Beta prime 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>Beta prime 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":"059861df-089c-4167-87fc-9822754fd8de","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Beta_prime_distribution","wgTitle":"Beta prime distribution","wgCurRevisionId":1258094556,"wgRevisionId":1258094556,"wgArticleId":3126406,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Continuous distributions","Compound probability distributions"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Beta_prime_distribution","wgRelevantArticleId":3126406,"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":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q180188","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","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/1/18/Beta_prime_pdf.svg/1200px-Beta_prime_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/1/18/Beta_prime_pdf.svg/800px-Beta_prime_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/1/18/Beta_prime_pdf.svg/640px-Beta_prime_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="Beta prime 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/Beta_prime_distribution"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Beta_prime_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/Beta_prime_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-Beta_prime_distribution rootpage-Beta_prime_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" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </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"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></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"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></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"><span>Recent changes</span></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"><span>Upload file</span></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"> <span class="mw-logo-container skin-invert"> <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;"> </span> </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"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </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" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </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" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </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=""><span>Donate</span></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=Beta+prime+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></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=Beta+prime+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></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" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </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"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Beta+prime+distribution" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Beta+prime+distribution" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></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"><span>learn more</span></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"><span>Contributions</span></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"><span>Talk</span></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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definitions</span> </div> </a> <button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definitions subsection</span> </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Alternative_parameterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_parameterization"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Alternative parameterization</span> </div> </a> <ul id="toc-Alternative_parameterization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Generalization</span> </div> </a> <ul id="toc-Generalization-sublist" class="vector-toc-list"> <li id="toc-Compound_gamma_distribution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Compound_gamma_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Compound gamma distribution</span> </div> </a> <ul id="toc-Compound_gamma_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Related distributions</span> </div> </a> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Beta prime distribution</span></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 6 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-6" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">6 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%8D%D1%82%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_%D0%B4%D1%80%D1%83%D0%B3%D0%BE%D0%B3%D0%B0_%D1%82%D1%8B%D0%BF%D1%83" title="Бэта-размеркаванне другога тыпу – Belarusian" lang="be" hreflang="be" data-title="Бэта-размеркаванне другога тыпу" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_beta_prima" title="Distribució beta prima – Catalan" lang="ca" hreflang="ca" data-title="Distribució beta prima" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Inverse_Betaverteilung" title="Inverse Betaverteilung – German" lang="de" hreflang="de" data-title="Inverse Betaverteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></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_%D8%A8%D8%AA%D8%A7_%D9%BE%D8%B1%DB%8C%D9%85" title="توزیع بتا پریم – Persian" lang="fa" hreflang="fa" data-title="توزیع بتا پریم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_b%C3%AAta_prime" title="Loi bêta prime – French" lang="fr" hreflang="fr" data-title="Loi bêta prime" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AC%AC2%E7%A8%AE%E3%83%99%E3%83%BC%E3%82%BF%E5%88%86%E5%B8%83" title="第2種ベータ分布 – Japanese" lang="ja" hreflang="ja" data-title="第2種ベータ分布" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180188#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></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/Beta_prime_distribution" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Beta_prime_distribution" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></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" ><span class="vector-dropdown-label-text">English</span> </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/Beta_prime_distribution"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></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" ><span class="vector-dropdown-label-text">Tools</span> </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/Beta_prime_distribution"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&action=history"><span>View history</span></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/Beta_prime_distribution" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Beta_prime_distribution" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></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"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&oldid=1258094556" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Beta_prime_distribution&id=1258094556&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></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%2FBeta_prime_distribution"><span>Get shortened URL</span></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%2FBeta_prime_distribution"><span>Download QR code</span></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=Beta_prime_distribution&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Beta_prime_distribution&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></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 id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180188" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></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">Beta prime</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability density function</div><span typeof="mw:File"><a href="/wiki/File:Beta_prime_pdf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Beta_prime_pdf.svg/325px-Beta_prime_pdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Beta_prime_pdf.svg/488px-Beta_prime_pdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Beta_prime_pdf.svg/650px-Beta_prime_pdf.svg.png 2x" data-file-width="512" data-file-height="384" /></a></span></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><span typeof="mw:File"><a href="/wiki/File:Beta_prime_cdf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Beta_prime_cdf.svg/325px-Beta_prime_cdf.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Beta_prime_cdf.svg/488px-Beta_prime_cdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Beta_prime_cdf.svg/650px-Beta_prime_cdf.svg.png 2x" data-file-width="512" data-file-height="384" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (<a href="/wiki/Real_number" title="Real number">real</a>)<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87dc52878418173659e6d0ff8e77ab2897eac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >0}"></span> shape (real)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty )\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be00d03e32dc2701e685f3f746dee4702dfdd208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:10.021ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty )\!}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}"> <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> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674325daefc80f42efb70be3875393c3b2a54ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-right: -0.108ex; width:24.713ex; height:6.676ex;" alt="{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2217f44848f7d5bddc592c386a568033c4e0a029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.291ex; height:3.843ex;" alt="{\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{x}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x}(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663054c3f3dc36c0c8c445386d9e52aea7e26b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.858ex; height:2.843ex;" alt="{\displaystyle I_{x}(\alpha ,\beta )}"></span> is the incomplete beta function</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8a1d375526db50bec8faf95956ce90134a5c4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.171ex; height:5.176ex;" alt="{\displaystyle {\frac {\alpha }{\beta -1}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1476a19e8f0d4cee97bf949617be8ed3cbc676e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >1}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}"> <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> <mrow> <mi>β</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>α</mi> <mo>≥</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>, 0 otherwise</mtext> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6317382f881af2a06b1acd41dbccc9f6effcc97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.32ex; width:27.076ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529d41e85cd4f8c95c12855cee292ab3d7d62970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.179ex; height:6.509ex;" alt="{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b1d933041e4bc4f80449e3f36c922351881e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >2}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>β</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>β</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad26bea87fd7f6148e507947475432fbc5a3123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.753ex; height:7.676ex;" alt="{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac718ddc26f96cbf7a4e80cf9b3230ac9020a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >3}"></span></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 colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>β</mi> <mo>−</mo> <mn>11</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>β</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>β</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1b4b9ab2e6c5d8f08bc38d35ca153769f484f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.61ex; height:6.676ex;" alt="{\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6f9845c29d3eb3bc581c78f016e9ce0dd76078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >4}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\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 /> <mtd> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a719cc33f2580089f9cbb683a817a6a151b761a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:64.084ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a>.</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> Does not exist</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 colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> <mo>,</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mspace width="thickmathspace" /> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02d27cf5d3f60e5dafdfa9384560412c29bc955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.055ex; height:6.676ex;" alt="{\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>beta prime distribution</b> (also known as <b>inverted beta distribution</b> or <b>beta distribution of the second kind</b><sup id="cite_ref-Johnson_et_al_1995,_p_248_1-0" class="reference"><a href="#cite_note-Johnson_et_al_1995,_p_248-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) is an <a href="/wiki/Probability_distribution#Continuous_probability_distribution" title="Probability distribution">absolutely continuous probability distribution</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c3a52aa7b2d00227e85c641cca67e85583c43c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.752ex; height:2.843ex;" alt="{\displaystyle p\in [0,1]}"></span> has a <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a>, then the <a href="/wiki/Odds" title="Odds">odds</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p}{1-p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{1-p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e443e799dc4674631696ea930a2fc037f35e71e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.008ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{1-p}}}"></span> has a beta prime distribution. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beta prime distribution is defined for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span> with two parameters <i>α</i> and <i>β</i>, having the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}"> <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> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4619c02b9ae50cd2180140e45bb44dec6d91756a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.991ex; height:6.676ex;" alt="{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}"></span></dd></dl> <p>where <i>B</i> is the <a href="/wiki/Beta_function" title="Beta function">Beta function</a>. </p><p>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\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 stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38285fb9c37e5ab4e4264fe836ee9a752735ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.51ex; height:4.009ex;" alt="{\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}"></span></dd></dl> <p>where <i>I</i> is the <a href="/wiki/Regularized_incomplete_beta_function" class="mw-redirect" title="Regularized incomplete beta function">regularized incomplete beta function</a>. </p><p>While the related <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> is the <a href="/wiki/Conjugate_prior_distribution" class="mw-redirect" title="Conjugate prior distribution">conjugate prior distribution</a> of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in <a href="/wiki/Odds" title="Odds">odds</a>. The distribution is a <a href="/wiki/Pearson_distribution#The_Pearson_type_VI_distribution" title="Pearson distribution">Pearson type VI</a> distribution.<sup id="cite_ref-Johnson_et_al_1995,_p_248_1-1" class="reference"><a href="#cite_note-Johnson_et_al_1995,_p_248-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The mode of a variate <i>X</i> distributed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c494b685f92c1ee92ea8a2fd3e21d2462c9b0253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.685ex; height:3.009ex;" alt="{\displaystyle \beta '(\alpha ,\beta )}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>β</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6150208a4c5a8b39a4f5028b58b12df6d03d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.405ex; height:5.676ex;" alt="{\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}"></span>. Its mean is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\beta -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8a1d375526db50bec8faf95956ce90134a5c4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.171ex; height:5.176ex;" alt="{\displaystyle {\frac {\alpha }{\beta -1}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1476a19e8f0d4cee97bf949617be8ed3cbc676e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >1}"></span> (if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbaaa6fe41fb9684556d3c1577e5558ca520e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta \leq 1}"></span> the mean is infinite, in other words it has no well defined mean) and its variance is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529d41e85cd4f8c95c12855cee292ab3d7d62970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.179ex; height:6.509ex;" alt="{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b1d933041e4bc4f80449e3f36c922351881e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >2}"></span>. </p><p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\alpha <k<\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>α</mi> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\alpha <k<\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3beacabe6f7bdce6d0ca24fbd05c081bf685c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.036ex; height:2.509ex;" alt="{\displaystyle -\alpha <k<\beta }"></span>, the <i>k</i>-th moment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X^{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X^{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd54fe6c5cb4afbfcd7bd94c4778d13b8bbc3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.155ex; height:3.176ex;" alt="{\displaystyle E[X^{k}]}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>k</mi> <mo>,</mo> <mi>β</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd9d05a47fe1f0f99394c92575308526827c714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.148ex; height:6.509ex;" alt="{\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}"></span></dd></dl> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<\beta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>β</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<\beta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d782023d4738dbc1e6fe599916def2dc0278113d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.289ex; height:2.509ex;" alt="{\displaystyle k<\beta ,}"></span> this simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <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>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>β</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f1689a0ef95460a83f9f53462da32a9b1e8f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.226ex; height:7.343ex;" alt="{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}"></span></dd></dl> <p>The cdf can also be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d182604564bdb3d05145382a745a607f79b01608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.419ex; height:6.509ex;" alt="{\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7506f01afc6742d7168ab0cb201337ca2a216ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.603ex; height:2.509ex;" alt="{\displaystyle {}_{2}F_{1}}"></span> is the Gauss's <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a> <sub>2</sub>F<sub>1</sub> . </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_parameterization">Alternative parameterization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=2" title="Edit section: Alternative parameterization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The beta prime distribution may also be reparameterized in terms of its mean <i>μ</i> > 0 and precision <i>ν</i> > 0 parameters (<sup id="cite_ref-Bourguignon_2-0" class="reference"><a href="#cite_note-Bourguignon-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> p. 36). </p><p>Consider the parameterization <i>μ</i> = <i>α</i>/(<i>β</i>-1) and <i>ν</i> = <i>β</i>- 2, i.e., <i>α</i> = <i>μ</i>( 1 + <i>ν</i>) and <i>β</i> = 2 + <i>ν</i>. Under this parameterization E[Y] = <i>μ</i> and Var[Y] = <i>μ</i>(1 + <i>μ</i>)/<i>ν</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Generalization">Generalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=3" title="Edit section: Generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two more parameters can be added to form the <b>generalized beta prime distribution</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(\alpha ,\beta ,p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(\alpha ,\beta ,p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4808e6978e9ccb005f021e6ef9cd824062444104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.991ex; height:3.009ex;" alt="{\displaystyle \beta '(\alpha ,\beta ,p,q)}"></span>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (<a href="/wiki/Real_number" title="Real number">real</a>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> scale (<a href="/wiki/Real_number" title="Real number">real</a>)</li></ul> <p>having the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{q\mathrm {B} (\alpha ,\beta )}}}"> <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>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{q\mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0729778042432fda7e36baffca20ee811721d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.652ex; height:9.009ex;" alt="{\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{q\mathrm {B} (\alpha ,\beta )}}}"></span></dd></dl> <p>with <a href="/wiki/Mean" title="Mean">mean</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>β</mi> <mi>p</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e22ef0e756cac49fc4b9774ad62337cecb88a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.021ex; height:8.343ex;" alt="{\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}"></span></dd></dl> <p>and <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>β</mi> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>α</mi> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c8881c6a2ff5bc28f423af9d2996610d196468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.062ex; height:7.676ex;" alt="{\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}"></span></dd></dl> <p>Note that if <i>p</i> = <i>q</i> = 1 then the generalized beta prime distribution reduces to the <b>standard beta prime distribution</b>. </p><p>This generalization can be obtained via the following invertible transformation. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim \beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5a820906e27dd945f82b869211e4d9554e3e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.939ex; height:3.009ex;" alt="{\displaystyle y\sim \beta '(\alpha ,\beta )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=qy^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>q</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=qy^{1/p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843a015ad7cde1a582ab39be430f1ed563f552b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.361ex; height:3.176ex;" alt="{\displaystyle x=qy^{1/p}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q,p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>,</mo> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q,p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10bd2987a8392a7142157eea677bb60f8a2ba824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.534ex; height:2.509ex;" alt="{\displaystyle q,p>0}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21dbe0868d09c67bee84412f18ac21c4e7a4827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.42ex; height:3.009ex;" alt="{\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Compound_gamma_distribution">Compound gamma distribution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=4" title="Edit section: Compound gamma distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>compound gamma distribution</b><sup id="cite_ref-Dubey_3-0" class="reference"><a href="#cite_note-Dubey-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> is the generalization of the beta prime when the scale parameter, <i>q</i> is added, but where <i>p</i> = 1. It is so named because it is formed by <a href="/wiki/Compound_distribution" class="mw-redirect" title="Compound distribution">compounding</a> two <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distributions</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>q</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 mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>;</mo> <mi>β</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cad54ba3607d1cf04cc5ff382db4f3e9b6c4f91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.807ex; height:5.843ex;" alt="{\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x;a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x;a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80d4a7ba6e9bf506dc9f2e4411438fc67786c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.261ex; height:2.843ex;" alt="{\displaystyle G(x;a,b)}"></span> is the gamma pdf with shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and inverse scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p><p>The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by <i>q</i> and the variance by <i>q</i><sup>2</sup>. </p><p>Another way to express the compounding is if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\sim G(\beta ,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∼</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\sim G(\beta ,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45d173e9719c0f2a0129282e3221cb129ec88fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.219ex; height:2.843ex;" alt="{\displaystyle r\sim G(\beta ,q)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mid r\sim G(\alpha ,r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∣</mo> <mi>r</mi> <mo>∼</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mid r\sim G(\alpha ,r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c412336e5abce195ee60de57c39e7bf54ce8c3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.62ex; height:2.843ex;" alt="{\displaystyle x\mid r\sim G(\alpha ,r)}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acb95223a7d525db004e6035484959f02d1cf02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.413ex; height:3.009ex;" alt="{\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}"></span>. This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta )}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d016ffa2dfee395c14e11512e615205eaeb0d78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta )}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>X</mi> </mfrac> </mstyle> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ec158c341777a71753c032919fc399358299f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.019ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}"></span>.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416420b46dbc4edbbdf00a25066c5a3471b81269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.556ex; height:3.009ex;" alt="{\displaystyle Y\sim \beta '(\alpha ,\beta )}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=qY^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>q</mi> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=qY^{1/p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a2e20819bef8615b5033852eeed42828664fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.751ex; height:3.176ex;" alt="{\displaystyle X=qY^{1/p}}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eade216626a48cf6f6bd02935d3cbcfa85cb2ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.07ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"></span>.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eade216626a48cf6f6bd02935d3cbcfa85cb2ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.07ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>X</mi> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>k</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764444c84942945f2a955e9dc6dfb2d77b652a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.492ex; height:3.009ex;" alt="{\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a97057eff9376f94c89707d1e3bce0e0067b318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.86ex; height:3.009ex;" alt="{\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=6" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Beta</mtext> </mrow> </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 X\sim {\textrm {Beta}}(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419bd5ef77b17e89bf74f45fafff3ebe8cb30074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.486ex; height:2.843ex;" alt="{\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>X</mi> </mrow> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eef419e4b2d0d55e805018bc9cd69307c9bb0ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.602ex; height:5.343ex;" alt="{\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}"></span>. This property can be used to generate beta prime distributed variates.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta )}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d016ffa2dfee395c14e11512e615205eaeb0d78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta )}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>X</mi> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Beta</mtext> </mrow> </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 {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab30f4641008000fab5e828c3668c6f87b47c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.325ex; height:5.343ex;" alt="{\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}"></span>. This is a corollary from the property above.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim F(2\alpha ,2\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo>,</mo> <mn>2</mn> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim F(2\alpha ,2\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05c82685eb6dda5c748c2829b459cc58b6bee6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.807ex; height:2.843ex;" alt="{\displaystyle X\sim F(2\alpha ,2\beta )}"></span> has an <a href="/wiki/F-distribution" title="F-distribution"><i>F</i>-distribution</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>α</mi> <mi>β</mi> </mfrac> </mstyle> </mrow> <mi>X</mi> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91610cc4f7ff3d58b53d23b28136126f39a85454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.651ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}"></span>, or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>β</mi> <mi>α</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a74996ea875223514e42217310a50bfe967a2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.881ex; height:3.843ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}"></span>.</li> <li>For <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a> parametrization I: <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a010665036b0d8157050131e2e8ec4dc9bad45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.163ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}"></span> are independent, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <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"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f581d54fe69c033438a8f041a2e0721e31517ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.745ex; height:4.509ex;" alt="{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}"></span>. Note <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\alpha _{2},{\tfrac {\theta _{1}}{\theta _{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\alpha _{2},{\tfrac {\theta _{1}}{\theta _{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6494861dfe51a3ed6768796fe48bf26c096f0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.59ex; height:4.509ex;" alt="{\displaystyle \alpha _{1},\alpha _{2},{\tfrac {\theta _{1}}{\theta _{2}}}}"></span> are all scale parameters for their respective distributions.</li></ul></li> <li>For gamma distribution parametrization II: <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd7674f3b4e9d2ce7d8260c5f240cc5e127f58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.388ex; height:2.843ex;" alt="{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}"></span> are independent, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <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"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59418b1850acff2004e0ed6b67ec27a16de1b743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.904ex; height:4.509ex;" alt="{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}"></span>. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span> are rate parameters, while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20143ae2d12c7bd980ef6e9974940c3bb0c373c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:2.598ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}}"></span> is a scale parameter.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8519708fc32d973b3299c660615d0c3969bcd72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.676ex; height:2.843ex;" alt="{\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69595535cae40e26cb91020da40c5a8318678e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.592ex; height:2.843ex;" alt="{\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6e3aefed8b34adaa301cc915ca3f6a87826dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.626ex; height:3.009ex;" alt="{\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}"></span>. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span> are rate parameters for the gamma distributions, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeccd8b585b819e38f9c1fe5e9816a3ea01804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{1}}"></span> is the scale parameter for the beta prime.</li></ul></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Dagum</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e015a3f7100c7b8534eb1aa4eccb023047f0ba65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.193ex; height:3.009ex;" alt="{\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}"></span> the <a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum distribution</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SinghMaddala</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea8624cf62820955b65bdadf08e629f72baa15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.402ex; height:3.009ex;" alt="{\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}"></span> the <a href="/wiki/Singh%E2%80%93Maddala_distribution" class="mw-redirect" title="Singh–Maddala distribution">Singh–Maddala distribution</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>LL</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e9062d64476e0161246097953b16d7c8daec72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.289ex; height:3.009ex;" alt="{\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}"></span> the <a href="/wiki/Log_logistic_distribution" class="mw-redirect" title="Log logistic distribution">log logistic distribution</a>.</li> <li>The beta prime distribution is a special case of the type 6 <a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson distribution</a>.</li> <li>If <i>X</i> has a <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a> with minimum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12100e1dc5769ced8c9806b219abc06ab321d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.009ex;" alt="{\displaystyle x_{m}}"></span> and shape parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>X</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> <mo>∼</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1fbd9061d7eb15528b0bacf849b76378d545c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.457ex; height:5.509ex;" alt="{\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )}"></span>.</li> <li>If <i>X</i> has a <a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax distribution</a>, also known as a Pareto Type II distribution, with shape parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and scale parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mi>λ</mi> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260bf4cc5df97b622945905a2c8192d0508dfc7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.43ex; height:5.343ex;" alt="{\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}"></span>.</li> <li>If <i>X</i> has a standard <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto Type IV distribution</a> with shape parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and inequality parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>∼</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bdb3826585e538633b37af2118b08a7267e4a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.403ex; height:4.009ex;" alt="{\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}"></span>, or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3846970d2d6251d888b097fd9aa2af5b689f0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.553ex; height:3.676ex;" alt="{\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}"></span>.</li> <li>The <a href="/wiki/Inverted_Dirichlet_distribution" title="Inverted Dirichlet distribution">inverted Dirichlet distribution</a> is a generalization of the beta prime distribution.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta )}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d016ffa2dfee395c14e11512e615205eaeb0d78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta )}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85aab04c271b1afa060440276c5885ede5adb4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.306ex; height:2.176ex;" alt="{\displaystyle \ln X}"></span> has a <a href="/wiki/Generalized_logistic_distribution" title="Generalized logistic distribution">generalized logistic distribution</a>. More generally, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"> <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>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eade216626a48cf6f6bd02935d3cbcfa85cb2ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.07ex; height:3.009ex;" alt="{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85aab04c271b1afa060440276c5885ede5adb4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.306ex; height:2.176ex;" alt="{\displaystyle \ln X}"></span> has a <a href="/wiki/Sigmoid-beta_distribution#Generalization_with_location_and_scale_parameters" class="mw-redirect" title="Sigmoid-beta distribution">scaled and shifted</a> generalized logistic distribution.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Johnson_et_al_1995,_p_248-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Johnson_et_al_1995,_p_248_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Johnson_et_al_1995,_p_248_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Johnson et al (1995), p 248</span> </li> <li id="cite_note-Bourguignon-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bourguignon_2-0">^</a></b></span> <span class="reference-text"><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 id="CITEREFBourguignonSantos-Neto,_M.de_Castro,_M.2021" class="citation journal cs1">Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". <i>Metron</i>. <b>79</b>: 33–55. <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%2Fs40300-021-00203-y">10.1007/s40300-021-00203-y</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:233534544">233534544</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metron&rft.atitle=A+new+regression+model+for+positive+random+variables+with+skewed+and+long+tail&rft.volume=79&rft.pages=33-55&rft.date=2021&rft_id=info%3Adoi%2F10.1007%2Fs40300-021-00203-y&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A233534544%23id-name%3DS2CID&rft.aulast=Bourguignon&rft.aufirst=M.&rft.au=Santos-Neto%2C+M.&rft.au=de+Castro%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABeta+prime+distribution" class="Z3988"></span></span> </li> <li id="cite_note-Dubey-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dubey_3-0">^</a></b></span> <span class="reference-text"><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". <i>Metrika</i>. <b>16</b>: 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%3ABeta+prime+distribution" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_prime_distribution&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). <i>Continuous Univariate Distributions</i>, Volume 2 (2nd Edition), Wiley. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-58494-0" title="Special:BookSources/0-471-58494-0">0-471-58494-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourguignonSantos-Netode_Castro2021" class="citation cs2">Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", <i>Metron</i>, <b>79</b>: 33–55, <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%2Fs40300-021-00203-y">10.1007/s40300-021-00203-y</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:233534544">233534544</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metron&rft.atitle=A+new+regression+model+for+positive+random+variables+with+skewed+and+long+tail&rft.volume=79&rft.pages=33-55&rft.date=2021&rft_id=info%3Adoi%2F10.1007%2Fs40300-021-00203-y&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A233534544%23id-name%3DS2CID&rft.aulast=Bourguignon&rft.aufirst=M.&rft.au=Santos-Neto%2C+M.&rft.au=de+Castro%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABeta+prime+distribution" class="Z3988"></span></li></ul> <p><br /> </p> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BetaPrimeDistribution.html">MathWorld article</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 <br />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 <br />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 <br />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 <br />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 <br />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 <br />semi-infinite <br />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 class="mw-selflink selflink">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"><i>F</i></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 href="/wiki/Gamma_distribution" title="Gamma distribution">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 <i>T</i>-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 <br />on the whole <br />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 <i>z</i></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 <i>q</i></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 <i>S<sub>U</sub></i></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 <i>t</i></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 <i>t</i></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 <br />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 <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-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 <br />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-<br />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 <br />(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><span class="nobold"><i>Discrete: </i></span></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><span class="nobold"><i>Continuous: </i></span></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 <i>t</i></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><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></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 <i>t</i></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><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></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><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></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> <br />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><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></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><span class="noviewer" typeof="mw:File"><span title="Category"><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" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><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" /></span></span> <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=Beta_prime_distribution&oldid=1258094556">https://en.wikipedia.org/w/index.php?title=Beta_prime_distribution&oldid=1258094556</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:Compound_probability_distributions" title="Category:Compound probability distributions">Compound probability distributions</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:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</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 18 November 2024, at 03:05<span class="anonymous-show"> (UTC)</span>.</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=Beta_prime_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-f69cdc8f6-4xwft","wgBackendResponseTime":174,"wgPageParseReport":{"limitreport":{"cputime":"0.486","walltime":"0.812","ppvisitednodes":{"value":1795,"limit":1000000},"postexpandincludesize":{"value":78084,"limit":2097152},"templateargumentsize":{"value":2598,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":28322,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 422.861 1 -total"," 27.93% 118.100 4 Template:Navbox"," 27.81% 117.601 1 Template:ProbDistributions"," 26.19% 110.734 1 Template:Short_description"," 26.04% 110.122 1 Template:Reflist"," 21.45% 90.719 2 Template:Cite_journal"," 13.20% 55.813 2 Template:Pagetype"," 11.63% 49.190 1 Template:Probability_distribution"," 10.24% 43.320 5 Template:Main_other"," 9.61% 40.632 1 Template:SDcat"]},"scribunto":{"limitreport-timeusage":{"value":"0.235","limit":"10.000"},"limitreport-memusage":{"value":4819434,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-cdf8n","timestamp":"20241122143139","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Beta prime distribution","url":"https:\/\/en.wikipedia.org\/wiki\/Beta_prime_distribution","sameAs":"http:\/\/www.wikidata.org\/entity\/Q180188","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q180188","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":"2005-11-10T15:30:24Z","dateModified":"2024-11-18T03:05:15Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/1\/18\/Beta_prime_pdf.svg","headline":"absolutely continuous probability distribution"}</script> </body> </html>

CINXE.COM

Beta prime distribution - Wikipedia