genetics" title="Bacterial genetics">bacterial</a> <a href="/wiki/Gene_expression" title="Gene expression">gene expression</a>, the <a href="/wiki/Copy_number_analysis" title="Copy number analysis">copy number</a> of a <a href="/wiki/Constitutively_expressed" class="mw-redirect" title="Constitutively expressed">constitutively expressed</a> protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of <a href="/wiki/Protein_molecule" class="mw-redirect" title="Protein molecule">protein molecules</a> produced by a single mRNA during its lifetime.<a href="#cite_note-Friedman-40">[40]</a> In <a href="/wiki/Genomics" title="Genomics">genomics</a>, the gamma distribution was applied in <a href="/wiki/Peak_calling" title="Peak calling">peak calling</a> step (i.e., in recognition of signal) in <a href="/wiki/ChIP-chip" class="mw-redirect" title="ChIP-chip">ChIP-chip</a><a href="#cite_note-DJ_Reiss-41">[41]</a> and <a href="/wiki/ChIP-seq" class="mw-redirect" title="ChIP-seq">ChIP-seq</a><a href="#cite_note-MA_Mendoza-42">[42]</a> data analysis. In Bayesian statistics, the gamma distribution is widely used as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a>. It is the conjugate prior for the <a href="/wiki/Precision_(statistics)" title="Precision (statistics)">precision</a> (i.e. inverse of the variance) of a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. It is also the conjugate prior for the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>. In <a href="/wiki/Phylogenetics" title="Phylogenetics">phylogenetics</a>, the gamma distribution is the most commonly used approach to model among-sites rate variation<a href="#cite_note-43">[43]</a> when <a href="/wiki/Computational_phylogenetics#Maximum_likelihood" title="Computational phylogenetics">maximum likelihood</a>, <a href="/wiki/Bayesian_inference_in_phylogeny" title="Bayesian inference in phylogeny">Bayesian</a>, or <a href="/wiki/Distance_matrices_in_phylogeny" title="Distance matrices in phylogeny">distance matrix methods</a> are used to estimate phylogenetic trees. Phylogenetic analyzes that use the gamma distribution to model rate variation estimate a single parameter from the data because they limit consideration to distributions where α = β. This parameterization means that the mean of this distribution is 1 and the variance is 1/α. Maximum likelihood and Bayesian methods typically use a discrete approximation to the continuous gamma distribution.<a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a> <div class="mw-heading mw-heading2"><h2 id="Random_variate_generation">Random variate generation</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=29" title="Edit section: Random variate generation">edit</a>]</div> Given the scaling property above, it is enough to generate gamma variables with θ = 1, as we can later convert to any value of β with a simple division. Suppose we wish to generate random variables from Gamma(n + δ, 1), where n is a non-negative integer and 0 < δ < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of <a href="/wiki/Exponential_distribution#Random_variate_generation" title="Exponential distribution">generating exponential variables</a>, we conclude that if U is <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> on (0, 1], then −ln U is distributed Gamma(1, 1) (i.e. <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>). Now, using the "α-addition" property of gamma distribution, we expand this result: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a83a9843f29cf470dc2831bda24053d4262bb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.892ex; height:6.843ex;" alt="{\displaystyle -\sum _{k=1}^{n}\ln U_{k}\sim \Gamma (n,1)}"> where Uk are all uniformly distributed on (0, 1] and <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. All that is left now is to generate a variable distributed as Gamma(δ, 1) for 0 < δ < 1 and apply the "α-addition" property once more. This is the most difficult part. Random generation of gamma variates is discussed in detail by Devroye,<a href="#cite_note-Devroye-1986-46">[46]</a>: 401–428  noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.<a href="#cite_note-Devroye-1986-46">[46]</a>: 406  For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter<a href="#cite_note-Ahrens-1982-47">[47]</a> modified acceptance-rejection method Algorithm GD (shape k ≥ 1), or transformation method<a href="#cite_note-48">[48]</a> when 0 < k < 1. Also see Cheng and Feast Algorithm GKM 3<a href="#cite_note-49">[49]</a> or Marsaglia's squeeze method.<a href="#cite_note-50">[50]</a> The following is a version of the Ahrens-Dieter <a href="/wiki/Rejection_sampling" title="Rejection sampling">acceptance–rejection method</a>:<a href="#cite_note-Ahrens-1982-47">[47]</a> <ol><li>Generate U, V and W as <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> uniform (0, 1] variates.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\leq {\frac {e}{e+\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <mi>e</mi> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\leq {\frac {e}{e+\delta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78687aeaefa95f26e5c2eedf29a5e9d16e354292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.69ex; height:5.009ex;" alt="{\displaystyle U\leq {\frac {e}{e+\delta }}}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =V^{1/\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =V^{1/\delta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc600e64975b15e012887f0072dd610e9ee928ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.663ex; height:3.176ex;" alt="{\displaystyle \xi =V^{1/\delta }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =W\xi ^{\delta -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>W</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =W\xi ^{\delta -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9acf3fb974231242cb1622667fb3fef4dbd65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.811ex; height:3.176ex;" alt="{\displaystyle \eta =W\xi ^{\delta -1}}">. Otherwise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =1-\ln V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =1-\ln V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3bb894921726acc9b91410489f8db898cddd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.245ex; height:2.509ex;" alt="{\displaystyle \xi =1-\ln V}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =We^{-\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>W</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ξ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =We^{-\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b92760858b460290e1acc45654f8713b8770ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.026ex; height:3.176ex;" alt="{\displaystyle \eta =We^{-\xi }}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>></mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ξ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c999d5c04f901d0d448bed573cee6b42166dd46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.699ex; height:3.176ex;" alt="{\displaystyle \eta >\xi ^{\delta -1}e^{-\xi }}"> then go to step 1.</li> <li>ξ is distributed as Γ(δ, 1).</li></ol> A summary of this is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln U_{i}\right)\sim \Gamma (k,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln U_{i}\right)\sim \Gamma (k,\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6adf8db6aaf40d7b1cd9fa13983dac96ce6f2fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.181ex; height:7.843ex;" alt="{\displaystyle \theta \left(\xi -\sum _{i=1}^{\lfloor k\rfloor }\ln U_{i}\right)\sim \Gamma (k,\theta )}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \lfloor k\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \lfloor k\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f6297fc6736d117d930b473bc9d4325f8228ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.316ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \lfloor k\rfloor }"> is the integer part of k, ξ is generated via the algorithm above with δ = {k} (the fractional part of k) and the Uk are all independent. While the above approach is technically correct, Devroye notes that it is linear in the value of k and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context.<a href="#cite_note-Devroye-1986-46">[46]</a>: 401–428  For example, Marsaglia's simple transformation-rejection method relying on one normal variate X and one uniform variate U:<a href="#cite_note-Marsaglia-2000-25">[25]</a> <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=a-{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=a-{\frac {1}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b5f13d36077c9400af25d998bbe4bffa7772c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.383ex; height:5.176ex;" alt="{\displaystyle d=a-{\frac {1}{3}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\sqrt {9d}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>9</mn> <mi>d</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\sqrt {9d}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74964f3d4c7c5a61d85bd18e36b9c6bedf0509d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.256ex; height:6.176ex;" alt="{\displaystyle c={\frac {1}{\sqrt {9d}}}}">.</li> <li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=(1+cX)^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>c</mi> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=(1+cX)^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd133c8bb01c03e81f752a57d132ee59fcff2ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.079ex; height:3.176ex;" alt="{\displaystyle v=(1+cX)^{3}}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v>0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>U</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>d</mi> <mo>−</mo> <mi>d</mi> <mi>v</mi> <mo>+</mo> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae59d8c5427259a4388fd99891925d4d8db16821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.232ex; height:5.676ex;" alt="{\displaystyle \ln U<{\frac {X^{2}}{2}}+d-dv+d\ln v}"> return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dv}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5f695cb4931398dd078f0335e6de6905abf748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.343ex; height:2.176ex;" alt="{\displaystyle dv}">, else go back to step 2.</li></ol> With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq a=\alpha =k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>=</mo> <mi>α</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq a=\alpha =k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8d12ad014d8646f91b7db9d53dd0a4bf1afdab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.387ex; height:2.343ex;" alt="{\displaystyle 1\leq a=\alpha =k}"> generates a gamma distributed random number in time that is approximately constant with k. The acceptance rate does depend on k, with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4. For k < 1, one can use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>α</mi> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a438686f43515309dc233e2bb3b2c8344357c7ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.945ex; height:3.343ex;" alt="{\displaystyle \gamma _{\alpha }=\gamma _{1+\alpha }U^{1/\alpha }}"> to boost k to be usable with this method. In <a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a> numbers can be generated using the function gamrnd(), which uses the k, θ representation. <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=30" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/gamma-distribution">"Gamma distribution | Probability, Statistics, Distribution | Britannica"</a>. www.britannica.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240519084458/https://www.britannica.com/science/gamma-distribution">Archived</a> from the original on 2024-05-19. Retrieved 2024-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.britannica.com&rft.atitle=Gamma+distribution+%7C+Probability%2C+Statistics%2C+Distribution+%7C+Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fgamma-distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GammaDistribution.html">"Gamma Distribution"</a>. mathworld.wolfram.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240528053806/https://mathworld.wolfram.com/GammaDistribution.html">Archived</a> from the original on 2024-05-28. Retrieved 2024-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Gamma+Distribution&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGammaDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php">"Gamma Distribution | Gamma Function | Properties | PDF"</a>. www.probabilitycourse.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240613044322/https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php">Archived</a> from the original on 2024-06-13. Retrieved 2024-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.probabilitycourse.com&rft.atitle=Gamma+Distribution+%7C+Gamma+Function+%7C+Properties+%7C+PDF&rft_id=https%3A%2F%2Fwww.probabilitycourse.com%2Fchapter4%2F4_2_4_Gamma_distribution.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/4:_Continuous_Random_Variables/4.5:_Exponential_and_Gamma_Distributions">"4.5: Exponential and Gamma Distributions"</a>. Statistics LibreTexts. 2019-03-11. Retrieved 2024-10-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Statistics+LibreTexts&rft.atitle=4.5%3A+Exponential+and+Gamma+Distributions&rft.date=2019-03-11&rft_id=https%3A%2F%2Fstats.libretexts.org%2FCourses%2FSaint_Mary%27s_College_Notre_Dame%2FMATH_345__-_Probability_%28Kuter%29%2F4%3A_Continuous_Random_Variables%2F4.5%3A_Exponential_and_Gamma_Distributions&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParkBera2009" class="citation journal cs1">Park, Sung Y.; Bera, Anil K. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf">"Maximum entropy autoregressive conditional heteroskedasticity model"</a> (PDF). Journal of Econometrics. 150 (2): 219–230. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.511.9750">10.1.1.511.9750</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jeconom.2008.12.014">10.1016/j.jeconom.2008.12.014</a>. Archived from <a rel="nofollow" class="external text" href="http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf">the original</a> (PDF) on 2016-03-07. Retrieved 2011-06-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Econometrics&rft.atitle=Maximum+entropy+autoregressive+conditional+heteroskedasticity+model&rft.volume=150&rft.issue=2&rft.pages=219-230&rft.date=2009&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.511.9750%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2Fj.jeconom.2008.12.014&rft.aulast=Park&rft.aufirst=Sung+Y.&rft.au=Bera%2C+Anil+K.&rft_id=http%3A%2F%2Fwww.wise.xmu.edu.cn%2FMaster%2FDownload%2F..%255C..%255CUploadFiles%255Cpaper-masterdownload%255C2009519932327055475115776.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoggCraig1978" class="citation book cs1"><a href="/wiki/Robert_V._Hogg" title="Robert V. Hogg">Hogg, R. V.</a>; Craig, A. T. (1978). Introduction to Mathematical Statistics (4th ed.). New York: Macmillan. pp. Remark 3.3.1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0023557109" title="Special:BookSources/0023557109"><bdi>0023557109</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Statistics&rft.place=New+York&rft.pages=Remark+3.3.1&rft.edition=4th&rft.pub=Macmillan&rft.date=1978&rft.isbn=0023557109&rft.aulast=Hogg&rft.aufirst=R.+V.&rft.au=Craig%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGopalanHofmanBlei2013" class="citation arxiv cs1">Gopalan, Prem; Hofman, Jake M.; <a href="/wiki/David_Blei" title="David Blei">Blei, David M.</a> (2013). "Scalable Recommendation with Poisson Factorization". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.1704">1311.1704</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.IR">cs.IR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Scalable+Recommendation+with+Poisson+Factorization&rft.date=2013&rft_id=info%3Aarxiv%2F1311.1704&rft.aulast=Gopalan&rft.aufirst=Prem&rft.au=Hofman%2C+Jake+M.&rft.au=Blei%2C+David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Papoulis-8">^ <a href="#cite_ref-Papoulis_8-0">a</a> <a href="#cite_ref-Papoulis_8-1">b</a> Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, Fourth Edition </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Jeesen Chen, <a href="/w/index.php?title=Herman_Rubin&action=edit&redlink=1" class="new" title="Herman Rubin (page does not exist)">Herman Rubin</a>, Bounds for the difference between median and mean of gamma and Poisson distributions, Statistics & Probability Letters, Volume 4, Issue 6, October 1986, Pages 281–283, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0167-7152">0167-7152</a>, <a rel="nofollow" class="external autonumber" href="https://dx.doi.org/10.1016/0167-7152(86)90044-1">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203229/https://www.sciencedirect.com/unsupported_browser">Archived</a> 2024-10-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Choi, K. P. <a rel="nofollow" class="external text" href="https://www.ams.org/journals/proc/1994-121-01/S0002-9939-1994-1195477-8/S0002-9939-1994-1195477-8.pdf">"On the Medians of the Gamma Distributions and an Equation of Ramanujan"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210123121523/https://www.ams.org/journals/proc/1994-121-01/S0002-9939-1994-1195477-8/S0002-9939-1994-1195477-8.pdf">Archived</a> 2021-01-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Proceedings of the American Mathematical Society, Vol. 121, No. 1 (May, 1994), pp. 245–251. </li> <li id="cite_note-Pedersen,_Henrik_L.-2006-11">^ <a href="#cite_ref-Pedersen,_Henrik_L.-2006_11-0">a</a> <a href="#cite_ref-Pedersen,_Henrik_L.-2006_11-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerg,_ChristianPedersen,_Henrik_L.2006" class="citation journal cs1">Berg, Christian & Pedersen, Henrik L. (March 2006). <a rel="nofollow" class="external text" href="https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2006/0013/0001/MAA-2006-0013-0001-a004.pdf">"The Chen–Rubin conjecture in a continuous setting"</a> (PDF). Methods and Applications of Analysis. 13 (1): 63–88. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FMAA.2006.v13.n1.a4">10.4310/MAA.2006.v13.n1.a4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6704865">6704865</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210116114105/https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2006/0013/0001/MAA-2006-0013-0001-a004.pdf">Archived</a> (PDF) from the original on 16 January 2021. Retrieved 1 April 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Methods+and+Applications+of+Analysis&rft.atitle=The+Chen%E2%80%93Rubin+conjecture+in+a+continuous+setting&rft.volume=13&rft.issue=1&rft.pages=63-88&rft.date=2006-03&rft_id=info%3Adoi%2F10.4310%2FMAA.2006.v13.n1.a4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6704865%23id-name%3DS2CID&rft.au=Berg%2C+Christian&rft.au=Pedersen%2C+Henrik+L.&rft_id=https%3A%2F%2Fwww.intlpress.com%2Fsite%2Fpub%2Ffiles%2F_fulltext%2Fjournals%2Fmaa%2F2006%2F0013%2F0001%2FMAA-2006-0013-0001-a004.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Berg-12">^ <a href="#cite_ref-Berg_12-0">a</a> <a href="#cite_ref-Berg_12-1">b</a> Berg, Christian and Pedersen, Henrik L. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0609442">"Convexity of the median in the gamma distribution"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230526181721/https://arxiv.org/abs/math/0609442">Archived</a> 2023-05-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaunt,_Robert_E.,_and_Milan_Merkle2021" class="citation journal cs1">Gaunt, Robert E., and Milan Merkle (2021). "On bounds for the mode and median of the generalized hyperbolic and related distributions". Journal of Mathematical Analysis and Applications. 493 (1): 124508. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2002.01884">2002.01884</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmaa.2020.124508">10.1016/j.jmaa.2020.124508</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221103640">221103640</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Analysis+and+Applications&rft.atitle=On+bounds+for+the+mode+and+median+of+the+generalized+hyperbolic+and+related+distributions&rft.volume=493&rft.issue=1&rft.pages=124508&rft.date=2021&rft_id=info%3Aarxiv%2F2002.01884&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221103640%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.jmaa.2020.124508&rft.au=Gaunt%2C+Robert+E.%2C+and+Milan+Merkle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) </li> <li id="cite_note-Lyon-2021a-14">^ <a href="#cite_ref-Lyon-2021a_14-0">a</a> <a href="#cite_ref-Lyon-2021a_14-1">b</a> <a href="#cite_ref-Lyon-2021a_14-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLyon2021" class="citation journal cs1">Lyon, Richard F. (13 May 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8118309">"On closed-form tight bounds and approximations for the median of a gamma distribution"</a>. <a href="/wiki/PLOS_One" title="PLOS One">PLOS One</a>. 16 (5): e0251626. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2011.04060">2011.04060</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PLoSO..1651626L">2021PLoSO..1651626L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0251626">10.1371/journal.pone.0251626</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8118309">8118309</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33984053">33984053</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+One&rft.atitle=On+closed-form+tight+bounds+and+approximations+for+the+median+of+a+gamma+distribution&rft.volume=16&rft.issue=5&rft.pages=e0251626&rft.date=2021-05-13&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8118309%23id-name%3DPMC&rft_id=info%3Abibcode%2F2021PLoSO..1651626L&rft_id=info%3Aarxiv%2F2011.04060&rft_id=info%3Apmid%2F33984053&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0251626&rft.aulast=Lyon&rft.aufirst=Richard+F.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8118309&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Lyon-2021b-15">^ <a href="#cite_ref-Lyon-2021b_15-0">a</a> <a href="#cite_ref-Lyon-2021b_15-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLyon2021" class="citation journal cs1">Lyon, Richard F. (13 May 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10490949">"Tight bounds for the median of a gamma distribution"</a>. <a href="/wiki/PLOS_One" title="PLOS One">PLOS One</a>. 18 (9): e0288601. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0288601">10.1371/journal.pone.0288601</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10490949">10490949</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37682854">37682854</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+One&rft.atitle=Tight+bounds+for+the+median+of+a+gamma+distribution&rft.volume=18&rft.issue=9&rft.pages=e0288601&rft.date=2021-05-13&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10490949%23id-name%3DPMC&rft_id=info%3Apmid%2F37682854&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0288601&rft.aulast=Lyon&rft.aufirst=Richard+F.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10490949&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMathai1982" class="citation journal cs1">Mathai, A. M. (1982). "Storage capacity of a dam with gamma type inputs". Annals of the Institute of Statistical Mathematics. 34 (3): 591–597. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02481056">10.1007/BF02481056</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-3157">0020-3157</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122537756">122537756</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+Institute+of+Statistical+Mathematics&rft.atitle=Storage+capacity+of+a+dam+with+gamma+type+inputs&rft.volume=34&rft.issue=3&rft.pages=591-597&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122537756%23id-name%3DS2CID&rft.issn=0020-3157&rft_id=info%3Adoi%2F10.1007%2FBF02481056&rft.aulast=Mathai&rft.aufirst=A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoschopoulos1985" class="citation journal cs1">Moschopoulos, P. G. (1985). "The distribution of the sum of independent gamma random variables". Annals of the Institute of Statistical Mathematics. 37 (3): 541–544. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02481123">10.1007/BF02481123</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120066454">120066454</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+Institute+of+Statistical+Mathematics&rft.atitle=The+distribution+of+the+sum+of+independent+gamma+random+variables&rft.volume=37&rft.issue=3&rft.pages=541-544&rft.date=1985&rft_id=info%3Adoi%2F10.1007%2FBF02481123&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120066454%23id-name%3DS2CID&rft.aulast=Moschopoulos&rft.aufirst=P.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenny" class="citation web cs1">Penny, W. D. <a rel="nofollow" class="external text" href="https://www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps">"KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=KL-Divergences+of+Normal%2C+Gamma%2C+Dirichlet%2C+and+Wishart+densities&rft.aulast=Penny&rft.aufirst=W.+D.&rft_id=https%3A%2F%2Fwww.fil.ion.ucl.ac.uk%2F~wpenny%2Fpublications%2Fdensities.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/ExpGammaDistribution.html">"ExpGammaDistribution—Wolfram Language Documentation"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ExpGammaDistribution%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FExpGammaDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loggamma.html#scipy.stats.loggamma">"scipy.stats.loggamma — SciPy v1.8.0 Manual"</a>. docs.scipy.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.scipy.org&rft.atitle=scipy.stats.loggamma+%E2%80%94+SciPy+v1.8.0+Manual&rft_id=https%3A%2F%2Fdocs.scipy.org%2Fdoc%2Fscipy%2Freference%2Fgenerated%2Fscipy.stats.loggamma.html%23scipy.stats.loggamma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSunKongPal2021" class="citation journal cs1">Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). <a rel="nofollow" class="external text" href="https://figshare.com/articles/journal_contribution/The_Modified-Half-Normal_distribution_Properties_and_an_efficient_sampling_scheme/14825266/1/files/28535884.pdf">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a> (PDF). Communications in Statistics - Theory and Methods. 52 (5): 1591–1613. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>. Retrieved 2 September 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&rft.atitle=The+Modified-Half-Normal+distribution%3A+Properties+and+an+efficient+sampling+scheme&rft.volume=52&rft.issue=5&rft.pages=1591-1613&rft.date=2021-06-22&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A237919587%23id-name%3DS2CID&rft.issn=0361-0926&rft_id=info%3Adoi%2F10.1080%2F03610926.2021.1934700&rft.aulast=Sun&rft.aufirst=Jingchao&rft.au=Kong%2C+Maiying&rft.au=Pal%2C+Subhadip&rft_id=https%3A%2F%2Ffigshare.com%2Farticles%2Fjournal_contribution%2FThe_Modified-Half-Normal_distribution_Properties_and_an_efficient_sampling_scheme%2F14825266%2F1%2Ffiles%2F28535884.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Dubey-1970-22"><a href="#cite_ref-Dubey-1970_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDubey1970" class="citation journal cs1">Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02613934">10.1007/BF02613934</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123366328">123366328</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrika&rft.atitle=Compound+gamma%2C+beta+and+F+distributions&rft.volume=16&rft.pages=27-31&rft.date=1970-12&rft_id=info%3Adoi%2F10.1007%2FBF02613934&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123366328%23id-name%3DS2CID&rft.aulast=Dubey&rft.aufirst=Satya+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinka2002" class="citation web cs1">Minka, Thomas P. (2002). <a rel="nofollow" class="external text" href="https://tminka.github.io/papers/minka-gamma.pdf">"Estimating a Gamma distribution"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Estimating+a+Gamma+distribution&rft.date=2002&rft.aulast=Minka&rft.aufirst=Thomas+P.&rft_id=https%3A%2F%2Ftminka.github.io%2Fpapers%2Fminka-gamma.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChoiWette1969" class="citation journal cs1">Choi, S. C.; Wette, R. (1969). "Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias". Technometrics. 11 (4): 683–690. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00401706.1969.10490731">10.1080/00401706.1969.10490731</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technometrics&rft.atitle=Maximum+Likelihood+Estimation+of+the+Parameters+of+the+Gamma+Distribution+and+Their+Bias&rft.volume=11&rft.issue=4&rft.pages=683-690&rft.date=1969&rft_id=info%3Adoi%2F10.1080%2F00401706.1969.10490731&rft.aulast=Choi&rft.aufirst=S.+C.&rft.au=Wette%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Marsaglia-2000-25">^ <a href="#cite_ref-Marsaglia-2000_25-0">a</a> <a href="#cite_ref-Marsaglia-2000_25-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarsagliaTsang2000" class="citation journal cs1">Marsaglia, G.; Tsang, W. W. (2000). "A simple method for generating gamma variables". ACM Transactions on Mathematical Software. 26 (3): 363–372. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358407.358414">10.1145/358407.358414</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2634158">2634158</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Mathematical+Software&rft.atitle=A+simple+method+for+generating+gamma+variables&rft.volume=26&rft.issue=3&rft.pages=363-372&rft.date=2000&rft_id=info%3Adoi%2F10.1145%2F358407.358414&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2634158%23id-name%3DS2CID&rft.aulast=Marsaglia&rft.aufirst=G.&rft.au=Tsang%2C+W.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYeChen2017" class="citation journal cs1">Ye, Zhi-Sheng; Chen, Nan (2017). <a rel="nofollow" class="external text" href="https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1209129">"Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations"</a>. The American Statistician. 71 (2): 177–181. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2016.1209129">10.1080/00031305.2016.1209129</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124682698">124682698</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230526181723/https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1209129">Archived</a> from the original on 2023-05-26. Retrieved 2019-07-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Closed-Form+Estimators+for+the+Gamma+Distribution+Derived+from+Likelihood+Equations&rft.volume=71&rft.issue=2&rft.pages=177-181&rft.date=2017&rft_id=info%3Adoi%2F10.1080%2F00031305.2016.1209129&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124682698%23id-name%3DS2CID&rft.aulast=Ye&rft.aufirst=Zhi-Sheng&rft.au=Chen%2C+Nan&rft_id=https%3A%2F%2Famstat.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.2016.1209129&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLouzadaRamosRamos2019" class="citation journal cs1">Louzada, Francisco; Ramos, Pedro L.; Ramos, Eduardo (2019). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1513376">"A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations"</a>. The American Statistician. 73 (2): 195–199. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2018.1513376">10.1080/00031305.2018.1513376</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126086375">126086375</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230526181723/https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1513376">Archived</a> from the original on 2023-05-26. Retrieved 2019-07-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=A+Note+on+Bias+of+Closed-Form+Estimators+for+the+Gamma+Distribution+Derived+from+Likelihood+Equations&rft.volume=73&rft.issue=2&rft.pages=195-199&rft.date=2019&rft_id=info%3Adoi%2F10.1080%2F00031305.2018.1513376&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126086375%23id-name%3DS2CID&rft.aulast=Louzada&rft.aufirst=Francisco&rft.au=Ramos%2C+Pedro+L.&rft.au=Ramos%2C+Eduardo&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.2018.1513376&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Fink-28"><a href="#cite_ref-Fink_28-0">^</a> Fink, D. 1995 <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.157.5540&rep=rep1&type=pdf">A Compendium of Conjugate Priors</a>. In progress report: Extension and enhancement of methods for setting data quality objectives. (DOE contract 95‑831). </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJessica.2001" class="citation book cs1">Jessica., Scheiner, Samuel M., 1956- Gurevitch (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AgsTDAAAQBAJ&dq=gamma+distribution+failure+waiting+time&pg=PA235">"13. Failure-time analysis"</a>. <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/43694448">Design and analysis of ecological experiments</a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513187-8" title="Special:BookSources/0-19-513187-8"><bdi>0-19-513187-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/43694448">43694448</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203230/https://search.worldcat.org/title/43694448">Archived</a> from the original on 2024-10-09. Retrieved 2022-05-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=13.+Failure-time+analysis&rft.btitle=Design+and+analysis+of+ecological+experiments&rft.pub=Oxford+University+Press&rft.date=2001&rft_id=info%3Aoclcnum%2F43694448&rft.isbn=0-19-513187-8&rft.aulast=Jessica.&rft.aufirst=Scheiner%2C+Samuel+M.%2C+1956-+Gurevitch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAgsTDAAAQBAJ%26dq%3Dgamma%2Bdistribution%2Bfailure%2Bwaiting%2Btime%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>) </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubev2016" class="citation journal cs1">Golubev, A. (March 2016). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/j.jtbi.2015.12.027">"Applications and implications of the exponentially modified gamma distribution as a model for time variabilities related to cell proliferation and gene expression"</a>. Journal of Theoretical Biology. 393: 203–217. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016JThBi.393..203G">2016JThBi.393..203G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jtbi.2015.12.027">10.1016/j.jtbi.2015.12.027</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-5193">0022-5193</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26780652">26780652</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203230/https://www.sciencedirect.com/unsupported_browser">Archived</a> from the original on 2024-10-09. Retrieved 2022-05-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Theoretical+Biology&rft.atitle=Applications+and+implications+of+the+exponentially+modified+gamma+distribution+as+a+model+for+time+variabilities+related+to+cell+proliferation+and+gene+expression&rft.volume=393&rft.pages=203-217&rft.date=2016-03&rft_id=info%3Adoi%2F10.1016%2Fj.jtbi.2015.12.027&rft.issn=0022-5193&rft_id=info%3Apmid%2F26780652&rft_id=info%3Abibcode%2F2016JThBi.393..203G&rft.aulast=Golubev&rft.aufirst=A.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1016%2Fj.jtbi.2015.12.027&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoonDavisChao2005" class="citation journal cs1">Poon, Art; Davis, Bradley H; Chao, Lin (2005-07-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1534/genetics.104.037259">"The Coupon Collector and the Suppressor Mutation"</a>. Genetics. 170 (3): 1323–1332. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1534%2Fgenetics.104.037259">10.1534/genetics.104.037259</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1943-2631">1943-2631</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1451182">1451182</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15879511">15879511</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203231/https://academic.oup.com/genetics/article/170/3/1323/6060337">Archived</a> from the original on 2024-10-09. Retrieved 2022-05-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Genetics&rft.atitle=The+Coupon+Collector+and+the+Suppressor+Mutation&rft.volume=170&rft.issue=3&rft.pages=1323-1332&rft.date=2005-07-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1451182%23id-name%3DPMC&rft.issn=1943-2631&rft_id=info%3Apmid%2F15879511&rft_id=info%3Adoi%2F10.1534%2Fgenetics.104.037259&rft.aulast=Poon&rft.aufirst=Art&rft.au=Davis%2C+Bradley+H&rft.au=Chao%2C+Lin&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1534%2Fgenetics.104.037259&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVineyardAmoako-GyampahMeredith1999" class="citation journal cs1">Vineyard, Michael; Amoako-Gyampah, Kwasi; Meredith, Jack R (July 1999). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/s0377-2217(98)00096-4">"Failure rate distributions for flexible manufacturing systems: An empirical study"</a>. European Journal of Operational Research. 116 (1): 139–155. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0377-2217%2898%2900096-4">10.1016/s0377-2217(98)00096-4</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0377-2217">0377-2217</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203230/https://www.sciencedirect.com/unsupported_browser">Archived</a> from the original on 2024-10-09. Retrieved 2022-05-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Operational+Research&rft.atitle=Failure+rate+distributions+for+flexible+manufacturing+systems%3A+An+empirical+study&rft.volume=116&rft.issue=1&rft.pages=139-155&rft.date=1999-07&rft_id=info%3Adoi%2F10.1016%2Fs0377-2217%2898%2900096-4&rft.issn=0377-2217&rft.aulast=Vineyard&rft.aufirst=Michael&rft.au=Amoako-Gyampah%2C+Kwasi&rft.au=Meredith%2C+Jack+R&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1016%2Fs0377-2217%2898%2900096-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiefRockMehtaMooseker2000" class="citation journal cs1">Rief, Matthias; Rock, Ronald S.; Mehta, Amit D.; Mooseker, Mark S.; Cheney, Richard E.; Spudich, James A. (2000-08-15). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC16890">"Myosin-V stepping kinetics: A molecular model for processivity"</a>. Proceedings of the National Academy of Sciences. 97 (17): 9482–9486. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PNAS...97.9482R">2000PNAS...97.9482R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.97.17.9482">10.1073/pnas.97.17.9482</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC16890">16890</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10944217">10944217</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Myosin-V+stepping+kinetics%3A+A+molecular+model+for+processivity&rft.volume=97&rft.issue=17&rft.pages=9482-9486&rft.date=2000-08-15&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC16890%23id-name%3DPMC&rft_id=info%3Abibcode%2F2000PNAS...97.9482R&rft_id=info%3Apmid%2F10944217&rft_id=info%3Adoi%2F10.1073%2Fpnas.97.17.9482&rft.issn=0027-8424&rft.aulast=Rief&rft.aufirst=Matthias&rft.au=Rock%2C+Ronald+S.&rft.au=Mehta%2C+Amit+D.&rft.au=Mooseker%2C+Mark+S.&rft.au=Cheney%2C+Richard+E.&rft.au=Spudich%2C+James+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC16890&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> p. 43, Philip J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall CRC 2007 </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilks1990" class="citation journal cs1">Wilks, Daniel S. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0442%281990%29003%3C1495%3AMLEFTG%3E2.0.CO%3B2">"Maximum Likelihood Estimation for the Gamma Distribution Using Data Containing Zeros"</a>. Journal of Climate. 3 (12): 1495–1501. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990JCli....3.1495W">1990JCli....3.1495W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0442%281990%29003%3C1495%3AMLEFTG%3E2.0.CO%3B2">10.1175/1520-0442(1990)003<1495:MLEFTG>2.0.CO;2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0894-8755">0894-8755</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/26196366">26196366</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Climate&rft.atitle=Maximum+Likelihood+Estimation+for+the+Gamma+Distribution+Using+Data+Containing+Zeros&rft.volume=3&rft.issue=12&rft.pages=1495-1501&rft.date=1990&rft_id=info%3Adoi%2F10.1175%2F1520-0442%281990%29003%3C1495%3AMLEFTG%3E2.0.CO%3B2&rft.issn=0894-8755&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F26196366%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1990JCli....3.1495W&rft.aulast=Wilks&rft.aufirst=Daniel+S.&rft_id=https%3A%2F%2Fdoi.org%2F10.1175%252F1520-0442%25281990%2529003%253C1495%253AMLEFTG%253E2.0.CO%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelikov2017" class="citation journal cs1">Belikov, Aleksey V. (22 September 2017). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5610194">"The number of key carcinogenic events can be predicted from cancer incidence"</a>. Scientific Reports. 7 (1): 12170. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017NatSR...712170B">2017NatSR...712170B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-017-12448-7">10.1038/s41598-017-12448-7</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5610194">5610194</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28939880">28939880</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=The+number+of+key+carcinogenic+events+can+be+predicted+from+cancer+incidence&rft.volume=7&rft.issue=1&rft.pages=12170&rft.date=2017-09-22&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5610194%23id-name%3DPMC&rft_id=info%3Apmid%2F28939880&rft_id=info%3Adoi%2F10.1038%2Fs41598-017-12448-7&rft_id=info%3Abibcode%2F2017NatSR...712170B&rft.aulast=Belikov&rft.aufirst=Aleksey+V.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5610194&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelikovVyatkinLeonov2021" class="citation journal cs1">Belikov, Aleksey V.; Vyatkin, Alexey; Leonov, Sergey V. (2021-08-06). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8351573">"The Erlang distribution approximates the age distribution of incidence of childhood and young adulthood cancers"</a>. PeerJ. 9: e11976. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7717%2Fpeerj.11976">10.7717/peerj.11976</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2167-8359">2167-8359</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8351573">8351573</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34434669">34434669</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PeerJ&rft.atitle=The+Erlang+distribution+approximates+the+age+distribution+of+incidence+of+childhood+and+young+adulthood+cancers&rft.volume=9&rft.pages=e11976&rft.date=2021-08-06&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8351573%23id-name%3DPMC&rft.issn=2167-8359&rft_id=info%3Apmid%2F34434669&rft_id=info%3Adoi%2F10.7717%2Fpeerj.11976&rft.aulast=Belikov&rft.aufirst=Aleksey+V.&rft.au=Vyatkin%2C+Alexey&rft.au=Leonov%2C+Sergey+V.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8351573&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Robson-38"><a href="#cite_ref-Robson_38-0">^</a> J. G. Robson and J. B. Troy, "Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat", J. Opt. Soc. Am. A 4, 2301–2307 (1987) </li> <li id="cite_note-Wright,_2015-39"><a href="#cite_ref-Wright,_2015_39-0">^</a> M.C.M. Wright, I.M. Winter, J.J. Forster, S. Bleeck "Response to best-frequency tone bursts in the ventral cochlear nucleus is governed by ordered inter-spike interval statistics", Hearing Research 317 (2014) </li> <li id="cite_note-Friedman-40"><a href="#cite_ref-Friedman_40-0">^</a> N. Friedman, L. Cai and X. S. Xie (2006) "Linking stochastic dynamics to population distribution: An analytical framework of gene expression", Phys. Rev. Lett. 97, 168302. </li> <li id="cite_note-DJ_Reiss-41"><a href="#cite_ref-DJ_Reiss_41-0">^</a> DJ Reiss, MT Facciotti and NS Baliga (2008) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121117144623/http://bioinformatics.oxfordjournals.org/content/24/3/396.full.pdf+html">"Model-based deconvolution of genome-wide DNA binding"</a>, Bioinformatics, 24, 396–403 </li> <li id="cite_note-MA_Mendoza-42"><a href="#cite_ref-MA_Mendoza_42-0">^</a> MA Mendoza-Parra, M Nowicka, W Van Gool, H Gronemeyer (2013) <a rel="nofollow" class="external text" href="http://www.biomedcentral.com/1471-2164/14/834">"Characterising ChIP-seq binding patterns by model-based peak shape deconvolution"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203230/https://bmcgenomics.biomedcentral.com/articles/10.1186/1471-2164-14-834">Archived</a> 2024-10-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, BMC Genomics, 14:834 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYang1996" class="citation journal cs1">Yang, Ziheng (September 1996). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/0169534796100410">"Among-site rate variation and its impact on phylogenetic analyses"</a>. Trends in Ecology & Evolution. 11 (9): 367–372. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996TEcoE..11..367Y">1996TEcoE..11..367Y</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.99">10.1.1.19.99</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0169-5347%2896%2910041-0">10.1016/0169-5347(96)10041-0</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21237881">21237881</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240412001342/https://linkinghub.elsevier.com/retrieve/pii/0169534796100410">Archived</a> from the original on 2024-04-12. Retrieved 2023-09-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trends+in+Ecology+%26+Evolution&rft.atitle=Among-site+rate+variation+and+its+impact+on+phylogenetic+analyses&rft.volume=11&rft.issue=9&rft.pages=367-372&rft.date=1996-09&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.19.99%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F21237881&rft_id=info%3Adoi%2F10.1016%2F0169-5347%2896%2910041-0&rft_id=info%3Abibcode%2F1996TEcoE..11..367Y&rft.aulast=Yang&rft.aufirst=Ziheng&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2F0169534796100410&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYang1994" class="citation journal cs1">Yang, Ziheng (September 1994). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/BF00160154">"Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: Approximate methods"</a>. Journal of Molecular Evolution. 39 (3): 306–314. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994JMolE..39..306Y">1994JMolE..39..306Y</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.6626">10.1.1.19.6626</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00160154">10.1007/BF00160154</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2844">0022-2844</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/7932792">7932792</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17911050">17911050</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203844/https://link.springer.com/article/10.1007/BF00160154">Archived</a> from the original on 2024-10-09. Retrieved 2023-09-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Molecular+Evolution&rft.atitle=Maximum+likelihood+phylogenetic+estimation+from+DNA+sequences+with+variable+rates+over+sites%3A+Approximate+methods&rft.volume=39&rft.issue=3&rft.pages=306-314&rft.date=1994-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17911050%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1994JMolE..39..306Y&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.19.6626%23id-name%3DCiteSeerX&rft.issn=0022-2844&rft_id=info%3Adoi%2F10.1007%2FBF00160154&rft_id=info%3Apmid%2F7932792&rft.aulast=Yang&rft.aufirst=Ziheng&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2FBF00160154&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelsenstein2001" class="citation journal cs1">Felsenstein, Joseph (2001-10-01). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/s002390010234">"Taking Variation of Evolutionary Rates Between Sites into Account in Inferring Phylogenies"</a>. Journal of Molecular Evolution. 53 (4–5): 447–455. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JMolE..53..447F">2001JMolE..53..447F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002390010234">10.1007/s002390010234</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2844">0022-2844</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11675604">11675604</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9791493">9791493</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241009203754/https://link.springer.com/article/10.1007/s002390010234">Archived</a> from the original on 2024-10-09. Retrieved 2023-09-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Molecular+Evolution&rft.atitle=Taking+Variation+of+Evolutionary+Rates+Between+Sites+into+Account+in+Inferring+Phylogenies&rft.volume=53&rft.issue=4%E2%80%935&rft.pages=447-455&rft.date=2001-10-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9791493%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2001JMolE..53..447F&rft.issn=0022-2844&rft_id=info%3Adoi%2F10.1007%2Fs002390010234&rft_id=info%3Apmid%2F11675604&rft.aulast=Felsenstein&rft.aufirst=Joseph&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2Fs002390010234&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-Devroye-1986-46">^ <a href="#cite_ref-Devroye-1986_46-0">a</a> <a href="#cite_ref-Devroye-1986_46-1">b</a> <a href="#cite_ref-Devroye-1986_46-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevroye1986" class="citation book cs1">Devroye, Luc (1986). <a rel="nofollow" class="external text" href="http://luc.devroye.org/rnbookindex.html">Non-Uniform Random Variate Generation</a>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96305-1" title="Special:BookSources/978-0-387-96305-1"><bdi>978-0-387-96305-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120717112308/http://luc.devroye.org/rnbookindex.html">Archived</a> from the original on 2012-07-17. Retrieved 2012-02-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Uniform+Random+Variate+Generation&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1986&rft.isbn=978-0-387-96305-1&rft.aulast=Devroye&rft.aufirst=Luc&rft_id=http%3A%2F%2Fluc.devroye.org%2Frnbookindex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> See Chapter 9, Section 3. </li> <li id="cite_note-Ahrens-1982-47">^ <a href="#cite_ref-Ahrens-1982_47-0">a</a> <a href="#cite_ref-Ahrens-1982_47-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhrensDieter1982" class="citation journal cs1">Ahrens, J. H.; Dieter, U (January 1982). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358315.358390">"Generating gamma variates by a modified rejection technique"</a>. Communications of the ACM. 25 (1): 47–54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358315.358390">10.1145/358315.358390</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15128188">15128188</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Generating+gamma+variates+by+a+modified+rejection+technique&rft.volume=25&rft.issue=1&rft.pages=47-54&rft.date=1982-01&rft_id=info%3Adoi%2F10.1145%2F358315.358390&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15128188%23id-name%3DS2CID&rft.aulast=Ahrens&rft.aufirst=J.+H.&rft.au=Dieter%2C+U&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F358315.358390&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988">. See Algorithm GD, p. 53. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhrensDieter1974" class="citation journal cs1">Ahrens, J. H.; Dieter, U. (1974). "Computer methods for sampling from gamma, beta, Poisson and binomial distributions". Computing. 12 (3): 223–246. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.93.3828">10.1.1.93.3828</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02293108">10.1007/BF02293108</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:37484126">37484126</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computing&rft.atitle=Computer+methods+for+sampling+from+gamma%2C+beta%2C+Poisson+and+binomial+distributions&rft.volume=12&rft.issue=3&rft.pages=223-246&rft.date=1974&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.93.3828%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A37484126%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02293108&rft.aulast=Ahrens&rft.aufirst=J.+H.&rft.au=Dieter%2C+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChengFeast1979" class="citation journal cs1">Cheng, R. C. H.; Feast, G. M. (1979). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347200">"Some Simple Gamma Variate Generators"</a>. Journal of the Royal Statistical Society. Series C (Applied Statistics). 28 (3): 290–295. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2347200">10.2307/2347200</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347200">2347200</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society.+Series+C+%28Applied+Statistics%29&rft.atitle=Some+Simple+Gamma+Variate+Generators&rft.volume=28&rft.issue=3&rft.pages=290-295&rft.date=1979&rft_id=info%3Adoi%2F10.2307%2F2347200&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2347200%23id-name%3DJSTOR&rft.aulast=Cheng&rft.aufirst=R.+C.+H.&rft.au=Feast%2C+G.+M.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2347200&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> Marsaglia, G. The squeeze method for generating gamma variates. Comput, Math. Appl. 3 (1977), 321–325. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Gamma_distribution&action=edit&section=31" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Statistics" class="extiw" title="wikibooks:Statistics">Statistics</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Statistics/Distributions/Gamma" class="extiw" title="wikibooks:Statistics/Distributions/Gamma">Gamma distribution</a></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Gamma-distribution">"Gamma-distribution"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Gamma-distribution&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DGamma-distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GammaDistribution.html">"Gamma distribution"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Gamma+distribution&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGammaDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGamma+distribution" class="Z3988"></li> <li>ModelAssist (2017) <a rel="nofollow" class="external text" href="http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Continuous_distributions/Gamma.htm">Uses of the gamma distribution in risk modeling, including applied examples in Excel</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170509042425/http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Continuous_distributions/Gamma.htm">Archived</a> 2017-05-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm">Engineering Statistics Handbook</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Probability_distributions_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a semi-infinite interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution">F</a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a class="mw-selflink selflink">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's T-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on the whole real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's z</a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q</a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU</a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral t</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t</a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis κ-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis κ-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis κ-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis κ-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis κ-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution">q-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution">q-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous- discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate (joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Discrete: </li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li>Continuous: </li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate t</a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix t</a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt>Bivariate (spherical)</dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt>Bivariate (toroidal)</dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt>Multivariate</dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Degenerate</dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt>Singular</dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">Commons</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Gamma_distribution&oldid=1258598224">https://en.wikipedia.org/w/index.php?title=Gamma_distribution&oldid=1258598224</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Continuous_distributions" title="Category:Continuous distributions">Continuous distributions</a></li><li><a href="/wiki/Category:Factorial_and_binomial_topics" title="Category:Factorial and binomial topics">Factorial and binomial topics</a></li><li><a href="/wiki/Category:Conjugate_prior_distributions" title="Category:Conjugate prior distributions">Conjugate prior distributions</a></li><li><a href="/wiki/Category:Exponential_family_distributions" title="Category:Exponential family distributions">Exponential family distributions</a></li><li><a href="/wiki/Category:Infinitely_divisible_probability_distributions" title="Category:Infinitely divisible probability distributions">Infinitely divisible probability distributions</a></li><li><a href="/wiki/Category:Survival_analysis" title="Category:Survival analysis">Survival analysis</a></li><li><a href="/wiki/Category:Gamma_and_related_functions" title="Category:Gamma and related functions">Gamma and related functions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">CS1 maint: numeric names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2012" title="Category:Articles with unsourced statements from September 2012">Articles with unsourced statements from September 2012</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2019" title="Category:Articles with unsourced statements from May 2019">Articles with unsourced statements from May 2019</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 20 November 2024, at 16:07 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Gamma_distribution&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-zw2dd","wgBackendResponseTime":161,"wgPageParseReport":{"limitreport":{"cputime":"1.425","walltime":"1.748","ppvisitednodes":{"value":11112,"limit":1000000},"postexpandincludesize":{"value":213390,"limit":2097152},"templateargumentsize":{"value":13313,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":189766,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1200.794 1 -total"," 39.74% 477.181 1 Template:Reflist"," 15.46% 185.673 26 Template:Cite_journal"," 13.16% 157.996 8 Template:Cite_web"," 11.49% 137.971 1 Template:Short_description"," 11.13% 133.678 89 Template:Math"," 11.08% 133.037 1 Template:ProbDistributions"," 11.05% 132.634 4 Template:Navbox"," 9.14% 109.796 2 Template:Pagetype"," 5.85% 70.240 1 Template:Infobox_probability_distribution_2"]},"scribunto":{"limitreport-timeusage":{"value":"0.675","limit":"10.000"},"limitreport-memusage":{"value":7859125,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-4bspq","timestamp":"20241122140549","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Gamma distribution","url":"https:\/\/en.wikipedia.org\/wiki\/Gamma_distribution","sameAs":"http:\/\/www.wikidata.org\/entity\/Q117806","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q117806","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-04-07T21:17:16Z","dateModified":"2024-11-20T16:07:11Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e6\/Gamma_distribution_pdf.svg","headline":"probability distribution"}</script> </body> </html>

CINXE.COM

Gamma distribution - Wikipedia