Distribució normal - Viquipèdia, l'enciclopèdia lliure

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més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" 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="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" alt="l'Enciclopèdia Lliure" src="/static/images/mobile/copyright/wikipedia-tagline-ca.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </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/Especial:Cerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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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">Aparença</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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca" class=""><span>Donatius</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=Especial:Crea_compte&returnto=Distribuci%C3%B3+normal" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</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=Especial:Registre_i_entrada&returnto=Distribuci%C3%B3+normal" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</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="Més opcions" > <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="Eines personals" > <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">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a 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mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</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/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</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="Lloc"> <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="Contingut" 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">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Funció_de_densitat_de_probabilitat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funció_de_densitat_de_probabilitat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Funció de densitat de probabilitat</span> </div> </a> <ul id="toc-Funció_de_densitat_de_probabilitat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funció_de_distribució" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funció_de_distribució"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Funció de distribució</span> </div> </a> <ul id="toc-Funció_de_distribució-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcions_generadores" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcions_generadores"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Funcions generadores</span> </div> </a> <button aria-controls="toc-Funcions_generadores-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>Commuta la subsecció Funcions generadores</span> </button> <ul id="toc-Funcions_generadores-sublist" class="vector-toc-list"> <li id="toc-Funció_generadora_de_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funció_generadora_de_moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Funció generadora de moments</span> </div> </a> <ul id="toc-Funció_generadora_de_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funció_característica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funció_característica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Funció característica</span> </div> </a> <ul id="toc-Funció_característica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propietats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propietats"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propietats</span> </div> </a> <button aria-controls="toc-Propietats-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>Commuta la subsecció Propietats</span> </button> <ul id="toc-Propietats-sublist" class="vector-toc-list"> <li id="toc-Estandardització_de_variables_aleatòries_normals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estandardització_de_variables_aleatòries_normals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Estandardització de variables aleatòries normals</span> </div> </a> <ul id="toc-Estandardització_de_variables_aleatòries_normals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Moments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Moments</span> </div> </a> <button aria-controls="toc-Moments-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>Commuta la subsecció Moments</span> </button> <ul id="toc-Moments-sublist" class="vector-toc-list"> <li id="toc-Moments_d'una_variable_normal_centrada" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moments_d'una_variable_normal_centrada"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Moments d'una variable normal centrada</span> </div> </a> <ul id="toc-Moments_d'una_variable_normal_centrada-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cas_general" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_general"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Cas general</span> </div> </a> <ul id="toc-Cas_general-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliogafia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliogafia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliogafia</span> </div> </a> <ul id="toc-Bibliogafia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Distribució normal</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="Vés a un article en una altra llengua. Disponible en 73 llengües" > <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-73" 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">73 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung - alemany suís" lang="gsw" hreflang="gsw" data-title="Normalverteilung" data-language-autonym="Alemannisch" data-language-local-name="alemany suís" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D9%8A_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="توزيع احتمالي طبيعي - àrab" lang="ar" hreflang="ar" data-title="توزيع احتمالي طبيعي" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal - asturià" lang="ast" hreflang="ast" data-title="Distribución normal" data-language-autonym="Asturianu" data-language-local-name="asturià" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Normal_paylanma" title="Normal paylanma - azerbaidjanès" lang="az" hreflang="az" data-title="Normal paylanma" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%D9%88%D8%B1%D9%85%D8%A7%D9%84_%D8%AF%D8%A7%D8%BA%DB%8C%D9%84%DB%8C%D9%85" title="نورمال داغیلیم - South Azerbaijani" lang="azb" hreflang="azb" data-title="نورمال داغیلیم" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Нармальнае размеркаванне - belarús" lang="be" hreflang="be" data-title="Нармальнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормално разпределение - búlgar" lang="bg" hreflang="bg" data-title="Нормално разпределение" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela - bosnià" lang="bs" hreflang="bs" data-title="Normalna raspodjela" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD" title="Normální rozdělení - txec" lang="cs" hreflang="cs" data-title="Normální rozdělení" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B2%D0%B0%D0%BB%D0%B5%C3%A7%C4%95%D0%B2%C4%95" title="Гаусс валеçĕвĕ - txuvaix" lang="cv" hreflang="cv" data-title="Гаусс валеçĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dosraniad_normal" title="Dosraniad normal - gal·lès" lang="cy" hreflang="cy" data-title="Dosraniad normal" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling - danès" lang="da" hreflang="da" data-title="Normalfordeling" data-language-autonym="Dansk" data-language-local-name="danès" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung - alemany" lang="de" hreflang="de" data-title="Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CE%BD%CE%BF%CE%BD%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE" title="Κανονική κατανομή - grec" lang="el" hreflang="el" data-title="Κανονική κατανομή" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normal_distribution" title="Normal distribution - anglès" lang="en" hreflang="en" data-title="Normal distribution" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Normala_distribuo" title="Normala distribuo - esperanto" lang="eo" hreflang="eo" data-title="Normala distribuo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal - espanyol" lang="es" hreflang="es" data-title="Distribución normal" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Normaaljaotus" title="Normaaljaotus - estonià" lang="et" hreflang="et" data-title="Normaaljaotus" data-language-autonym="Eesti" data-language-local-name="estonià" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Banaketa_normal" title="Banaketa normal - basc" lang="eu" hreflang="eu" data-title="Banaketa normal" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</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_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="توزیع نرمال - persa" lang="fa" hreflang="fa" data-title="توزیع نرمال" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Normaalijakauma" title="Normaalijakauma - finès" lang="fi" hreflang="fi" data-title="Normaalijakauma" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="article bo"><a href="https://fr.wikipedia.org/wiki/Loi_normale" title="Loi normale - francès" lang="fr" hreflang="fr" data-title="Loi normale" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Normoolferdialang" title="Normoolferdialang - frisó septentrional" lang="frr" hreflang="frr" data-title="Normoolferdialang" data-language-autonym="Nordfriisk" data-language-local-name="frisó septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%A1ileadh_normalach" title="Dáileadh normalach - irlandès" lang="ga" hreflang="ga" data-title="Dáileadh normalach" data-language-autonym="Gaeilge" data-language-local-name="irlandès" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal - gallec" lang="gl" hreflang="gl" data-title="Distribución normal" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="התפלגות נורמלית - hebreu" lang="he" hreflang="he" data-title="התפלגות נורמלית" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%AC%E0%A4%82%E0%A4%9F%E0%A4%A8" title="प्रसामान्य बंटन - hindi" lang="hi" hreflang="hi" data-title="प्रसामान्य बंटन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela - croat" lang="hr" hreflang="hr" data-title="Normalna raspodjela" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Norm%C3%A1lis_eloszl%C3%A1s" title="Normális eloszlás - hongarès" lang="hu" hreflang="hu" data-title="Normális eloszlás" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B8%D6%80%D5%B4%D5%A1%D5%AC_%D5%A2%D5%A1%D5%B7%D5%AD%D5%B8%D6%82%D5%B4" title="Նորմալ բաշխում - armeni" lang="hy" hreflang="hy" data-title="Նորմալ բաշխում" data-language-autonym="Հայերեն" data-language-local-name="armeni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_normal" title="Distribusi normal - indonesi" lang="id" hreflang="id" data-title="Distribusi normal" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Normaldreifing" title="Normaldreifing - islandès" lang="is" hreflang="is" data-title="Normaldreifing" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_normale" title="Distribuzione normale - italià" lang="it" hreflang="it" data-title="Distribuzione normale" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83" title="正規分布 - japonès" lang="ja" hreflang="ja" data-title="正規分布" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%9D%E1%83%A0%E1%83%9B%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%9C%E1%83%90%E1%83%AC%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="ნორმალური განაწილება - georgià" lang="ka" hreflang="ka" data-title="ნორმალური განაწილება" data-language-autonym="ქართული" data-language-local-name="georgià" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%B0%D0%BB%D1%8B%D0%BF%D1%82%D1%8B_%D0%B4%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F" title="Қалыпты дисперсия - kazakh" lang="kk" hreflang="kk" data-title="Қалыпты дисперсия" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EA%B7%9C_%EB%B6%84%ED%8F%AC" title="정규 분포 - coreà" lang="ko" hreflang="ko" data-title="정규 분포" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Distributio_normalis" title="Distributio normalis - llatí" lang="la" hreflang="la" data-title="Distributio normalis" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Distribuzzion_normala" title="Distribuzzion normala - llombard" lang="lmo" hreflang="lmo" data-title="Distribuzzion normala" data-language-autonym="Lombard" data-language-local-name="llombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Normalusis_skirstinys" title="Normalusis skirstinys - lituà" lang="lt" hreflang="lt" data-title="Normalusis skirstinys" data-language-autonym="Lietuvių" data-language-local-name="lituà" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Norm%C4%81lais_sadal%C4%ABjums" title="Normālais sadalījums - letó" lang="lv" hreflang="lv" data-title="Normālais sadalījums" data-language-autonym="Latviešu" data-language-local-name="letó" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B1%D0%B0" title="Нормална распределба - macedoni" lang="mk" hreflang="mk" data-title="Нормална распределба" data-language-autonym="Македонски" data-language-local-name="macedoni" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A4%B0%E0%A4%A3" title="सामान्य वितरण - marathi" lang="mr" hreflang="mr" data-title="सामान्य वितरण" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Taburan_normal" title="Taburan normal - malai" lang="ms" hreflang="ms" data-title="Taburan normal" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Normale_verdeling" title="Normale verdeling - neerlandès" lang="nl" hreflang="nl" data-title="Normale verdeling" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling - noruec nynorsk" lang="nn" hreflang="nn" data-title="Normalfordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling - noruec bokmål" lang="nb" hreflang="nb" data-title="Normalfordeling" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_normalny" title="Rozkład normalny - polonès" lang="pl" hreflang="pl" data-title="Rozkład normalny" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Distribussion_%C3%ABd_Gauss" title="Distribussion ëd Gauss - piemontès" lang="pms" hreflang="pms" data-title="Distribussion ëd Gauss" data-language-autonym="Piemontèis" data-language-local-name="piemontès" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_normal" title="Distribuição normal - portuguès" lang="pt" hreflang="pt" data-title="Distribuição normal" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Distribu%C8%9Bia_Gauss" title="Distribuția Gauss - romanès" lang="ro" hreflang="ro" data-title="Distribuția Gauss" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормальное распределение - rus" lang="ru" hreflang="ru" data-title="Нормальное распределение" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Normalna_raspodela" title="Normalna raspodela - serbocroat" lang="sh" hreflang="sh" data-title="Normalna raspodela" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Normal_distribution" title="Normal distribution - Simple English" lang="en-simple" hreflang="en-simple" data-title="Normal distribution" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Norm%C3%A1lne_rozdelenie" title="Normálne rozdelenie - eslovac" lang="sk" hreflang="sk" data-title="Normálne rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Normalna_porazdelitev" title="Normalna porazdelitev - eslovè" lang="sl" hreflang="sl" data-title="Normalna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_normale" title="Shpërndarja normale - albanès" lang="sq" hreflang="sq" data-title="Shpërndarja normale" data-language-autonym="Shqip" data-language-local-name="albanès" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D0%BE%D0%B4%D0%B5%D0%BB%D0%B0" title="Нормална расподела - serbi" lang="sr" hreflang="sr" data-title="Нормална расподела" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_normal" title="Sebaran normal - sondanès" lang="su" hreflang="su" data-title="Sebaran normal" data-language-autonym="Sunda" data-language-local-name="sondanès" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Normalf%C3%B6rdelning" title="Normalfördelning - suec" lang="sv" hreflang="sv" data-title="Normalfördelning" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B2%E0%AF%8D%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="இயல்நிலைப் பரவல் - tàmil" lang="ta" hreflang="ta" data-title="இயல்நிலைப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%88%E0%B8%81%E0%B9%81%E0%B8%88%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%81%E0%B8%95%E0%B8%B4" title="การแจกแจงปรกติ - tai" lang="th" hreflang="th" data-title="การแจกแจงปรกติ" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_normal" title="Distribusyong normal - tagal" lang="tl" hreflang="tl" data-title="Distribusyong normal" data-language-autonym="Tagalog" data-language-local-name="tagal" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım - turc" lang="tr" hreflang="tr" data-title="Normal dağılım" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B1%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D1%88%D0%B5" title="Гаусс бүленеше - tàtar" lang="tt" hreflang="tt" data-title="Гаусс бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="tàtar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D1%80%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB" title="Нормальний розподіл - ucraïnès" lang="uk" hreflang="uk" data-title="Нормальний розподіл" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%A7%D8%B1%D9%85%D9%84_%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="نارمل تقسیم - urdú" lang="ur" hreflang="ur" data-title="نارمل تقسیم" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ph%E1%BB%91i_chu%E1%BA%A9n" title="Phân phối chuẩn - vietnamita" lang="vi" hreflang="vi" data-title="Phân phối chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 - xinès wu" lang="wuu" hreflang="wuu" data-title="正态分布" data-language-autonym="吴语" data-language-local-name="xinès wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%90%D7%A8%D7%9E%D7%90%D7%9C%D7%A2_%D7%A4%D7%90%D7%A8%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A0%D7%92" title="נארמאלע פארטיילונג - ídix" lang="yi" hreflang="yi" data-title="נארמאלע פארטיילונג" data-language-autonym="ייִדיש" data-language-local-name="ídix" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 - xinès" lang="zh" hreflang="zh" data-title="正态分布" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%C3%B4ng-th%C3%A0i_hun-p%C3%B2%CD%98" title="Siông-thài hun-pò͘ - xinès min del sud" lang="nan" hreflang="nan" data-title="Siông-thài hun-pò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈 - cantonès" lang="yue" hreflang="yue" data-title="常態分佈" data-language-autonym="粵語" data-language-local-name="cantonès" class="interlanguage-link-target"><span>粵語</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/Q133871#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</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="Espais de noms"> <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/Distribuci%C3%B3_normal" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussi%C3%B3:Distribuci%C3%B3_normal&action=edit&redlink=1" rel="discussion" class="new" title="Discussió sobre el contingut d'aquesta pàgina (encara no existeix) [t]" accesskey="t"><span>Discussió</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="Canvia la variant de llengua" > <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">català</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="Vistes"> <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/Distribuci%C3%B3_normal"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <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="Eines" > <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">Eines</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">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </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/Distribuci%C3%B3_normal"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=history"><span>Mostra l'historial</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/Especial:Enlla%C3%A7os/Distribuci%C3%B3_normal" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Distribuci%C3%B3_normal" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A0gines_especials" title="Llista totes les pàgines especials [q]" accesskey="q"><span>Pàgines especials</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&oldid=34002483" title="Enllaç permanent a aquesta revisió de la pàgina"><span>Enllaç permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=info" title="Més informació sobre aquesta pàgina"><span>Informació de la pàgina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citau&page=Distribuci%C3%B3_normal&id=34002483&wpFormIdentifier=titleform" title="Informació sobre com citar aquesta pàgina"><span>Citau aquest article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FDistribuci%25C3%25B3_normal"><span>Obtén una URL abreujada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FDistribuci%25C3%25B3_normal"><span>Descarrega el codi QR</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"> Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Distribuci%C3%B3+normal"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Distribuci%C3%B3_normal&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Distribuci%C3%B3_normal&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</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"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Normal_distribution" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q133871" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</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="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <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">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</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">De la Viquipèdia, l'enciclopèdia lliure</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="ca" dir="ltr"><table class="infobox" style="font-size:90%;width:25em"><caption style="font-weight:bold;background:#b0d1ad; font-size:120%;"><span style="float:left;margin-left: 3px;"><span typeof="mw:File"><span title="Infotaula distribució de probabilitat"><img alt="Infotaula distribució de probabilitat" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Probstats2.svg/22px-Probstats2.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Probstats2.svg/33px-Probstats2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Probstats2.svg/44px-Probstats2.svg.png 2x" data-file-width="115" data-file-height="89" /></span></span></span>Distribució normal</caption><tbody><tr><td colspan="2" class="infobox-full-data infobox-data" style="text-align:center"><div>Funció de densitat de probabilitat</div><span typeof="mw:File"><a href="/wiki/Fitxer:Normal_Distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/350px-Normal_Distribution_PDF.svg.png" decoding="async" width="350" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/525px-Normal_Distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/700px-Normal_Distribution_PDF.svg.png 2x" data-file-width="720" data-file-height="460" /></a></span><br /><small>La corba vermella és la <i>distribució normal estàndard</i></small></td></tr><tr><td colspan="2" class="infobox-full-data infobox-data" style="text-align:center"><div>Funció de distribució de probabilitat</div><span typeof="mw:File"><a href="/wiki/Fitxer:Normal_Distribution_CDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/350px-Normal_Distribution_CDF.svg.png" decoding="async" width="350" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/525px-Normal_Distribution_CDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/700px-Normal_Distribution_CDF.svg.png 2x" data-file-width="720" data-file-height="460" /></a></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Tipus</th><td class="infobox-data"><a href="/wiki/Distribucions_de_Tweedie" title="Distribucions de Tweedie">distribution de Tweedie</a>, <a href="/wiki/Distribuci%C3%B3_t_de_Student" title="Distribució t de Student">Distribució t de Student</a>, <a href="/wiki/Distribuci%C3%B3_normal_multivariable" title="Distribució normal multivariable">distribució normal multivariant</a>, <a href="/wiki/Fam%C3%ADlia_exponencial" title="Família exponencial">família exponencial</a>, <a href="/wiki/Distribuci%C3%B3_normal_esbiaixada" title="Distribució normal esbiaixada">Distribució normal esbiaixada</a>, <a href="/wiki/Distribuci%C3%B3_estable" title="Distribució estable">Distribució estable</a>, <a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1660125" class="extiw" title="d:Special:EntityPage/Q1660125">contaminated normal distribution</a> <sup>(en)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q1660125?uselang=ca" title="Tradueix"><img alt="Tradueix" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span>, <a href="/wiki/Distribuci%C3%B3_univariant" title="Distribució univariant">distribució univariant</a> i <a href="https://www.wikidata.org/wiki/Special:EntityPage/Q917918" class="extiw" title="d:Special:EntityPage/Q917918">distribució de probabilitat contínua</a> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P279" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Epònim</th><td class="infobox-data"><a href="/wiki/Carl_Friedrich_Gau%C3%9F" title="Carl Friedrich Gauß">Carl Friedrich Gauß</a> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P138" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Notació</th><td 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 {\mathcal {N}}(\mu ,\,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ce0ec18437bd01cec5958c39d732d4ddaf7530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:9.353ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(\mu ,\,\sigma ^{2})}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Paràmetres</th><td class="infobox-data"><span style="white-space:nowrap;"><i>μ</i> ∈ <b>R</b></span> — mitjana (<a href="/w/index.php?title=Par%C3%A0metre_de_posici%C3%B3&action=edit&redlink=1" class="new" title="Paràmetre de posició (encara no existeix)">posició</a>)<br /><span style="white-space:nowrap;"><i>σ</i>² > 0</span> — variància (<a href="/wiki/Par%C3%A0metre_d%27escala" title="Paràmetre d'escala">escala</a> al quadrat)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Suport_(matem%C3%A0tiques)" title="Suport (matemàtiques)">Suport</a></th><td 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10731" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Funci%C3%B3_de_densitat_de_probabilitat" title="Funció de densitat de probabilitat">fdp</a></th><td 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 {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceba7f5b0eec78de0f0c330406230ede67dbfc30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.859ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Funci%C3%B3_de_distribuci%C3%B3" title="Funció de distribució">FD</a></th><td 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 {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187f33664b79492eedf4406c66d67f9fe5f524ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.328ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Funci%C3%B3_quantil" title="Funció quantil">Quantil</a></th><td 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 \mu +\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(2F-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>erf</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>F</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(2F-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f3c46ea6af68612837bfe03580fe402401a57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.148ex; height:3.176ex;" alt="{\displaystyle \mu +\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(2F-1)}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Esperan%C3%A7a_matem%C3%A0tica" title="Esperança matemàtica">Esperança matemàtica</a></th><td 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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10738" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Mediana" title="Mediana">Mediana</a></th><td 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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10739" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Moda_(estad%C3%ADstica)" title="Moda (estadística)">Moda</a></th><td 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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10740" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Vari%C3%A0ncia" title="Variància">Variància</a></th><td 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 \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10743" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Coeficient_de_simetria" class="mw-redirect" title="Coeficient de simetria">Coeficient de simetria</a></th><td 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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10744" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Curtosi" title="Curtosi">Curtosi</a></th><td 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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10745" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Entropia_(informaci%C3%B3)" class="mw-redirect" title="Entropia (informació)">Entropia</a></th><td 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 {1}{2}}\ln(2\pi e\,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>e</mi> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\ln(2\pi e\,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283bb3954fd82cdd6a67108e7561d98593e459e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.484ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\ln(2\pi e\,\sigma ^{2})}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Funci%C3%B3_generadora_de_moments" class="mw-redirect" title="Funció generadora de moments">FGM</a></th><td 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 \mathrm {e} ^{\mu t+{\frac {\sigma ^{2}t^{2}}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mu t+{\frac {\sigma ^{2}t^{2}}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83ff6129970d09a5c44dd6100f8b7bfc956ddf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.811ex; height:4.009ex;" alt="{\displaystyle \mathrm {e} ^{\mu t+{\frac {\sigma ^{2}t^{2}}{2}}}}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10747" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Funci%C3%B3_caracter%C3%ADstica_(teoria_de_la_probabilitat)" title="Funció característica (teoria de la probabilitat)">FC</a></th><td 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 \mathrm {e} ^{\mathrm {i} \mu t-{\frac {\sigma ^{2}t^{2}}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>μ</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} \mu t-{\frac {\sigma ^{2}t^{2}}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a684908c73ca7c520e4132d537e02f92c54d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.269ex; height:4.009ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \mu t-{\frac {\sigma ^{2}t^{2}}{2}}}}"></span> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P10735" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee"><a href="/wiki/Informaci%C3%B3_de_Fisher" title="Informació de Fisher">Informació de Fisher</a></th><td 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{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262dc78a347dc277885993a5cc349a16547ae366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.886ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="text-align:left;background:#eeeeee">Mathworld</th><td class="infobox-data"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/NormalDistribution.html">NormalDistribution</a> <span class="penicon"><span class="mw-valign-baseline" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133871?uselang=ca#P2812" title="Modifica el valor a Wikidata"><img alt="Modifica el valor a Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/10px-Arbcom_ru_editing.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/15px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/20px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></td></tr></tbody></table> <p>La <b>distribució normal</b>, també coneguda com a <b>distribució gaussiana</b>, és una important família de distribucions de probabilitat contínues i és aplicable a molts camps. Cada membre de la família queda definit per dos paràmetres: la localització o mitjana <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> i l'escala o desviació estàndard <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>, i es denota per <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 \ {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747c8201577ac846094e6bc2c20e6a4a0c82760a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.485ex; height:3.176ex;" alt="{\displaystyle \ {\mathcal {N}}(\mu ,\sigma ^{2})}"></span>. Un cas particular és la <b>distribució normal estàndard</b>, per la qual la mitjana és 0 i la desviació estàndard és 1. Fou <a href="/wiki/Carl_Friedrich_Gauss" class="mw-redirect" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> qui descobrí la distribució normal quan analitzava dades astronòmiques, i definí l'equació de la seva <a href="/wiki/Funci%C3%B3_de_densitat_de_probabilitat" title="Funció de densitat de probabilitat">funció de densitat de probabilitat</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Aquesta distribució també s'anomena <b>campana de Gauss</b>, atès que el gràfic de la seva funció de densitat de probabilitat s'assembla a una campana. </p><p>La importància de la distribució normal en les ciències naturals i del comportament rau en el <a href="/wiki/Teorema_central_del_l%C3%ADmit" class="mw-redirect" title="Teorema central del límit">teorema central del límit</a>. Aquest teorema estableix que la suma d'un elevat nombre d'efectes independents segueix (aproximadament) una distribució normal. D'aquesta manera, és útil en processos en els quals hi ha errors de mesura que es deuen a un elevat nombre de factors, tots ells contribuint una petita porció a l'error total. En la teoria de probabilitat i d'inferència estadística, el <a href="/wiki/Teorema_central_del_l%C3%ADmit" class="mw-redirect" title="Teorema central del límit">teorema central del límit</a> garanteix que un llarg nombre d'estadístics segueixen la distribució normal, si més no aproximadament. Per exemple, la mitjana mostral o els estimadors màxim versemblants segueixen aproximadament una distribució normal sota certes condicions matemàtiques que són força generals.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Funció_de_densitat_de_probabilitat"><span id="Funci.C3.B3_de_densitat_de_probabilitat"></span>Funció de densitat de probabilitat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=1" title="Modifica la secció: Funció de densitat de probabilitat"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <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;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}={\frac {1}{\sigma }}\,\phi \left({\frac {x-\mu }{\sigma }}\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>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>σ</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}={\frac {1}{\sigma }}\,\phi \left({\frac {x-\mu }{\sigma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac4bdeecb12c453c71806103e1039cafb123308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.427ex; height:7.176ex;" alt="{\displaystyle f(x;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}={\frac {1}{\sigma }}\,\phi \left({\frac {x-\mu }{\sigma }}\right)}"></span></dd></dl> <p>on σ és la <a href="/wiki/Desviacio_est%C3%A0ndard" class="mw-redirect" title="Desviacio estàndard">desviacio estàndard</a>, μ és l'<a href="/wiki/Esperan%C3%A7a_matem%C3%A0tica" title="Esperança matemàtica">esperança matemàtica</a>, i </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 \phi (x)={\frac {1}{\sqrt {2\pi \,}}}\,e^{-{\frac {1}{2}}x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi \,}}}\,e^{-{\frac {1}{2}}x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3810bbf5e43f1d1b570604c9e3de8687241d733d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.533ex; height:6.176ex;" alt="{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi \,}}}\,e^{-{\frac {1}{2}}x^{2}}}"></span></dd></dl> <p>és la funció de densitat de probabilitat de la distribució normal estàndard, és a dir, la distribució normal amb μ = 0 i σ = 1. Per comprovar que la <a href="/wiki/Integral" class="mw-redirect" title="Integral">integral</a> de <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 \varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4046f1f2de7df04bde418ba2bc4d3898ac2385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.659ex; height:2.843ex;" alt="{\displaystyle \varphi (x)}"></span> sobre la recta real és igual a 1 vegeu la <a href="/wiki/Integral_de_Gau%C3%9F" title="Integral de Gauß">integral de Gauß</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Funció_de_distribució"><span id="Funci.C3.B3_de_distribuci.C3.B3"></span>Funció de distribució</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=2" title="Modifica la secció: Funció de distribució"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La funció de distribució d'una distribució normal <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 {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\mu ,\sigma )=\int _{-\infty }^{x}f(t;\mu ,\sigma )\,dt={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-(t-\mu )^{2}/(2\sigma ^{2})}\,dt,\quad x\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\sigma )=\int _{-\infty }^{x}f(t;\mu ,\sigma )\,dt={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-(t-\mu )^{2}/(2\sigma ^{2})}\,dt,\quad x\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e5fa019868da786e4f1d5ea25c6c67f817912c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:68.254ex; height:6.343ex;" alt="{\displaystyle F(x;\mu ,\sigma )=\int _{-\infty }^{x}f(t;\mu ,\sigma )\,dt={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-(t-\mu )^{2}/(2\sigma ^{2})}\,dt,\quad x\in \mathbb {R} .}"></span>Per a una distribució normal estàndard <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 {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eeb356405c0b33b680b5caa425ada4e9f53e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(0,1)}"></span> s'acostuma a utilitzar la notació <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 \Phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Phi (x)}"></span> per designar la seva funció de distribució. Concretament, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)=F(x;0,1)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt,\quad x\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x)=F(x;0,1)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt,\quad x\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670b7f938a76b594afb3c744d379b8a67ceaf362" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:48.732ex; height:6.343ex;" alt="{\displaystyle \Phi (x)=F(x;0,1)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt,\quad x\in \mathbb {R} .}"></span>Cal notar que <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\mu ,\sigma )=\Phi {\Big (}{\frac {x-\mu }{\sigma }}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\sigma )=\Phi {\Big (}{\frac {x-\mu }{\sigma }}{\Big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4f192a45d18c61bfb8cc79b8a1748b8b8d4ead" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.286ex; height:5.176ex;" alt="{\displaystyle F(x;\mu ,\sigma )=\Phi {\Big (}{\frac {x-\mu }{\sigma }}{\Big )}.}"></span> </p><p>(Vegeu mes avall l'apartat sobre estandardització de variables normals).<sup id="cite_ref-FOOTNOTEBogaert2020122_5-0" class="reference"><a href="#cite_note-FOOTNOTEBogaert2020122-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECramer201350_6-0" class="reference"><a href="#cite_note-FOOTNOTECramer201350-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Es important remarcar que la funció de distribució no és pot expressar en termes de <a href="/wiki/Funci%C3%B3_elemental" title="Funció elemental">funcions elementals</a> (polinomis, exponencials, funcions trigonomètriques,..) Vegeu un comentari sobre la demostració d'aquesta propietat a l'article.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Per aquest motiu, de cara a la utilització pràctica de les distribucions normals i els càlculs numèrics corresponents, les aproximacions a la funció de distribució són molt importants i s'han utilitzat tècniques d'<a href="/wiki/Integraci%C3%B3_num%C3%A8rica" title="Integració numèrica">integració numèrica</a>, <a href="/wiki/S%C3%A8ries_de_Taylor" class="mw-redirect" title="Sèries de Taylor">sèries de Taylor</a>, <a href="/w/index.php?title=S%C3%A8ries_asimpt%C3%B2tiques&action=edit&redlink=1" class="new" title="Sèries asimptòtiques (encara no existeix)">sèries asimptòtiques</a> o fraccions contínues. Vegeu Patel and Read per una revisió d'aquestes aproximacions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Funcions_generadores">Funcions generadores</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=3" title="Modifica la secció: Funcions generadores"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Funció_generadora_de_moments"><span id="Funci.C3.B3_generadora_de_moments"></span>Funció generadora de moments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=4" title="Modifica la secció: Funció generadora de moments"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Funci%C3%B3_generadora_de_moments" class="mw-redirect" title="Funció generadora de moments">funció generadora de moments</a> es defineix com a l'<a href="/wiki/Esperan%C3%A7a_matem%C3%A0tica" title="Esperança matemàtica">esperança matemàtica</a> de exp(<i>tX</i>). Per la distribució normal la funció generadora de moments és:<sup id="cite_ref-FOOTNOTECramer201350_6-1" class="reference"><a href="#cite_note-FOOTNOTECramer201350-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}(t)=\mathrm {E} \left[\exp {(tX)}\right]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp {\left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)}\exp {(tx)}\,dx=\exp {\left(\mu t+{\frac {\sigma ^{2}t^{2}}{2}}\right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\mathrm {E} \left[\exp {(tX)}\right]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp {\left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)}\exp {(tx)}\,dx=\exp {\left(\mu t+{\frac {\sigma ^{2}t^{2}}{2}}\right)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402b21160ecb721d43b22c70394f7adf3450d04a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.622ex; height:7.509ex;" alt="{\displaystyle M_{X}(t)=\mathrm {E} \left[\exp {(tX)}\right]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp {\left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)}\exp {(tx)}\,dx=\exp {\left(\mu t+{\frac {\sigma ^{2}t^{2}}{2}}\right)}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Funció_característica"><span id="Funci.C3.B3_caracter.C3.ADstica"></span>Funció característica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=5" title="Modifica la secció: Funció característica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Funci%C3%B3_caracter%C3%ADstica_(teoria_de_la_probabilitat)" title="Funció característica (teoria de la probabilitat)">funció característica</a> es defineix com a l'<a href="/wiki/Esperan%C3%A7a_matem%C3%A0tica" title="Esperança matemàtica">esperança matemàtica</a> de exp(<i>itX</i>), on i és el nombre imaginari, i <i>t</i> és un nombre real. Per la distribució normal la funció característica és:<sup id="cite_ref-FOOTNOTECramer201351_9-0" class="reference"><a href="#cite_note-FOOTNOTECramer201351-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBogaert2020123_10-0" class="reference"><a href="#cite_note-FOOTNOTEBogaert2020123-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)=E{\big [}e^{itX}]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\exp(itx)\,dx=\exp \left(i\mu t-{\frac {\sigma ^{2}t^{2}}{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>μ</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)=E{\big [}e^{itX}]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\exp(itx)\,dx=\exp \left(i\mu t-{\frac {\sigma ^{2}t^{2}}{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182562a184ad4c5089f94b2fb3d4349fb8daa7c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:82.768ex; height:7.509ex;" alt="{\displaystyle \varphi _{X}(t)=E{\big [}e^{itX}]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\exp(itx)\,dx=\exp \left(i\mu t-{\frac {\sigma ^{2}t^{2}}{2}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Propietats">Propietats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=6" title="Modifica la secció: Propietats"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algunes propietats: </p> <ol><li>Si <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 \,{\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace" /> <mo>∼</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\sim \,{\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b8cfbadc3eef86878ebcd66b6701fd7e2b45c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.757ex; height:3.176ex;" alt="{\displaystyle X\,\sim \,{\mathcal {N}}(\mu ,\sigma ^{2})}"></span> i <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> i <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> són nombres reals, aleshores <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 aX+b\,\sim \,N(a\mu +b,(a\sigma )^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>X</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mo>∼</mo> <mspace width="thinmathspace" /> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>μ</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>σ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aX+b\,\sim \,N(a\mu +b,(a\sigma )^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c94274681f82346e8a7271e527dd854ed8c16cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.72ex; height:3.176ex;" alt="{\displaystyle aX+b\,\sim \,N(a\mu +b,(a\sigma )^{2})}"></span> (veure <a href="/wiki/Esperan%C3%A7a_matem%C3%A0tica" title="Esperança matemàtica">esperança</a> i <a href="/wiki/Vari%C3%A0ncia" title="Variància">variància</a>).<sup id="cite_ref-:1_11-0" class="reference"><a href="#cite_note-:1-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li>Si <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 {\mathcal {N}}(\mu _{X},\sigma _{X}^{2})}"> <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"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu _{X},\sigma _{X}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4229a241196a3ba2c85552998bc880deb4f18742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.19ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(\mu _{X},\sigma _{X}^{2})}"></span> i <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 {\mathcal {N}}(\mu _{Y},\sigma _{Y}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\mathcal {N}}(\mu _{Y},\sigma _{Y}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a3967fcb7485c0265da89abcae6e075b4ed1d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.691ex; height:3.176ex;" alt="{\displaystyle Y\sim {\mathcal {N}}(\mu _{Y},\sigma _{Y}^{2})}"></span> són <a href="/wiki/Variable_aleat%C3%B2ria" title="Variable aleatòria">variables aleatòries</a> normals <a href="/wiki/Independ%C3%A8ncia_estad%C3%ADstica" title="Independència estadística">independents</a>, aleshores:<sup id="cite_ref-:12_12-0" class="reference"><a href="#cite_note-:12-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_11-1" class="reference"><a href="#cite_note-:1-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <ul><li>La seva suma segueix la distribució normal amb <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 U=X+Y\sim {\mathcal {N}}(\mu _{X}+\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=X+Y\sim {\mathcal {N}}(\mu _{X}+\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91c6adbba85770f28c652dfe73c655d808892b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.067ex; height:3.176ex;" alt="{\displaystyle U=X+Y\sim {\mathcal {N}}(\mu _{X}+\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}"></span>.</li> <li>La seva diferència segueix una distribució normal amb <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 V=X-Y\sim {\mathcal {N}}(\mu _{X}-\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>X</mi> <mo>−</mo> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=X-Y\sim {\mathcal {N}}(\mu _{X}-\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49a4b4c693b9e42deeaa23aa9125343f09d0856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.072ex; height:3.176ex;" alt="{\displaystyle V=X-Y\sim {\mathcal {N}}(\mu _{X}-\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2})}"></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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> i <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> són independents si i només si <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 \sigma _{X}=\sigma _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{X}=\sigma _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3c0863fd939cef6880102c1033b3ddb38698bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.872ex; height:2.009ex;" alt="{\displaystyle \sigma _{X}=\sigma _{Y}}"></span>.</li> <li>La <a href="/wiki/Diverg%C3%A8ncia_de_Kullback-Leibler" class="mw-redirect" title="Divergència de Kullback-Leibler">divergència de Kullback-Leibler</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <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 D_{\rm {KL}}(X\|Y)={1 \over 2}\left(\log \left({\sigma _{Y}^{2} \over \sigma _{X}^{2}}\right)+{\frac {\sigma _{X}^{2}}{\sigma _{Y}^{2}}}+{\frac {\left(\mu _{Y}-\mu _{X}\right)^{2}}{\sigma _{Y}^{2}}}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\rm {KL}}(X\|Y)={1 \over 2}\left(\log \left({\sigma _{Y}^{2} \over \sigma _{X}^{2}}\right)+{\frac {\sigma _{X}^{2}}{\sigma _{Y}^{2}}}+{\frac {\left(\mu _{Y}-\mu _{X}\right)^{2}}{\sigma _{Y}^{2}}}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3b2d975238bbf58ffed917dda72bad64e3f669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.777ex; height:7.509ex;" alt="{\displaystyle D_{\rm {KL}}(X\|Y)={1 \over 2}\left(\log \left({\sigma _{Y}^{2} \over \sigma _{X}^{2}}\right)+{\frac {\sigma _{X}^{2}}{\sigma _{Y}^{2}}}+{\frac {\left(\mu _{Y}-\mu _{X}\right)^{2}}{\sigma _{Y}^{2}}}-1\right).}"></span></li></ul></li> <li>Si <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 {\mathcal {N}}(0,\sigma _{X}^{2})}"> <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"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(0,\sigma _{X}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aae71b65b4519ab05d75bdf766ef732cd3db167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.319ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(0,\sigma _{X}^{2})}"></span> i <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 {\mathcal {N}}(0,\sigma _{Y}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\mathcal {N}}(0,\sigma _{Y}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d27a86fe8db14f34dff9affa7d647536435310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.966ex; height:3.176ex;" alt="{\displaystyle Y\sim {\mathcal {N}}(0,\sigma _{Y}^{2})}"></span> són variables aleatòries normals independents, aleshores:<sup id="cite_ref-:12_12-1" class="reference"><a href="#cite_note-:12-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <ul><li>El seu producte <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 XY}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XY}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95039ff8092deac9cb8fc7a9e0385a55128b4de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.753ex; height:2.176ex;" alt="{\displaystyle XY}"></span> segueix una distribució and funció de probabilitat de densitat <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> donada per <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 p(z)={\frac {1}{\pi \,\sigma _{X}\,\sigma _{Y}}}\;K_{0}\left({\frac {|z|}{\sigma _{X}\,\sigma _{Y}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(z)={\frac {1}{\pi \,\sigma _{X}\,\sigma _{Y}}}\;K_{0}\left({\frac {|z|}{\sigma _{X}\,\sigma _{Y}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d732a5bbd733b8089676f1f1132394d5b5a98454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:31.481ex; height:6.343ex;" alt="{\displaystyle p(z)={\frac {1}{\pi \,\sigma _{X}\,\sigma _{Y}}}\;K_{0}\left({\frac {|z|}{\sigma _{X}\,\sigma _{Y}}}\right),}"></span> on <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0af6cafb690d3dbb0f3f30a032631338dc476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{0}}"></span> és una <a href="/wiki/Funci%C3%B3_de_Bessel#funcions_Bessel_modificades" title="Funció de Bessel">funció de Bessel modificada de segon tipus</a>.</dd></dl></li> <li>El seu qüocient segueix una <a href="/wiki/Distribuci%C3%B3_de_Cauchy" title="Distribució de Cauchy">distribució de Cauchy</a> amb <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/Y\,\sim \,\mathrm {Cauchy} (0,\,\sigma _{X}/\sigma _{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Y</mi> <mspace width="thinmathspace" /> <mo>∼</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">y</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/Y\,\sim \,\mathrm {Cauchy} (0,\,\sigma _{X}/\sigma _{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed1cf72bf92bc584151adde754425c3aa605c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.802ex; height:2.843ex;" alt="{\displaystyle X/Y\,\sim \,\mathrm {Cauchy} (0,\,\sigma _{X}/\sigma _{Y})}"></span>.</li></ul></li> <li>Si <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_{1},\dots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ed92ce88f900210607bbb8f4d66e14d52d7a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n}}"></span> són variables aleatòries independents idènticament distribuïdes amb distribució normal estàndard, aleshores <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_{1}^{2}+\cdots +X_{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcb27043479e856fda9e71e61f5913db0d3e4b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.598ex; height:3.176ex;" alt="{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}}"></span> segueix una <a href="/wiki/Distribuci%C3%B3_khi_quadrat" title="Distribució khi quadrat">distribució khi quadrat</a> amb <i>n</i> graus de llibertat.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading3"><h3 id="Estandardització_de_variables_aleatòries_normals"><span id="Estandarditzaci.C3.B3_de_variables_aleat.C3.B2ries_normals"></span>Estandardització de variables aleatòries normals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=7" title="Modifica la secció: Estandardització de variables aleatòries normals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Com a conseqüència de la propietat 1, és possible relacionar totes les variables aleatòries normals amb la distribució normal estàndard; aquest procediment s'anomena estandardització d'una variable normal. </p><p>Si <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 {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dffd28dca2e7d8d6862b2dd19068ea97a99bd42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.564ex; height:3.176ex;" alt="{\displaystyle \ X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"></span>, aleshores </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 Z={\frac {X-\mu }{\sigma }}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={\frac {X-\mu }{\sigma }}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87736c83415dcd9fdcff9b416a246e72bf83fff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.108ex; width:11.558ex; height:5.343ex;" alt="{\displaystyle Z={\frac {X-\mu }{\sigma }}\!}"></span></dd></dl> <p>és una variable aleatòria normal estàndard: <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 \ Z\sim {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Z</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Z\sim {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3f96dcc3d8210710cd6e9d4d5b186de8c0f302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.803ex; height:3.009ex;" alt="{\displaystyle \ Z\sim {\mathcal {N}}(0,1)}"></span>. Una conseqüència important és que la funció de distribució de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96dd62d4cca19aa212ae1216891f4388ca4be24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.561ex; height:2.176ex;" alt="{\displaystyle \ X}"></span> és : </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;\mu ,\sigma )=\Pr(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\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>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\sigma )=\Pr(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90bad98a438ca8d630ab09eb5a712ff5476bef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.516ex; height:6.176ex;" alt="{\displaystyle F(x;\mu ,\sigma )=\Pr(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\right),}"></span></dd></dl> <p>on <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> és la funció de distribució normal estàndard: per a tot <a href="/wiki/Nombre_real" title="Nombre real">real</a> <i>t</i>, </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 \ \Phi (t)=\int _{-\infty }^{\,t}\phi (u)\,du=\int _{-\infty }^{\,t}{\frac {1}{\sqrt {2\,\pi }}}\,\mathrm {e} ^{-{\frac {u^{2}}{2}}}\,du={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {t}{\sqrt {2}}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mspace width="thinmathspace" /> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi (t)=\int _{-\infty }^{\,t}\phi (u)\,du=\int _{-\infty }^{\,t}{\frac {1}{\sqrt {2\,\pi }}}\,\mathrm {e} ^{-{\frac {u^{2}}{2}}}\,du={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {t}{\sqrt {2}}}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64ddf3bc348fea792fbb5eaa4bcb0297f8013e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:66.099ex; height:6.676ex;" alt="{\displaystyle \ \Phi (t)=\int _{-\infty }^{\,t}\phi (u)\,du=\int _{-\infty }^{\,t}{\frac {1}{\sqrt {2\,\pi }}}\,\mathrm {e} ^{-{\frac {u^{2}}{2}}}\,du={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {t}{\sqrt {2}}}\right)\right).}"></span></dd></dl> <p>D'altra banda, si <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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> és una variable aleatòria normal estàndard, <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 \ Z\sim {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Z</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Z\sim {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3f96dcc3d8210710cd6e9d4d5b186de8c0f302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.803ex; height:3.009ex;" alt="{\displaystyle \ Z\sim {\mathcal {N}}(0,1)}"></span>, aleshores </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 X=\sigma \,Z+\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>σ</mi> <mspace width="thinmathspace" /> <mi>Z</mi> <mo>+</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\sigma \,Z+\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd705daa33c9c05f1f1c4ca426ed7883fafd98eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.718ex; height:2.676ex;" alt="{\displaystyle X=\sigma \,Z+\mu }"></span></dd></dl> <p>és una variable aleatòria normal amb esperança <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> i variància <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 \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span>. </p><p>La funció de distribució normal estàndard <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> ha estat tabulada, i les altres funcions de distribució normals en són simples transformacions, tal com hem explicat anteriorment. Per tant, un pot emprar valors tabulats de la funció de distribució normal estàndard per a trobar el valor de la funció de distribució de qualsevol altra distribució normal. </p> <div class="mw-heading mw-heading2"><h2 id="Moments">Moments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=8" title="Modifica la secció: Moments"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alguns dels primers <a href="/w/index.php?title=Moments_(matem%C3%A0tiques)&action=edit&redlink=1" class="new" title="Moments (matemàtiques) (encara no existeix)">moments</a> de la distribució normal són: </p> <table class="wikitable"> <tbody><tr bgcolor="#CCCCCC"> <th>Número</th> <th>Moment</th> <th>Moment central</th> <th>Cumulant </th></tr> <tr> <td>0</td> <td>1</td> <td>1</td> <td> </td></tr> <tr> <td>1</td> <td><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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span></td> <td>0</td> <td><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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> </td></tr> <tr> <td>2</td> <td><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 \mu ^{2}+\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{2}+\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49112e161897039a88a162ff2ad10ea4a8c9e8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:3.176ex;" alt="{\displaystyle \mu ^{2}+\sigma ^{2}}"></span></td> <td><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 \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span></td> <td><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 \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> </td></tr> <tr> <td>3</td> <td><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 \mu ^{3}+3\mu \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>μ</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{3}+3\mu \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036df43f1879ad3e1e36a3394d3428475217891c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.245ex; height:3.176ex;" alt="{\displaystyle \mu ^{3}+3\mu \sigma ^{2}}"></span></td> <td>0</td> <td>0 </td></tr> <tr> <td>4</td> <td><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 \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0c44283f14f944c968ea3c5c9fd20cd905a7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.687ex; height:3.176ex;" alt="{\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}"></span></td> <td><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 3\sigma ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\sigma ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60711695348caa7b632267f93d69be5c1e70993e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.547ex; height:2.676ex;" alt="{\displaystyle 3\sigma ^{4}}"></span></td> <td>0 </td></tr></tbody></table> <p>Tots els <a href="/w/index.php?title=Cumulant&action=edit&redlink=1" class="new" title="Cumulant (encara no existeix)">cumulants</a> de la distribució normal a partir del segon són zero. </p> <div class="mw-heading mw-heading3"><h3 id="Moments_d'una_variable_normal_centrada"><span id="Moments_d.27una_variable_normal_centrada"></span>Moments d'una variable normal centrada</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=9" title="Modifica la secció: Moments d'una variable normal centrada"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per a les variables aleatòries normals centrades tenim la següent fórmula per als moments de qualsevol ordre. Si <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 Z\sim {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\sim {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8886c9cf1d7deacceafa86817f44ba217446b0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.222ex; height:3.009ex;" alt="{\displaystyle Z\sim {\mathcal {N}}(0,1)}"></span> aleshores <sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\{\dfrac {(2n)!}{2^{n}\,n!}},&{\text{si}}\ k=2n.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>és senar</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\{\dfrac {(2n)!}{2^{n}\,n!}},&{\text{si}}\ k=2n.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9324213830733b3ac182a1c755f2eee47b329455" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:33.412ex; height:12.176ex;" alt="{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\{\dfrac {(2n)!}{2^{n}\,n!}},&{\text{si}}\ k=2n.\end{cases}}}"></span> Notem que <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(2n)!}{2^{n}\,n!}}={\frac {(2n-1)!}{2^{n-1}\,(n-1)!}}=(2n-1)(2n-3)\cdots 1=(2n-1)!!=(k-1)!!,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(2n)!}{2^{n}\,n!}}={\frac {(2n-1)!}{2^{n-1}\,(n-1)!}}=(2n-1)(2n-3)\cdots 1=(2n-1)!!=(k-1)!!,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ce80a095a32b83181705e076b81b517f9fe62c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:71.827ex; height:6.676ex;" alt="{\displaystyle {\frac {(2n)!}{2^{n}\,n!}}={\frac {(2n-1)!}{2^{n-1}\,(n-1)!}}=(2n-1)(2n-3)\cdots 1=(2n-1)!!=(k-1)!!,}"></span> on <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 m!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67a953debb08bc72c09d7d66c41a8c4e4aaad4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.334ex; height:2.176ex;" alt="{\displaystyle m!!}"></span> denota el <a href="/wiki/Doble_factorial" title="Doble factorial">doble factorial</a> de <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>. Així, de forma més compacta podem escriure <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\(k-1)!!,&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>és senar</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>és parell</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\(k-1)!!,&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557d3825378b7f5c2a2ac4f3b4b9f36dfd223a2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.045ex; margin-bottom: -0.293ex; width:36.257ex; height:9.843ex;" alt="{\displaystyle E[Z^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\\\(k-1)!!,&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}"></span> </p><p><br /> </p><p>Alternativament, usant la relació del doble factorial amb la <a href="/wiki/Funci%C3%B3_gamma" title="Funció gamma">funció gamma</a>, per 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> parell, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)!!={\frac {2^{k/2}}{\sqrt {\pi }}}\,\Gamma {\Big (}{\frac {k+1}{2}}{\Big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k-1)!!={\frac {2^{k/2}}{\sqrt {\pi }}}\,\Gamma {\Big (}{\frac {k+1}{2}}{\Big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4417473d4c142628c44e22f6b9c9610a5b0e43f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.459ex; height:6.843ex;" alt="{\displaystyle (k-1)!!={\frac {2^{k/2}}{\sqrt {\pi }}}\,\Gamma {\Big (}{\frac {k+1}{2}}{\Big )},}"></span>on <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 (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec077ba0bdbf87c0d66173bc4d98598fe582ac37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle \Gamma (x)}"></span> és la funció gamma. </p><p>De la fórmula pels moments de <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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> és dedueix que si <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 {\mathcal {N}}(0,\sigma ^{2})}"> <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"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(0,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec48dfbea884c8c63858726b151d52037383037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.744ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(0,\sigma ^{2})}"></span>, aleshores </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\(k-1)!!\,\sigma ^{k},&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}"> <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"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>és senar</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si</mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>és parell</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\(k-1)!!\,\sigma ^{k},&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2ae2b1474048b2a37eddbb18fac9586a2fa2e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.159ex; height:7.509ex;" alt="{\displaystyle E[X^{k}]={\begin{cases}0,&{\text{si}}\ k\ {\text{és senar}},\\(k-1)!!\,\sigma ^{k},&{\text{si}}\ k\ {\text{és parell}}.\end{cases}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Cas_general">Cas general</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=10" title="Modifica la secció: Cas general"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sigui <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 {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dffd28dca2e7d8d6862b2dd19068ea97a99bd42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.564ex; height:3.176ex;" alt="{\displaystyle \ X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"></span>, i designem per <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 m_{k}=E[X^{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <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 m_{k}=E[X^{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d09da10ffa60f0489d04c8a901e5dfd25398b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.382ex; height:3.176ex;" alt="{\displaystyle m_{k}=E[X^{k}]}"></span> el moment d'ordre <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> . Aleshores <sup id="cite_ref-:0_16-0" class="reference"><a href="#cite_note-:0-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}=\sigma ^{k}k!\sum _{j=0}^{[k/2]}{\frac {(\mu /\sigma )^{k-2j}}{2^{j}j!(k-2j)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>k</mi> <mo>!</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>σ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> <mi>j</mi> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mi>j</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k}=\sigma ^{k}k!\sum _{j=0}^{[k/2]}{\frac {(\mu /\sigma )^{k-2j}}{2^{j}j!(k-2j)!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48027ee12021cc0ecb8df2eb60cca928bc025905" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.483ex; height:8.009ex;" alt="{\displaystyle m_{k}=\sigma ^{k}k!\sum _{j=0}^{[k/2]}{\frac {(\mu /\sigma )^{k-2j}}{2^{j}j!(k-2j)!}},}"></span>on <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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>r</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [r]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a2bcc2aac5f01558c1fdd11d9445b1a1ab2294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.342ex; height:2.843ex;" alt="{\displaystyle [r]}"></span> designa la part entera del nombre <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. </p><p><b>Recurrència pels moments d'una variable normal</b> </p><p><br /> </p><p>Amb les notacions anteriors tenim <sup id="cite_ref-:0_16-1" class="reference"><a href="#cite_note-:0-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k+1}=\mu \,m_{k}+k\sigma ^{2}m_{k-1},\quad k\geq 1.\qquad \qquad (*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>k</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>≥</mo> <mn>1.</mn> <mspace width="2em" /> <mspace width="2em" /> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k+1}=\mu \,m_{k}+k\sigma ^{2}m_{k-1},\quad k\geq 1.\qquad \qquad (*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683095a5d18fc63f2ec47d3663ebf2706676ac10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.65ex; height:3.176ex;" alt="{\displaystyle m_{k+1}=\mu \,m_{k}+k\sigma ^{2}m_{k-1},\quad k\geq 1.\qquad \qquad (*)}"></span> </p><p><b>Expressió compacta dels moments</b> </p><p><br /> </p><p>Suposem que <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 \sigma =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f759e9b01b4c117d116da9f6d0e635b2247ee502" 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 \sigma =1}"></span>. Aleshores <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k}=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f9c5881bb04cba68cd42533cafd44a23b5fab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.248ex; height:6.843ex;" alt="{\displaystyle m_{k}=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}"></span>on <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 d^{n}f(x)/dx^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{n}f(x)/dx^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9844278eaac9883b89fbbdde9a2eae9d8ed273a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.781ex; height:2.843ex;" alt="{\displaystyle d^{n}f(x)/dx^{n}}"></span> designa la derivada d'ordre <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-èssim de la funció <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, amb el conveni <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 d^{0}f(x)/dx^{0}=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{0}f(x)/dx^{0}=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266bf79287a0201e31bc60c4a430e7f139519fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.968ex; height:3.176ex;" alt="{\displaystyle d^{0}f(x)/dx^{0}=f(x)}"></span> . Aquesta fórmula és demostra mitjançant la <a href="/wiki/Regla_de_Leibniz_(regla_del_producte_generalitzada)" title="Regla de Leibniz (regla del producte generalitzada)">regla de Leibniz</a> per provar que la funció <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k}(\mu )=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{k}(\mu )=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda9adb6e692c92778fc1f2cc652a0eb40c7b354" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.911ex; height:6.843ex;" alt="{\displaystyle P_{k}(\mu )=e^{-\mu ^{2}/2}\,{\frac {d^{k}e^{\mu ^{2}/2}}{d\mu ^{k}}},}"></span>compleix la recurrència (*), amb <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}(\mu )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}(\mu )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42af657eed7f369a9d81adf7d795485c58e97bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.018ex; height:2.843ex;" alt="{\displaystyle P_{0}(\mu )=1}"></span> i <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_{1}(\mu )=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}(\mu )=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d299d90fd0b959bef4d2b537d4bc1c90f371de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle P_{1}(\mu )=\mu }"></span>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=11" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFHavil2003"><span style="font-variant: small-caps;">Havil</span>, Julian. <a rel="nofollow" class="external text" href="https://www.google.cat/books/edition/Gamma/Z3YAwQEACAAJ?hl=ca"><i>Gamma: exploring Euler's constant</i></a>.  Princeton, N.J.: Princeton Univ. Press, 2003. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-691-09983-5" title="Especial:Fonts bibliogràfiques/978-0-691-09983-5">ISBN 978-0-691-09983-5</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gamma%3A+exploring+Euler%27s+constant&rft.aulast=Havil&rft.aufirst=Julian&rft.date=2003&rft.pub=Princeton+Univ.+Press&rft.place=Princeton%2C+N.J.&rft.isbn=978-0-691-09983-5&rft_id=https%3A%2F%2Fwww.google.cat%2Fbooks%2Fedition%2FGamma%2FZ3YAwQEACAAJ%3Fhl%3Dca"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> «<a rel="nofollow" class="external text" href="https://www.uv.es/webgid/DescriptivaV/31_normal.html">3.1. Normal</a>». A: <a rel="nofollow" class="external text" href="https://www.uv.es/webgid/DescriptivaV/index.html"><i>Estadística I</i></a>.  Universitat de València [Consulta: 7 febrer 2024].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Estad%C3%ADstica+I&rft.atitle=3.1.+Normal&rft.pub=Universitat+de+Val%C3%A8ncia&rft_id=https%3A%2F%2Fwww.uv.es%2Fwebgid%2FDescriptivaV%2Findex.html"><span style="display: none;"> </span></span><a rel="nofollow" class="external text" href="https://web.archive.org/web/20240207094201/https://www.uv.es/webgid/DescriptivaV/31_normal.html">Arxivat</a> 2024-02-07 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFDodge2004"><span style="font-variant: small-caps;">Dodge</span>, Yadolah. <a rel="nofollow" class="external text" href="https://books.google.cat/books/about/Statistique.html?id=PyDEP3M-T4cC&redir_esc=y"><i>Statistique: Dictionnaire encyclopédique</i></a> (en francès).  Springer Science & Business Media, 2004, p. 309. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-2-287-21325-0" title="Especial:Fonts bibliogràfiques/978-2-287-21325-0">ISBN 978-2-287-21325-0</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistique%3A+Dictionnaire+encyclop%C3%A9dique&rft.aulast=Dodge&rft.aufirst=Yadolah&rft.date=2004&rft.pub=Springer+Science+%26+Business+Media&rft.pages=309&rft.isbn=978-2-287-21325-0&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%2Fabout%2FStatistique.html%3Fid%3DPyDEP3M-T4cC%26redir_esc%3Dy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFLifshits1995"><span style="font-variant: small-caps;">Lifshits</span>, M. A.. <a rel="nofollow" class="external text" href="https://books.google.cat/books/about/Gaussian_Random_Functions.html?id=vNh6_n-K9_4C&redir_esc=y"><i>Gaussian Random Functions</i></a> (en anglès).  Springer Science & Business Media, 1995-02-28, p. 2. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-7923-3385-2" title="Especial:Fonts bibliogràfiques/978-0-7923-3385-2">ISBN 978-0-7923-3385-2</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gaussian+Random+Functions&rft.aulast=Lifshits&rft.aufirst=M.+A.&rft.date=1995-02-28&rft.pub=Springer+Science+%26+Business+Media&rft.pages=2&rft.isbn=978-0-7923-3385-2&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%2Fabout%2FGaussian_Random_Functions.html%3Fid%3DvNh6_n-K9_4C%26redir_esc%3Dy"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-FOOTNOTEBogaert2020122-5"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTEBogaert2020122_5-0">↑</a></span> <span class="reference-text"><a href="#CITEREFBogaert2020">Bogaert, 2020</a>, p. 122.</span> </li> <li id="cite_note-FOOTNOTECramer201350-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-FOOTNOTECramer201350_6-0">6,0</a></sup> <sup><a href="#cite_ref-FOOTNOTECramer201350_6-1">6,1</a></sup></span> <span class="reference-text"><a href="#CITEREFCramer2013">Cramer, 2013</a>, p. 50.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFGasullUtzet2014"><span style="font-variant: small-caps;">Gasull</span>, Armengol; <span style="font-variant: small-caps;">Utzet</span>, Frederic «<a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S0022247X14004764">Approximating Mills ratio</a>» (en anglès). <i>Journal of Mathematical Analysis and Applications</i>, 420, 2, 12-2014, pàg. 1832–1853, Remark 4. <a href="/wiki/DOI" title="DOI">DOI</a>: <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.jmaa.2014.05.034">10.1016/j.jmaa.2014.05.034</a>.</span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFPatelRead1996"><span style="font-variant: small-caps;">Patel</span>, Jagdish K.; <span style="font-variant: small-caps;">Read</span>, Campbell B. <a rel="nofollow" class="external text" href="https://www.google.cat/books/edition/Handbook_of_the_Normal_Distribution_Seco/zoVLF0VF9UYC?hl=ca&gbpv=0"><i>Handbook of the normal distribution</i></a>. 2nd ed., rev. and expanded.  New York Basel Hong Kong: M. Dekker, 1996. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-8247-9342-5" title="Especial:Fonts bibliogràfiques/978-0-8247-9342-5">ISBN 978-0-8247-9342-5</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+the+normal+distribution&rft.aulast=Patel&rft.aufirst=Jagdish+K.&rft.date=1996&rft.edition=2nd+ed.%2C+rev.+and+expanded&rft.pub=M.+Dekker&rft.place=New+York+Basel+Hong+Kong&rft.isbn=978-0-8247-9342-5&rft_id=https%3A%2F%2Fwww.google.cat%2Fbooks%2Fedition%2FHandbook_of_the_Normal_Distribution_Seco%2FzoVLF0VF9UYC%3Fhl%3Dca%26gbpv%3D0"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-FOOTNOTECramer201351-9"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTECramer201351_9-0">↑</a></span> <span class="reference-text"><a href="#CITEREFCramer2013">Cramer, 2013</a>, p. 51.</span> </li> <li id="cite_note-FOOTNOTEBogaert2020123-10"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTEBogaert2020123_10-0">↑</a></span> <span class="reference-text"><a href="#CITEREFBogaert2020">Bogaert, 2020</a>, p. 123.</span> </li> <li id="cite_note-:1-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:1_11-0">11,0</a></sup> <sup><a href="#cite_ref-:1_11-1">11,1</a></sup></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFRoss2007"><span style="font-variant: small-caps;">Ross</span>, Sheldon M. <i>Initiation aux Probabilités</i>.  Presses Polytechniques et Universitaires Romandes, 2007, p. 235.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Initiation+aux+Probabilit%C3%A9s&rft.aulast=Ross&rft.aufirst=Sheldon+M.&rft.date=2007&rft.pub=Presses+Polytechniques+et+Universitaires+Romandes&rft.pages=235"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-:12-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:12_12-0">12,0</a></sup> <sup><a href="#cite_ref-:12_12-1">12,1</a></sup></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFFuchs1995"><span style="font-variant: small-caps;">Fuchs</span>, Aimé «<a rel="nofollow" class="external text" href="https://web.archive.org/web/20141006084358/http://www-irma.u-strasbg.fr/~foata/fuchs/FuchsNormale.pdf">Plaidoyer pour la Loi Normale</a>» (<style data-mw-deduplicate="TemplateStyles:r33780657">.mw-parser-output .linkformat{position:relative;font-family:sans-serif;font-size:0.85em;font-weight:bold;cursor:default;color:#808080;background-color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .linkformat{background-color:inherit;color:#009400}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .linkformat{background-color:inherit;color:#009400}}</style><span class="linkformat" title="És PDF"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/PDF_icon_bold.svg/14px-PDF_icon_bold.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/PDF_icon_bold.svg/21px-PDF_icon_bold.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/PDF_icon_bold.svg/28px-PDF_icon_bold.svg.png 2x" data-file-width="512" data-file-height="585" /></span></span> PDF</span>) (en francès). <i>Pour la Science</i>, 1995, pàg. 17. Arxivat de l'<a rel="nofollow" class="external text" href="http://www-irma.u-strasbg.fr/~foata/fuchs/FuchsNormale.pdf">original</a> el 2014-10-06 [Consulta: 7 febrer 2024].</span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFAllison2012"><span style="font-variant: small-caps;">Allison</span>, Lloyd. «<a rel="nofollow" class="external text" href="https://web.archive.org/web/20231002202942/http://www.allisons.org/ll/MML/KL/Normal/">Normal, Gaussian</a>» (en anglès), 2012. Arxivat de l'<a rel="nofollow" class="external text" href="http://www.allisons.org/ll/MML/KL/Normal/">original</a> el 2023-10-02. [Consulta: 2 març 2017].</span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFBussabMorettin2010"><span style="font-variant: small-caps;">Bussab</span>, Wilton de O.; <span style="font-variant: small-caps;">Morettin</span>, Pedro A. <a rel="nofollow" class="external text" href="https://www.google.cat/books/edition/ESTAT%C3%8DSTICA_B%C3%81SICA/vDhnDwAAQBAJ?hl=ca&gbpv=0"><i>Estatística Básica</i></a> (en portuguès).  São Paulo: Saraiva, 2010, p. 77. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/9788502207172" title="Especial:Fonts bibliogràfiques/9788502207172">ISBN 9788502207172</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Estat%C3%ADstica+B%C3%A1sica&rft.aulast=Bussab&rft.aufirst=Wilton+de+O.&rft.date=2010&rft.pub=Saraiva&rft.place=S%C3%A3o+Paulo&rft.pages=77&rft.isbn=9788502207172&rft_id=https%3A%2F%2Fwww.google.cat%2Fbooks%2Fedition%2FESTAT%25C3%258DSTICA_B%25C3%2581SICA%2FvDhnDwAAQBAJ%3Fhl%3Dca%26gbpv%3D0"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFNualartSanz1990"><span style="font-variant: small-caps;">Nualart</span>, David; <span style="font-variant: small-caps;">Sanz</span>, Marta. <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/801861669"><i>Curs de probabilitats</i></a>.  Barcelona: PPU, 1990, p. 116. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/84-7665-718-8" title="Especial:Fonts bibliogràfiques/84-7665-718-8">ISBN 84-7665-718-8</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Curs+de+probabilitats&rft.aulast=Nualart&rft.aufirst=David&rft.date=1990&rft.pub=PPU&rft.place=Barcelona&rft.pages=116&rft.isbn=84-7665-718-8&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F801861669"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-:0-16"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_16-0">16,0</a></sup> <sup><a href="#cite_ref-:0_16-1">16,1</a></sup></span> <span class="reference-text"><span class="citation" style="font-style:normal" id="CITEREFWillink2005"><span style="font-variant: small-caps;">Willink</span>, R. «<a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S0167715205001124">Normal moments and Hermite polynomials</a>» (en anglès). <i>Statistics & Probability Letters</i>, 73, 3, 7-2005, pàg. 271–275. <a href="/wiki/DOI" title="DOI">DOI</a>: <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.spl.2005.03.015">10.1016/j.spl.2005.03.015</a>.</span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFOlver2000"><span style="font-variant: small-caps;">Olver</span>, Peter J. <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=sI2bAxgLMXYC&pg=PA318"><i>Applications of Lie Groups to Differential Equations</i></a>.  Springer, 2000, p. 318–319. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/9780387950006" title="Especial:Fonts bibliogràfiques/9780387950006">ISBN 9780387950006</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Lie+Groups+to+Differential+Equations&rft.aulast=Olver&rft.aufirst=Peter+J.&rft.date=2000&rft.pub=Springer&rft.pages=318%E2%80%93319&rft.isbn=9780387950006&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DsI2bAxgLMXYC%26pg%3DPA318"><span style="display: none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliogafia">Bibliogafia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=12" title="Modifica la secció: Bibliogafia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFBogaert2020"><span style="font-variant: small-caps;">Bogaert</span>, Patrick. <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=Jh3RDwAAQBAJ&newbks=0&hl=ca"><i>Probabilités pour scientifiques et ingénieurs: Introduction au calcul des probabilités</i></a> (en francès).  De Boeck Superieur, juliol 2020. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-2-8073-2655-2" title="Especial:Fonts bibliogràfiques/978-2-8073-2655-2">ISBN 978-2-8073-2655-2</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probabilit%C3%A9s+pour+scientifiques+et+ing%C3%A9nieurs%3A+Introduction+au+calcul+des+probabilit%C3%A9s&rft.aulast=Bogaert&rft.aufirst=Patrick&rft.date=2020&rft.pub=De+Boeck+Superieur&rft.isbn=978-2-8073-2655-2&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DJh3RDwAAQBAJ%26newbks%3D0%26hl%3Dca"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFCramer2013"><span style="font-variant: small-caps;">Cramer</span>, Harald. <a rel="nofollow" class="external text" href="https://books.google.cat/books/about/Random_Variables_and_Probability_Distrib.html?id=46wuygEACAAJ&redir_esc=y"><i>Random Variables and Probability Distributions</i></a> (en anglès).  Cambridge University Press, 2013.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Variables+and+Probability+Distributions&rft.aulast=Cramer&rft.aufirst=Harald&rft.date=2013&rft.pub=Cambridge+University+Press&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%2Fabout%2FRandom_Variables_and_Probability_Distrib.html%3Fid%3D46wuygEACAAJ%26redir_esc%3Dy"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distribuci%C3%B3_normal&action=edit&section=13" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Normal_distribution" class="extiw" title="commons:Category:Normal distribution">Distribució normal</a></b></i></div></div> </div> <p><span style="display: none;" class="interProject"><a href="https://ca.wiktionary.org/wiki/distribuci%C3%B3_normal" class="extiw" title="wikt:distribució normal">Viccionari</a></span> </p> <ul><li><a href="/wiki/M%C3%A8tode_dels_m%C3%ADnims_quadrats" title="Mètode dels mínims quadrats">Mètode dels mínims quadrats</a></li></ul> <div role="navigation" class="navbox" aria-labelledby="Distribucions_de_probabilitat" style="padding:3px"><table class="nowraplinks collapsible collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><span typeof="mw:File"><a href="/wiki/Plantilla:Distribucions_de_probabilitat" title="Plantilla:Distribucions de probabilitat"><img alt="Vegeu aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/18px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/27px-Commons-emblem-notice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/36px-Commons-emblem-notice.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div><div id="Distribucions_de_probabilitat" style="font-size:114%;margin:0 4em"><a href="/wiki/Distribuci%C3%B3_de_probabilitat" title="Distribució de probabilitat">Distribucions de probabilitat</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div id="Llista"><a href="/wiki/Llista_de_distribucions_de_probabilitat" title="Llista de distribucions de probabilitat">Llista</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions discretes<br />amb suport finit</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Llei_de_Benford" title="Llei de Benford">Benford</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Bernoulli" title="Distribució de Bernoulli">Bernoulli</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_Beta-binomial&action=edit&redlink=1" class="new" title="Distribució Beta-binomial (encara no existeix)">Beta-binomial</a></li> <li><a href="/wiki/Distribuci%C3%B3_binomial" title="Distribució binomial">Binomial</a></li> <li><a href="/wiki/Distribuci%C3%B3_binomial_de_Poisson" title="Distribució binomial de Poisson">Binomial de Poisson</a></li> <li><a href="/wiki/Distribuci%C3%B3_categ%C3%B2rica" title="Distribució categòrica">Categòrica</a></li> <li><a href="/wiki/Distribuci%C3%B3_hipergeom%C3%A8trica" title="Distribució hipergeomètrica">Hipergeomètrica</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Rademacher" title="Distribució de Rademacher">Rademacher</a></li> <li><a href="/wiki/Distribuci%C3%B3_uniforme_discreta" title="Distribució uniforme discreta">Uniforme discreta</a></li> <li><a href="/wiki/Llei_de_Zipf" title="Llei de Zipf">Zipf</a></li> <li><a href="/wiki/Llei_de_Zipf-Mandelbrot" title="Llei de Zipf-Mandelbrot">Zipf-Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions discretes<br />amb suport infinit</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_beta-binomial_negativa" title="Distribució beta-binomial negativa">Beta-binomial negativa</a></li> <li><a href="/wiki/Distribuci%C3%B3_binomial_negativa" title="Distribució binomial negativa">Binomial negativa</a> <ul><li><a href="/wiki/Distribuci%C3%B3_binomial_negativa_estesa" title="Distribució binomial negativa estesa">estesa</a></li></ul></li> <li><a href="/wiki/Distribuci%C3%B3_de_Borel" title="Distribució de Borel">Borel</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Conway-Maxwell-Poisson" class="mw-redirect" title="Distribució de Conway-Maxwell-Poisson">Conway-Maxwell-Poisson</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Delaporte&action=edit&redlink=1" class="new" title="Distribució de Delaporte (encara no existeix)">Delaporte</a></li> <li><a href="/wiki/Distribuci%C3%B3_discreta_de_tipus_fase" title="Distribució discreta de tipus fase">Tipus fase</a></li> <li><a href="/wiki/Distribuci%C3%B3_fractal_parab%C3%B2lica" title="Distribució fractal parabòlica">Fractal parabòlica</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gauss-Kuzmin" title="Distribució de Gauss-Kuzmin">Gauss-Kuzmin</a></li> <li><a href="/wiki/Distribuci%C3%B3_geom%C3%A8trica" title="Distribució geomètrica">Geomètrica</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_logar%C3%ADtmica&action=edit&redlink=1" class="new" title="Distribució logarítmica (encara no existeix)">Logarítmica</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Poisson" title="Distribució de Poisson">Poisson</a> <ul><li><a href="/wiki/Distribuci%C3%B3_de_Poisson_mixta" title="Distribució de Poisson mixta">mixta</a></li></ul></li> <li><a href="/wiki/Distribuci%C3%B3_de_Skellam" title="Distribució de Skellam">Skellam</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Yule-Simon" title="Distribució de Yule-Simon">Yule-Simon</a></li> <li><a href="/wiki/Distribuci%C3%B3_zeta" title="Distribució zeta">Zeta</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions contínues<br />suportades sobre un interval acotat</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_arcsinus" title="Distribució arcsinus">Arcsinus</a></li> <li><a href="/wiki/Distribuci%C3%B3_ARGUS" title="Distribució ARGUS">ARGUS</a></li> <li><a href="/wiki/Model_de_Balding-Nichols" class="mw-redirect" title="Model de Balding-Nichols">Balding-Nichols</a></li> <li><a href="/wiki/Distribuci%C3%B3_Bates" class="mw-redirect" title="Distribució Bates">Bates</a></li> <li><a href="/wiki/Distribuci%C3%B3_beta" title="Distribució beta">Beta</a> <ul><li><a href="/wiki/Distribuci%C3%B3_beta_no_central" title="Distribució beta no central">no central</a></li> <li><a href="/wiki/Distribuci%C3%B3_beta_rectangular" class="mw-redirect" title="Distribució beta rectangular">rectangular</a></li></ul></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_del_cosinus_elevat&action=edit&redlink=1" class="new" title="Distribució del cosinus elevat (encara no existeix)">Cosinus elevat</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_Irwin-Hall&action=edit&redlink=1" class="new" title="Distribució Irwin-Hall (encara no existeix)">Irwin-Hall</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_Kumaraswamy&action=edit&redlink=1" class="new" title="Distribució Kumaraswamy (encara no existeix)">Kumaraswamy</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_Logit-normal&action=edit&redlink=1" class="new" title="Distribució Logit-normal (encara no existeix)">Logit-normal</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_parab%C3%B2lica&action=edit&redlink=1" class="new" title="Distribució parabòlica (encara no existeix)">Parabòlica</a></li> <li><a href="/wiki/Distribuci%C3%B3_PERT" title="Distribució PERT">PERT</a></li> <li><a href="/wiki/Distribuci%C3%B3_rec%C3%ADproca" title="Distribució recíproca">Recíproca</a></li> <li><a href="/wiki/Distribuci%C3%B3_triangular" title="Distribució triangular">Triangular</a></li> <li><a href="/wiki/Distribuci%C3%B3_uniforme_cont%C3%ADnua" title="Distribució uniforme contínua">Uniforme</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Wigner&action=edit&redlink=1" class="new" title="Distribució de Wigner (encara no existeix)">Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions contínues<br />suportades sobre un interval semi-infinit</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_de_Benini" title="Distribució de Benini">Benini</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Benktander" title="Distribució de Benktander">Benktander</a></li> <li><a href="/wiki/Distribuci%C3%B3_beta_prima" title="Distribució beta prima">Beta prima</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Burr" title="Distribució de Burr">Burr</a></li> <li><a href="/wiki/Distribuci%C3%B3_khi" title="Distribució khi">χ</a></li> <li><a href="/wiki/Distribuci%C3%B3_khi_quadrat" title="Distribució khi quadrat">χ2</a> <ul><li><a href="/wiki/Distribuci%C3%B3_khi_quadrat_inversa" title="Distribució khi quadrat inversa">inversa</a></li> <li><a href="/wiki/Distribuci%C3%B3_khi_quadrat_inversa_escalada" title="Distribució khi quadrat inversa escalada">inversa escalada</a></li> <li><a href="/wiki/Distribuci%C3%B3_khi_quadrat_no_central" title="Distribució khi quadrat no central">no central</a></li></ul></li> <li><a href="/wiki/Distribuci%C3%B3_de_Dagum" title="Distribució de Dagum">Dagum</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Davis" title="Distribució de Davis">Davis</a></li> <li><a href="/wiki/Distribuci%C3%B3_d%27Erlang" title="Distribució d'Erlang">Erlang</a></li> <li><a href="/wiki/Distribuci%C3%B3_exponencial" title="Distribució exponencial">Exponencial</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_exponencial-logar%C3%ADtmic&action=edit&redlink=1" class="new" title="Distribució exponencial-logarítmic (encara no existeix)">Exponencial-logarítmic</a></li> <li><a href="/wiki/Distribuci%C3%B3_F" title="Distribució F">F</a> <ul><li><a href="/wiki/Distribuci%C3%B3_F_no_central" title="Distribució F no central">no central</a></li></ul></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Flory-Schulz&action=edit&redlink=1" class="new" title="Distribució de Flory-Schulz (encara no existeix)">Flory-Schulz</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Fr%C3%A9chet" title="Distribució de Fréchet">Fréchet</a></li> <li><a href="/wiki/Distribuci%C3%B3_gamma" title="Distribució gamma">Gamma</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_Gamma/Gompertz&action=edit&redlink=1" class="new" title="Distribució Gamma/Gompertz (encara no existeix)">Gamma/Gompertz</a></li> <li><a href="/wiki/Distribuci%C3%B3_gamma_inversa" title="Distribució gamma inversa">Gamma inversa</a></li> <li><a href="/wiki/Distribuci%C3%B3_gaussiana_inversa" title="Distribució gaussiana inversa">Gaussiana inversa</a></li> <li><a href="/wiki/Distribuci%C3%B3_gaussiana_inversa_generalitzada" title="Distribució gaussiana inversa generalitzada">Gaussiana inversa generalitzada</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gompertz" title="Distribució de Gompertz">Gompertz</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gompertz_despla%C3%A7ada" title="Distribució de Gompertz desplaçada">Gompertz desplaçada</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gumbel_de_tipus_II" title="Distribució de Gumbel de tipus II">Gumbel de tipus II</a></li> <li><a href="/wiki/Distribuci%C3%B3_hiper-Erlang" title="Distribució hiper-Erlang">hiper-Erlang</a></li> <li><a href="/wiki/Distribuci%C3%B3_hiperexponencial" title="Distribució hiperexponencial">Hiperexponencial</a></li> <li><a href="/wiki/Distribuci%C3%B3_hipoexponencial" title="Distribució hipoexponencial">Hipoexponencial</a></li> <li><a href="/wiki/Prova_de_Kolmog%C3%B3rov-Smirnov" title="Prova de Kolmogórov-Smirnov">Kolmogórov-Smirnov</a></li> <li><a href="/wiki/Distribuci%C3%B3_lambda_de_Wilks" title="Distribució lambda de Wilks">Lambda de Wilks</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_L%C3%A9vy" title="Distribució de Lévy">Lévy</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_log-Cauchy&action=edit&redlink=1" class="new" title="Distribució log-Cauchy (encara no existeix)">Log-Cauchy</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_log-Laplace&action=edit&redlink=1" class="new" title="Distribució log-Laplace (encara no existeix)">Log-Laplace</a></li> <li><a href="/wiki/Distribuci%C3%B3_log-log%C3%ADstica" title="Distribució log-logística">Log-logística</a></li> <li><a href="/wiki/Distribuci%C3%B3_log-normal" title="Distribució log-normal">Log-normal</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Lomax&action=edit&redlink=1" class="new" title="Distribució de Lomax (encara no existeix)">Lomax</a></li> <li><a href="/wiki/Distribuci%C3%B3_matriu_exponencial" class="mw-redirect" title="Distribució matriu exponencial">Matriu exponencial</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Maxwell-Boltzmann" title="Distribució de Maxwell-Boltzmann">Maxwell-Boltzmann</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Maxwell-J%C3%BCttner" title="Distribució de Maxwell-Jüttner">Maxwell-Jüttner</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_mig-log%C3%ADstica&action=edit&redlink=1" class="new" title="Distribució mig-logística (encara no existeix)">Mig-logística</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Mittag-Leffler&action=edit&redlink=1" class="new" title="Distribució de Mittag-Leffler (encara no existeix)">Mittag-Leffler</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Nakagami&action=edit&redlink=1" class="new" title="Distribució de Nakagami (encara no existeix)">Nakagami</a></li> <li><a href="/wiki/Distribuci%C3%B3_normal_plegada" title="Distribució normal plegada">Normal plegada</a></li> <li><a href="/wiki/Distribuci%C3%B3_normal_truncada" title="Distribució normal truncada">Normal truncada</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Pareto" title="Distribució de Pareto">Pareto</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Poly-Weibull&action=edit&redlink=1" class="new" title="Distribució de Poly-Weibull (encara no existeix)">Poly-Weibull</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Rayleigh" title="Distribució de Rayleigh">Rayleigh</a></li> <li><a href="/wiki/Distribuci%C3%B3_relativista_de_Breit-Wigner" title="Distribució relativista de Breit-Wigner">Relativista de Breit-Wigner</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Rice" title="Distribució de Rice">Rice</a></li> <li><a href="/wiki/Distribuci%C3%B3_seminormal" title="Distribució seminormal">Seminormal</a></li> <li><a href="/wiki/Distribuci%C3%B3_T_quadrat_de_Hotelling" title="Distribució T quadrat de Hotelling">T<sup>2</sup> de Hotelling</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_tipus_fase" title="Distribució de tipus fase">Tipus fase</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Weibull" title="Distribució de Weibull">Weibull</a> <ul><li><a href="/wiki/Distribuci%C3%B3_discreta_de_Weibull" title="Distribució discreta de Weibull">Discreta de Weibull</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions contínues<br />suportades en tota la recta real</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_asim%C3%A8trica_de_Laplace" title="Distribució asimètrica de Laplace">Asimètrica de Laplace</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Cauchy" title="Distribució de Cauchy">Cauchy</a></li> <li><a href="/wiki/Distribuci%C3%B3_estable" title="Distribució estable">Estable</a></li> <li><a href="/wiki/Distribuci%C3%B3_geom%C3%A8trica_estable" title="Distribució geomètrica estable">Geomètrica estable</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gumbel" title="Distribució de Gumbel">Gumbel</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Gumbel_de_tipus_I" title="Distribució de Gumbel de tipus I">Gumbel de tipus I</a></li> <li><a href="/wiki/Distribuci%C3%B3_hiperb%C3%B2lica_generalitzada" title="Distribució hiperbòlica generalitzada">Hiperbòlica generalitzada</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_hiperb%C3%B2lica_secant&action=edit&redlink=1" class="new" title="Distribució hiperbòlica secant (encara no existeix)">Hiperbòlica secant</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Holtsmark" title="Distribució de Holtsmark">Holtsmark</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Landau" title="Distribució de Landau">Landau</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Laplace" title="Distribució de Laplace">Laplace</a></li> <li><a href="/wiki/Distribuci%C3%B3_log%C3%ADstica" title="Distribució logística">Logística</a></li> <li><a class="mw-selflink selflink">Normal</a> <ul><li><a href="/wiki/Distribuci%C3%B3_normal_generalitzada" title="Distribució normal generalitzada">generalitzada</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_normal_inversa&action=edit&redlink=1" class="new" title="Distribució normal inversa (encara no existeix)">Normalinversa</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_normal_de_Skew&action=edit&redlink=1" class="new" title="Distribució normal de Skew (encara no existeix)">de Skew</a></li></ul></li> <li><a href="/wiki/Distribuci%C3%B3_q_gaussiana" title="Distribució q gaussiana"><i>Q</i> gaussiana</a></li> <li><a href="/wiki/Distribuci%C3%B3_SU_de_Johnson" title="Distribució SU de Johnson"><i>S<sub>U</sub></i> de Johnson</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Slash" title="Distribució de Slash">Slash</a></li> <li><a href="/wiki/Distribuci%C3%B3_t_no_central" title="Distribució t no central"><i>t</i> no central</a></li> <li><a href="/wiki/Distribuci%C3%B3_t_de_Student" title="Distribució t de Student"><i>t</i> de Student</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Tracy-Widom" title="Distribució de Tracy-Widom">Tracy-Widom</a></li> <li><a href="/wiki/Distribuci%C3%B3_vari%C3%A0ncia-gamma" title="Distribució variància-gamma">Variància-gamma</a></li> <li><a href="/wiki/Perfil_de_Voigt" title="Perfil de Voigt">Voigt</a></li> <li><a href="/wiki/Distribuci%C3%B3_z_de_Fisher" title="Distribució z de Fisher">Z de Fisher</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Distribucions contínues<br />amb el suport de varis tipus</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_lambda_de_Tukey" title="Distribució lambda de Tukey">Lambda de Tukey</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_log-log%C3%ADstica_despla%C3%A7ada&action=edit&redlink=1" class="new" title="Distribució log-logística desplaçada (encara no existeix)">Log-logística desplaçada</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Marchenko-Pastur" title="Distribució de Marchenko-Pastur">Marchenko-Pastur</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Pareto_generalitzada" title="Distribució de Pareto generalitzada">Pareto generalitzada</a></li> <li><a href="/wiki/Distribuci%C3%B3_q_gaussiana" title="Distribució q gaussiana"><i>q</i> gaussiana</a></li> <li><a href="/wiki/Distribuci%C3%B3_Q-exponencial" title="Distribució Q-exponencial"><i>q</i> exponencial</a></li> <li><a href="/wiki/Distribuci%C3%B3_q_de_Weibull" title="Distribució q de Weibull"><i>q</i> de Weibull</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_valor_extrem_generalitzada&action=edit&redlink=1" class="new" title="Distribució de valor extrem generalitzada (encara no existeix)">Valor extrem generalitzada</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Barreja de distribució variable-contínua</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Distribuci%C3%B3_rectificada_gaussiana" title="Distribució rectificada gaussiana">Rectificada gaussiana</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Distribuci%C3%B3_conjunta" title="Distribució conjunta">Distribució conjunta</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <dl><dt><span style="font-weight:normal;"><i>Discreta</i></span></dt> <dd><a href="/wiki/F%C3%B3rmula_de_mostreig_d%27Ewens" title="Fórmula de mostreig d'Ewens">Ewens</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_multinomial" title="Distribució multinomial">Multinomial</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_multinomial_de_Dirichlet" title="Distribució multinomial de Dirichlet">Multinomial de Dirichlet</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_multinomial_negativa&action=edit&redlink=1" class="new" title="Distribució multinomial negativa (encara no existeix)">Multinomial negativa</a></dd> <dt><span style="font-weight:normal;"><i>Contínua</i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_de_Dirichlet" title="Distribució de Dirichlet">Dirichlet</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_de_Dirichlet_generalitzada&action=edit&redlink=1" class="new" title="Distribució de Dirichlet generalitzada (encara no existeix)">Dirichlet generalitzada</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_estable_multivariant" title="Distribució estable multivariant">Estable multivariant</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_gamma_normal" title="Distribució gamma normal">Gamma normal</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_gamma_normal_inversa&action=edit&redlink=1" class="new" title="Distribució gamma normal inversa (encara no existeix)">Gamma normal inversa</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_normal_multivariable" title="Distribució normal multivariable">Normal multivariable</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_t_multivariable&action=edit&redlink=1" class="new" title="Distribució t multivariable (encara no existeix)"><i>t</i> multivariable</a></dd> <dt><span style="font-weight:normal;"><i><a href="/wiki/Matriu_aleat%C3%B2ria" title="Matriu aleatòria">Matriu de valor</a></i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_matriu_gamma" title="Distribució matriu gamma">Matriu gamma</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_matriu_gamma_inversa" title="Distribució matriu gamma inversa">Matriu gamma inversa</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_matriu_normal&action=edit&redlink=1" class="new" title="Distribució matriu normal (encara no existeix)">Matriu normal</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_normal_de_Wishart&action=edit&redlink=1" class="new" title="Distribució normal de Wishart (encara no existeix)">Normal de Wishart</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_normal_de_Wishart_inversa&action=edit&redlink=1" class="new" title="Distribució normal de Wishart inversa (encara no existeix)">Normal de Wishart inversa</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_t_matriu&action=edit&redlink=1" class="new" title="Distribució t matriu (encara no existeix)"><i>t</i> matriu</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_de_Wishart" title="Distribució de Wishart">Wishart</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_de_Wishart_inversa" title="Distribució de Wishart inversa">Wishart inversa</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Estad%C3%ADstiques_direccionals&action=edit&redlink=1" class="new" title="Estadístiques direccionals (encara no existeix)">Direccionals</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <dl><dt><span style="font-weight:normal;"><i>Univariada (circular)</i></span></dt> <dd><a href="/w/index.php?title=Distribuci%C3%B3_asim%C3%A8trica_de_Laplace_envoltada&action=edit&redlink=1" class="new" title="Distribució asimètrica de Laplace envoltada (encara no existeix)">Asimètrica de Laplace envoltada</a></dd> <dd><a href="/w/index.php?title=Distribuci%C3%B3_de_Cauchy_enviltada&action=edit&redlink=1" class="new" title="Distribució de Cauchy enviltada (encara no existeix)">Cauchy envoltada</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_exponencial_envoltada" title="Distribució exponencial envoltada">Exponencial envoltada</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_de_L%C3%A9vy_envoltada" title="Distribució de Lévy envoltada">Lévy envoltada</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_normal_envoltada" title="Distribució normal envoltada">Normal envoltada</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_circular_uniforme" title="Distribució circular uniforme">Circular uniforme</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_de_von_Mises" title="Distribució de von Mises">Univariada de von Mises</a></dd> <dt><span style="font-weight:normal;"><i>Bivariada (esfèrica)</i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_de_Kent" title="Distribució de Kent">Kent</a></dd> <dt><span style="font-weight:normal;"><i>Bivariada (toroidal)</i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_bivariada_de_von_Mises" title="Distribució bivariada de von Mises">Bivariada de von Mises</a></dd> <dt><span style="font-weight:normal;"><i>Multivariada</i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_de_von_Mises-Fisher" title="Distribució de von Mises-Fisher">von Mises-Fisher</a></dd> <dd><a href="/wiki/Distribuci%C3%B3_de_Bingham" title="Distribució de Bingham">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Distribuci%C3%B3_degenerada" title="Distribució degenerada">Degenerada</a> i <a href="/wiki/Distribuci%C3%B3_singular" title="Distribució singular">singular</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <dl><dt><span style="font-weight:normal;"><i>Degenerada</i></span></dt> <dd><a href="/wiki/Delta_de_Dirac" title="Delta de Dirac">Delta de Dirac</a></dd> <dt><span style="font-weight:normal;"><i>Singular</i></span></dt> <dd><a href="/wiki/Distribuci%C3%B3_de_Cantor" title="Distribució de Cantor">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Famílies</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/w/index.php?title=Distribuci%C3%B3_de_barreja&action=edit&redlink=1" class="new" title="Distribució de barreja (encara no existeix)">Barreja</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_circular&action=edit&redlink=1" class="new" title="Distribució circular (encara no existeix)">Circular</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_composta_de_Poisson&action=edit&redlink=1" class="new" title="Distribució composta de Poisson (encara no existeix)">Composta de Poisson</a></li> <li><a href="/wiki/Distribuci%C3%B3_el%C2%B7l%C3%ADptica" title="Distribució el·líptica">El·líptica</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_envoltada&action=edit&redlink=1" class="new" title="Distribució envoltada (encara no existeix)">Envoltada</a></li> <li><a href="/wiki/Fam%C3%ADlia_exponencial" title="Família exponencial">Exponencial</a></li> <li><a href="/w/index.php?title=Fam%C3%ADlia_exponencial_natural&action=edit&redlink=1" class="new" title="Família exponencial natural (encara no existeix)">Exponencial natural</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_probabilitat_de_m%C3%A0xima_entropia&action=edit&redlink=1" class="new" title="Distribució de probabilitat de màxima entropia (encara no existeix)">Màxima entropia</a></li> <li><a href="/wiki/Distribuci%C3%B3_de_Pearson" title="Distribució de Pearson">Pearson</a></li> <li><a href="/w/index.php?title=Distribuci%C3%B3_de_Tweedie&action=edit&redlink=1" class="new" title="Distribució de Tweedie (encara no existeix)">Tweedie</a></li> <li><a href="/w/index.php?title=Fam%C3%ADlia_d%27ubicaci%C3%B3-escala&action=edit&redlink=1" class="new" title="Família d'ubicació-escala (encara no existeix)">Ubicació-escala</a></li></ul> </div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks 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