Çokdeğişirli normal dağılım

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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="İçindekiler" 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">İçindekiler</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">gizle</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">Giriş</div> </a> </li> <li id="toc-Genel_hal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Genel_hal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Genel hal</span> </div> </a> <button aria-controls="toc-Genel_hal-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>Genel hal alt bölümünü aç/kapa</span> </button> <ul id="toc-Genel_hal-sublist" class="vector-toc-list"> <li id="toc-Yığmalı_dağılım_fonksiyonu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Yığmalı_dağılım_fonksiyonu"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Yığmalı dağılım fonksiyonu</span> </div> </a> <ul id="toc-Yığmalı_dağılım_fonksiyonu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bir_karşıt_örneğin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bir_karşıt_örneğin"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Bir karşıt örneğin</span> </div> </a> <ul id="toc-Bir_karşıt_örneğin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal_dağılım_gösterme_ve_bağımsızlık" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_dağılım_gösterme_ve_bağımsızlık"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Normal dağılım gösterme ve bağımsızlık</span> </div> </a> <ul id="toc-Normal_dağılım_gösterme_ve_bağımsızlık-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-İki_değişirli_hal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#İki_değişirli_hal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>İki değişirli hal</span> </div> </a> <ul id="toc-İki_değişirli_hal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Afin_dönüşümü" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Afin_dönüşümü"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Afin dönüşümü</span> </div> </a> <ul id="toc-Afin_dönüşümü-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometrik_açıklama" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometrik_açıklama"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geometrik açıklama</span> </div> </a> <ul id="toc-Geometrik_açıklama-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Korelasyonlar_ve_bağımsızlık" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Korelasyonlar_ve_bağımsızlık"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Korelasyonlar ve bağımsızlık</span> </div> </a> <ul id="toc-Korelasyonlar_ve_bağımsızlık-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Daha_yüksek_momentler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Daha_yüksek_momentler"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Daha yüksek momentler</span> </div> </a> <ul id="toc-Daha_yüksek_momentler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Koşullu_dağılımlar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Koşullu_dağılımlar"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Koşullu dağılımlar</span> </div> </a> <ul id="toc-Koşullu_dağılımlar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fisher'in_enformasyon_matrisi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fisher'in_enformasyon_matrisi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Fisher'in enformasyon matrisi</span> </div> </a> <ul id="toc-Fisher'in_enformasyon_matrisi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kullback-Leibler_ayrılımı" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kullback-Leibler_ayrılımı"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Kullback-Leibler ayrılımı</span> </div> </a> <ul id="toc-Kullback-Leibler_ayrılımı-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametrelerin_kestrimi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parametrelerin_kestrimi"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Parametrelerin kestrimi</span> </div> </a> <ul id="toc-Parametrelerin_kestrimi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Entropi"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Entropi</span> </div> </a> <ul id="toc-Entropi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Çokdeğişirli_normallik_sınamaları" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Çokdeğişirli_normallik_sınamaları"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Çokdeğişirli normallik sınamaları</span> </div> </a> <ul id="toc-Çokdeğişirli_normallik_sınamaları-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dağılımdan_değerlerin_bulunması" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dağılımdan_değerlerin_bulunması"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Dağılımdan değerlerin bulunması</span> </div> </a> <ul id="toc-Dağılımdan_değerlerin_bulunması-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kaynakça" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kaynakça"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Kaynakça</span> </div> </a> <ul id="toc-Kaynakça-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="İçindekiler" 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="İçindekiler tablosunu değiştir" > <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">İçindekiler tablosunu değiştir</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">Çokdeğişirli normal dağılım</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="Başka bir dildeki sayfaya gidin. 25 dilde mevcut" > <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-25" 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">25 dil</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A_%D9%85%D8%AA%D8%B9%D8%AF%D8%AF_%D8%A7%D9%84%D9%85%D8%AA%D8%BA%D9%8A%D8%B1%D8%A7%D8%AA" title="توزيع طبيعي متعدد المتغيرات - Arapça" lang="ar" hreflang="ar" data-title="توزيع طبيعي متعدد المتغيرات" data-language-autonym="العربية" data-language-local-name="Arapça" 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_multivariante" title="Distribución normal multivariante - Asturyasça" lang="ast" hreflang="ast" data-title="Distribución normal multivariante" data-language-autonym="Asturianu" data-language-local-name="Asturyasça" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%B0%D0%B2%D1%8B%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D0%B5_%D0%BD%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="Многавымернае нармальнае размеркаванне - Belarusça" lang="be" hreflang="be" data-title="Многавымернае нармальнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="Belarusça" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_normal_multivariable" title="Distribució normal multivariable - Katalanca" lang="ca" hreflang="ca" data-title="Distribució normal multivariable" data-language-autonym="Català" data-language-local-name="Katalanca" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Flerdimensional_normalfordeling" title="Flerdimensional normalfordeling - Danca" lang="da" hreflang="da" data-title="Flerdimensional normalfordeling" data-language-autonym="Dansk" data-language-local-name="Danca" 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/Mehrdimensionale_Normalverteilung" title="Mehrdimensionale Normalverteilung - Almanca" lang="de" hreflang="de" data-title="Mehrdimensionale Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="Almanca" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution - İngilizce" lang="en" hreflang="en" data-title="Multivariate normal distribution" data-language-autonym="English" data-language-local-name="İngilizce" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal_multivariada" title="Distribución normal multivariada - İspanyolca" lang="es" hreflang="es" data-title="Distribución normal multivariada" data-language-autonym="Español" data-language-local-name="İspanyolca" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aldagai_anitzeko_banaketa_normal" title="Aldagai anitzeko banaketa normal - Baskça" lang="eu" hreflang="eu" data-title="Aldagai anitzeko banaketa normal" data-language-autonym="Euskara" data-language-local-name="Baskça" 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_%DA%86%D9%86%D8%AF%D9%85%D8%AA%D8%BA%DB%8C%D8%B1%D9%87" title="توزیع نرمال چندمتغیره - Farsça" lang="fa" hreflang="fa" data-title="توزیع نرمال چندمتغیره" data-language-autonym="فارسی" data-language-local-name="Farsça" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_normale_multidimensionnelle" title="Loi normale multidimensionnelle - Fransızca" lang="fr" hreflang="fr" data-title="Loi normale multidimensionnelle" data-language-autonym="Français" data-language-local-name="Fransızca" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_normal_multivariante" title="Distribución normal multivariante - Galiçyaca" lang="gl" hreflang="gl" data-title="Distribución normal multivariante" data-language-autonym="Galego" data-language-local-name="Galiçyaca" 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%A8%D7%91-%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="התפלגות רב-נורמלית - İbranice" lang="he" hreflang="he" data-title="התפלגות רב-נורמלית" data-language-autonym="עברית" data-language-local-name="İbranice" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_normale_multivariata" title="Distribuzione normale multivariata - İtalyanca" lang="it" hreflang="it" data-title="Distribuzione normale multivariata" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target"><span>İtaliano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E5%A4%89%E9%87%8F%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83" title="多変量正規分布 - Japonca" lang="ja" hreflang="ja" data-title="多変量正規分布" data-language-autonym="日本語" data-language-local-name="Japonca" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EB%B3%80%EB%9F%89_%EC%A0%95%EA%B7%9C%EB%B6%84%ED%8F%AC" title="다변량 정규분포 - Korece" lang="ko" hreflang="ko" data-title="다변량 정규분포" data-language-autonym="한국어" data-language-local-name="Korece" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Multivariate_normale_verdeling" title="Multivariate normale verdeling - Felemenkçe" lang="nl" hreflang="nl" data-title="Multivariate normale verdeling" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wielowymiarowy_rozk%C5%82ad_normalny" title="Wielowymiarowy rozkład normalny - Lehçe" lang="pl" hreflang="pl" data-title="Wielowymiarowy rozkład normalny" data-language-autonym="Polski" data-language-local-name="Lehçe" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D0%BD%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="Многомерное нормальное распределение - Rusça" lang="ru" hreflang="ru" data-title="Многомерное нормальное распределение" data-language-autonym="Русский" data-language-local-name="Rusça" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Multivarijantna_normalna_raspodela" title="Multivarijantna normalna raspodela - Sırpça" lang="sr" hreflang="sr" data-title="Multivarijantna normalna raspodela" data-language-autonym="Српски / srpski" data-language-local-name="Sırpça" class="interlanguage-link-target"><span>Српски / srpski</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%E0%B8%AB%E0%B8%A5%E0%B8%B2%E0%B8%A2%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B9%81%E0%B8%9B%E0%B8%A3" title="การแจกแจงปรกติหลายตัวแปร - Tayca" lang="th" hreflang="th" data-title="การแจกแจงปรกติหลายตัวแปร" data-language-autonym="ไทย" data-language-local-name="Tayca" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D0%B3%D0%B0%D1%82%D0%BE%D0%B2%D0%B8%D0%BC%D1%96%D1%80%D0%BD%D0%B8%D0%B9_%D0%BD%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="Багатовимірний нормальний розподіл - Ukraynaca" lang="uk" hreflang="uk" data-title="Багатовимірний нормальний розподіл" data-language-autonym="Українська" data-language-local-name="Ukraynaca" class="interlanguage-link-target"><span>Українська</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_nhi%E1%BB%81u_chi%E1%BB%81u" title="Phân phối chuẩn nhiều chiều - Vietnamca" lang="vi" hreflang="vi" data-title="Phân phối chuẩn nhiều chiều" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamca" 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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">Görünüm</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">gizle</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">Vikipedi, özgür ansiklopedi</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="tr" dir="ltr"><table class="infobox bordered wikitable" style="width: 325px; font-size: 95%; margin-left: 1em; margin-bottom: 0.5em;"> <caption>Çokdeğişirli normal </caption> <tbody><tr style="text-align: center;"> <td colspan="2">Olasılık yoğunluk fonksiyonu<br /> </td></tr> <tr style="text-align: center;"> <td colspan="2">Yığmalı dağılım fonksiyonu<br /> </td></tr> <tr valign="top"> <th>Parametreler </th> <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 =[\mu _{1},\dots ,\mu _{N}]^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =[\mu _{1},\dots ,\mu _{N}]^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f3a817da688416d58f085f9784629863edd6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.91ex; height:3.176ex;" alt="{\displaystyle \mu =[\mu _{1},\dots ,\mu _{N}]^{T}}"></span> <a href="/w/index.php?title=Konum_parametresi&action=edit&redlink=1" class="new" title="Konum parametresi (sayfa mevcut değil)">konum parametresi</a> (<a href="/wiki/Reel_say%C4%B1" class="mw-redirect" title="Reel sayı">reel</a> <a href="/wiki/Vekt%C3%B6r" title="Vektör">vektör</a>)<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> <a href="/wiki/Kovaryans_matrisi" title="Kovaryans matrisi">kovaryans matrisi</a> (<a href="/w/index.php?title=Pozitif-kesin&action=edit&redlink=1" class="new" title="Pozitif-kesin (sayfa mevcut değil)">pozitif-kesin</a> reel <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\times N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="{\displaystyle N\times N}"></span> <a href="/wiki/Matris" class="mw-disambig" title="Matris">matris</a>) </td></tr> <tr> <th><a href="/w/index.php?title=Destek_(matematik)&action=edit&redlink=1" class="new" title="Destek (matematik) (sayfa mevcut değil)">Destek</a> </th> <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 x\in \mathbb {R} ^{N}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{N}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ad8fc3881ace51c65d094fda22c890f3610c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:7.54ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{N}\!}"></span> </td></tr> <tr> <th><a href="/wiki/Olas%C4%B1l%C4%B1k_yo%C4%9Funluk_fonksiyonu" title="Olasılık yoğunluk fonksiyonu">Olasılık yoğunluk fonksiyonu</a> (OYF) </th> <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 f_{X}(x_{1},\dots ,x_{N})={\frac {1}{(2\pi )^{N/2}\left|\Sigma \right|^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>|</mo> <mi mathvariant="normal">Σ</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x_{1},\dots ,x_{N})={\frac {1}{(2\pi )^{N/2}\left|\Sigma \right|^{1/2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce955cf83857c0d5c54a40d981ab1081d5487b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.408ex; height:6.509ex;" alt="{\displaystyle f_{X}(x_{1},\dots ,x_{N})={\frac {1}{(2\pi )^{N/2}\left|\Sigma \right|^{1/2}}}}"></span><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(-{\frac {1}{2}}(x-\mu )^{\top }\Sigma ^{-1}(x-\mu )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left(-{\frac {1}{2}}(x-\mu )^{\top }\Sigma ^{-1}(x-\mu )\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02643694b6bf2d0116db087096bffb45fbd6a955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.064ex; height:6.176ex;" alt="{\displaystyle \exp \left(-{\frac {1}{2}}(x-\mu )^{\top }\Sigma ^{-1}(x-\mu )\right)}"></span> </td></tr> <tr> <th><a href="/wiki/Birikimli_da%C4%9F%C4%B1l%C4%B1m_fonksiyonu" title="Birikimli dağılım fonksiyonu">Birikimli dağılım fonksiyonu</a> (YDF) </th> <td> </td></tr> <tr> <th><a href="/wiki/Beklenen_de%C4%9Fer" title="Beklenen değer">Ortalama</a> </th> <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> <th><a href="/wiki/Medyan" title="Medyan">Medyan</a> </th> <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> <th><a href="/wiki/Mod" title="Mod">Mod</a> </th> <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> <th><a href="/wiki/Varyans" title="Varyans">Varyans</a> </th> <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> (kovaryans matrisi) </td></tr> <tr> <th><a href="/wiki/%C3%87arp%C4%B1kl%C4%B1k" title="Çarpıklık">Çarpıklık</a> </th> <td>0 </td></tr> <tr> <th>Fazladan <a href="/wiki/Bas%C4%B1kl%C4%B1k" title="Basıklık">basıklık</a> </th> <td>0 </td></tr> <tr> <th><a href="/wiki/Entropi_(bilgi_kuram%C4%B1)" class="mw-redirect" title="Entropi (bilgi kuramı)">Entropi</a> </th> <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 \ln \left({\sqrt {(2\,\pi \,e)^{N}\left|\Sigma \right|}}\right)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mrow> <mo>|</mo> <mi mathvariant="normal">Σ</mi> <mo>|</mo> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left({\sqrt {(2\,\pi \,e)^{N}\left|\Sigma \right|}}\right)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89268a5da0e118ac232544317278a93b2296d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.509ex; height:6.176ex;" alt="{\displaystyle \ln \left({\sqrt {(2\,\pi \,e)^{N}\left|\Sigma \right|}}\right)\!}"></span> </td></tr> <tr> <th><a href="/wiki/Moment_%C3%BCreten_fonksiyon" title="Moment üreten fonksiyon">Moment üreten fonksiyon</a> (mf) </th> <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 M_{X}(t)=\exp \left(\mu ^{\top }t+{\frac {1}{2}}t^{\top }\Sigma t\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> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="normal">Σ</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\exp \left(\mu ^{\top }t+{\frac {1}{2}}t^{\top }\Sigma t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83cdae4115b4da5c9a5c5ef8e41fc07a190b7bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.067ex; height:6.176ex;" alt="{\displaystyle M_{X}(t)=\exp \left(\mu ^{\top }t+{\frac {1}{2}}t^{\top }\Sigma t\right)}"></span> </td></tr> <tr> <th><a href="/wiki/Karakteristik_fonksiyon" title="Karakteristik fonksiyon">Karakteristik fonksiyon</a> </th> <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 \phi _{X}(t;\mu ,\Sigma )=\exp \left(i\mu ^{\top }t-{\frac {1}{2}}t^{\top }\Sigma t\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>;</mo> <mi>μ</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="normal">Σ</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{X}(t;\mu ,\Sigma )=\exp \left(i\mu ^{\top }t-{\frac {1}{2}}t^{\top }\Sigma t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc6c324dc4480802b1379990aba0bbed8b9142b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.148ex; height:6.176ex;" alt="{\displaystyle \phi _{X}(t;\mu ,\Sigma )=\exp \left(i\mu ^{\top }t-{\frac {1}{2}}t^{\top }\Sigma t\right)}"></span> </td></tr></tbody></table> <p><a href="/wiki/Olas%C4%B1l%C4%B1k_kuram%C4%B1" class="mw-redirect" title="Olasılık kuramı">Olasılık kuramı</a> ve <a href="/wiki/%C4%B0statistik" title="İstatistik">istatistik</a> bilim kollarında, <b>çokdeğişirli normal dağılım</b> veya <b>çokdeğişirli Gauss-tipi dağılım</b>, tek değişirli bir dağılım olan <a href="/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım">normal dağılımın</a> (veya Gauss-tipi dağılımın) çoklu değişirli hallere genelleştirilmesidir. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Genel_hal">Genel hal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=1" title="Değiştirilen bölüm: Genel hal" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=1" title="Bölümün kaynak kodunu değiştir: Genel hal"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Yığmalı_dağılım_fonksiyonu"><span id="Y.C4.B1.C4.9Fmal.C4.B1_da.C4.9F.C4.B1l.C4.B1m_fonksiyonu"></span>Yığmalı dağılım fonksiyonu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=2" title="Değiştirilen bölüm: Yığmalı dağılım fonksiyonu" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=2" title="Bölümün kaynak kodunu değiştir: Yığmalı dağılım fonksiyonu"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Genel bir tanımla, <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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00816772e8dff4e6733c478ec77fab0382264a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}"></span> olarak ifade edilen <a href="/wiki/Y%C4%B1%C4%9Fmal%C4%B1_da%C4%9F%C4%B1l%C4%B1m_fonksiyonu" class="mw-redirect" title="Yığmalı dağılım fonksiyonu">yığmalı dağılım fonksiyonu</a>, bir rassal vektörun, <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"> <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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> vektörüne eşit veya bu vektör değerlerden daha az olduğu zaman karşıtı olarak bulunan bütün olasılıkların toplamını ifade eden bir fonksiyondur. Çokdeğişirli normal dağılım için bir cebirsel kapalı eşitlik şeklinde bir <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> ifadesi bulunmamaktadır. Ancak bu fonksiyonun sayısal değerlerini tahmin etmek için birkaç <a href="/wiki/Algoritma" title="Algoritma">algoritma</a> bulunmaktadır. Bu algoritma kullanışına bir örnek için verilen referanslarda MVNDST adlı algoritmaya bakınız. (<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> veya <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> <div class="mw-heading mw-heading3"><h3 id="Bir_karşıt_örneğin"><span id="Bir_kar.C5.9F.C4.B1t_.C3.B6rne.C4.9Fin"></span>Bir karşıt örneğin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=3" title="Değiştirilen bölüm: Bir karşıt örneğin" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=3" title="Bölümün kaynak kodunu değiştir: Bir karşıt örneğin"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>İki <a href="/wiki/Rassal_de%C4%9Fi%C5%9Fken" title="Rassal değişken">rassal değişken</a> olan <i>X</i> ve <i>Y</i> tek tek normal dağılım gösterseler bile bu iki rassal değişkenin bileşik olarak (<i>X</i>, <i>Y</i>) bir çoklunormal dağılım göstereceği anlamına gelmez. Buna basit bir örnekte eğer |<i>X</i>| > 1 ise <i>Y</i>=<i>X</i> olması ve eğer |<i>X</i>| < 1 ise <i>Y</i> = -<i>X</i> olmasıdır. Bu gerçek ikiden fazla sayıda rassal değişken içinde doğrudur. </p><p>Buna benzer bir karşıt örneğin için <a href="/w/index.php?title=Normal_olarak_da%C4%9F%C4%B1l%C4%B1ml%C4%B1_olup_ve_korrelasyon_olmamas%C4%B1_ba%C4%9F%C4%B1ms%C4%B1zl%C4%B1k_ifade_etmez&action=edit&redlink=1" class="new" title="Normal olarak dağılımlı olup ve korrelasyon olmaması bağımsızlık ifade etmez (sayfa mevcut değil)">normal olarak dağılımlı olup ve korrelasyon olmaması bağımsızlık ifade etmez</a> maddesine bakınız. </p> <div class="mw-heading mw-heading3"><h3 id="Normal_dağılım_gösterme_ve_bağımsızlık"><span id="Normal_da.C4.9F.C4.B1l.C4.B1m_g.C3.B6sterme_ve_ba.C4.9F.C4.B1ms.C4.B1zl.C4.B1k"></span>Normal dağılım gösterme ve bağımsızlık</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=4" title="Değiştirilen bölüm: Normal dağılım gösterme ve bağımsızlık" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=4" title="Bölümün kaynak kodunu değiştir: Normal dağılım gösterme ve bağımsızlık"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eğer <i>X</i> ve <i>Y</i> rassal değişkenleri tek tek normal dağılım gösterirlerse ve birbirlerinden istatistiksel olarak <a href="/wiki/Ba%C4%9F%C4%B1ms%C4%B1z_siyaset%C3%A7i" title="Bağımsız siyasetçi">Bağımsızlarsa</a>, o halde bu iki rassal değişken bileşiği (yani rassal vektörü) ikideğişirli normal dağılım gösterir veya diğer bir ifade ile <i>ortaklaşa normal dağılımlı</i>lardır. Ancak ortaklaşa normal dağılım gösteren her iki rassal değişkenin birbirinden bağımsız olduğu gerçek değildir. </p> <div class="mw-heading mw-heading2"><h2 id="İki_değişirli_hal"><span id=".C4.B0ki_de.C4.9Fi.C5.9Firli_hal"></span>İki değişirli hal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=5" title="Değiştirilen bölüm: İki değişirli hal" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=5" title="Bölümün kaynak kodunu değiştir: İki değişirli hal"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>İki boyutlu singuler olmayan halde, ikideğişirli normal dağılım için (ortalamalar (0,0)da ise) <a href="/wiki/Olas%C4%B1l%C4%B1k_yo%C4%9Funluk_fonksiyonu" title="Olasılık yoğunluk fonksiyonu">olasılık yoğunluk fonksiyonu</a> şöyle tanımlanır: </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,y)={\frac {1}{2\pi \sigma _{x}\sigma _{y}{\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {x^{2}}{\sigma _{x}^{2}}}+{\frac {y^{2}}{\sigma _{y}^{2}}}-{\frac {2\rho xy}{(\sigma _{x}\sigma _{y})}}\right)\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>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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 class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ρ</mi> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)={\frac {1}{2\pi \sigma _{x}\sigma _{y}{\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {x^{2}}{\sigma _{x}^{2}}}+{\frac {y^{2}}{\sigma _{y}^{2}}}-{\frac {2\rho xy}{(\sigma _{x}\sigma _{y})}}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa2ceb52a2b5b117e073d876bdb7e396c3f9d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:69.819ex; height:7.509ex;" alt="{\displaystyle f(x,y)={\frac {1}{2\pi \sigma _{x}\sigma _{y}{\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {x^{2}}{\sigma _{x}^{2}}}+{\frac {y^{2}}{\sigma _{y}^{2}}}-{\frac {2\rho xy}{(\sigma _{x}\sigma _{y})}}\right)\right)}"></span></dd></dl> <p>Burada <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> terimi <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"> <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> ve <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> arasındaki <a href="/w/index.php?title=Korelasyonu&action=edit&redlink=1" class="new" title="Korelasyonu (sayfa mevcut değil)">korelasyonu</a> gösterir ve şu ifade <a href="/wiki/Kovaryans_matrisi" title="Kovaryans matrisi">kovaryans matrisi</a> olur: </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 \Sigma ={\begin{bmatrix}\sigma _{x}^{2}&\rho \sigma _{x}\sigma _{y}\\\rho \sigma _{x}\sigma _{y}&\sigma _{y}^{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi>ρ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>ρ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ={\begin{bmatrix}\sigma _{x}^{2}&\rho \sigma _{x}\sigma _{y}\\\rho \sigma _{x}\sigma _{y}&\sigma _{y}^{2}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faffac5091a99e5ccd05b4aa34f3bfa9d167cfd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.462ex; height:6.509ex;" alt="{\displaystyle \Sigma ={\begin{bmatrix}\sigma _{x}^{2}&\rho \sigma _{x}\sigma _{y}\\\rho \sigma _{x}\sigma _{y}&\sigma _{y}^{2}\end{bmatrix}}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Afin_dönüşümü"><span id="Afin_d.C3.B6n.C3.BC.C5.9F.C3.BCm.C3.BC"></span>Afin dönüşümü</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=6" title="Değiştirilen bölüm: Afin dönüşümü" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=6" title="Bölümün kaynak kodunu değiştir: Afin dönüşümü"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Geometrik_açıklama"><span id="Geometrik_a.C3.A7.C4.B1klama"></span>Geometrik açıklama</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=7" title="Değiştirilen bölüm: Geometrik açıklama" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=7" title="Bölümün kaynak kodunu değiştir: Geometrik açıklama"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bir singuler olmayan çokdeğişirli normal dağılım için aynı yoğunluk gösteren kontur eğrileri <a href="/wiki/Elipsoit" title="Elipsoit">elipsoitlerdir</a>; yani ortalamada merkezleşmiş çok-boyutlu-kürelerin doğrusal dönüşümleridir.<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> Bu elipsoitlerin esas eksenlerinin yönleri kovaryans matrisinin özvektörleri (eigenvector) olarak verilmiştir. Esas eksenlerin orantılı uzunluklarının karesi bunlara karşıt olan özdeğerler (eigenvalues) olurlar. Bu halde şu ifade ortaya çıkar: </p> <dl><dd><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\ \sim N(\mu ,\Sigma )\iff X\ \sim \mu +U\Lambda ^{1/2}N(0,I)\iff X\ \sim \mu +UN(0,\Lambda ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mtext> </mtext> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>X</mi> <mtext> </mtext> <mo>∼</mo> <mi>μ</mi> <mo>+</mo> <mi>U</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>X</mi> <mtext> </mtext> <mo>∼</mo> <mi>μ</mi> <mo>+</mo> <mi>U</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\ \sim N(\mu ,\Sigma )\iff X\ \sim \mu +U\Lambda ^{1/2}N(0,I)\iff X\ \sim \mu +UN(0,\Lambda ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6be1f835f6cc0d44a16f8e1c1855ffb0fa9b416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.689ex; height:3.343ex;" alt="{\displaystyle X\ \sim N(\mu ,\Sigma )\iff X\ \sim \mu +U\Lambda ^{1/2}N(0,I)\iff X\ \sim \mu +UN(0,\Lambda ).}"></span></dd></dl></dd></dl> <p>Bunun yanında, <i>U</i> bir <a href="/w/index.php?title=Rotasyon_matrisi&action=edit&redlink=1" class="new" title="Rotasyon matrisi (sayfa mevcut değil)">rotasyon matrisi</a> olarak seçilebilir; çünkü bu eksenin tersini alınca <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(0,\Lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(0,\Lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aa118e168e585f0cb5fa74cfbada9d758fe91b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.682ex; height:2.843ex;" alt="{\displaystyle N(0,\Lambda )}"></span> hiç etkilenmemektedir; buna karşıt olarak bir matris sütûnunun tersi alınırsa <i>u</i>nun determinantının işaretleri değişir. <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(\mu ,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\mu ,\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1ba7e7ac5825628439be82c5d525af1e4f502d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.987ex; height:2.843ex;" alt="{\displaystyle N(\mu ,\Sigma )}"></span> ile özetlenen dağılım böylelikle <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(0,I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(0,I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb35ed5536ed6be56d780239548a0541f73376df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.241ex; height:2.843ex;" alt="{\displaystyle N(0,I)}"></span> ifadesinin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f503b27ec077be7eaf5d7b095c4d26a924f1da57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.311ex; height:2.843ex;" alt="{\displaystyle \Lambda ^{1/2}}"></span> ile ölçeğinin değiştirilmesi, <i>u</i> ile rotasyon yapılması ve <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> ile çevrilmesi ile ortaya çıkar. </p><p>Bunun aksine bakılırsa, <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> ve tam ranklı <i>U</i> matrisi ve pozitif çapraz girdiler olan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e324b19e0ce91e4b838c079d54dbb7ff4ae0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle \Lambda _{i}}"></span> değerleri için yapılan herhangi bir seçim, bir singuler olmayan çokdeğişirli normal dağılım ortaya çıkartır. Eğer herhangi bir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e324b19e0ce91e4b838c079d54dbb7ff4ae0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle \Lambda _{i}}"></span> sıfıra eşitse ve <i>u</i> <a href="/wiki/Kare_matris" title="Kare matris">kare matris</a> ise, bunun sonucunda ortaya çıkan <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\Lambda U^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi mathvariant="normal">Λ</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\Lambda U^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2f4023e28cab7f84c23676985c80010ce2ba8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.626ex; height:2.676ex;" alt="{\displaystyle U\Lambda U^{T}}"></span> kovaryans matrisi bir <a href="/w/index.php?title=Singuler_matris&action=edit&redlink=1" class="new" title="Singuler matris (sayfa mevcut değil)">singuler matris</a> olur. Geometrik olarak bunun açıklaması her kontur elipsoitin sonsuz olarak inceleşmesi ve <i>n</i>-boyutlu bir uzayda <i>0</i> bir <a href="/wiki/Hacim" title="Hacim">hacim</a> kapsamasıdır, çünkü en aşağı bir tane esas eksenin uzunluğu sıfır olmaktadır. </p> <div class="mw-heading mw-heading2"><h2 id="Korelasyonlar_ve_bağımsızlık"><span id="Korelasyonlar_ve_ba.C4.9F.C4.B1ms.C4.B1zl.C4.B1k"></span>Korelasyonlar ve bağımsızlık</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=8" title="Değiştirilen bölüm: Korelasyonlar ve bağımsızlık" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=8" title="Bölümün kaynak kodunu değiştir: Korelasyonlar ve bağımsızlık"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Genel olarak, rassal değişkenler birbirleriyle çok yüksek derecede bağımlı olabilirler ama hiç <a href="/wiki/Korelasyon" title="Korelasyon">korelasyon</a> göstermeyebilirler. Ama, eğer bir rassal vektör çokdeğişirli normal dağılım gösterirse o halde aralarında hiç korelasyon göstermeyen iki veya daha fazla sayıda vektör parçası istatistiksel olarak birbirinden bağımsızdır. Bundan da şu sonuc çıkartılabilir: eğer vektörün herhangi iki veya daha fazla parçası ikişer ikişer bağımsızlık gösteriyorsa, bu parçalar birbirinden bağımsızdırlar. </p><p>Fakat ayrı ayrı olarak ve marjinal olarak, iki rassal değişken normal dağılım gösterirlerse ve aralarında hiç korelasyon bulunmazsa, o halde bu iki değişkenler birbirinden bağımsızdır. Normal dağılım gösteren iki rassal değişken, ortaklaşa normal dağılım göstermeyebilirler; yani bir parçası oldukları vektör bir çokdeğişkenli normal dağılım göstermeyebilir. İki korelasyon göstermeyen ama normal dağılım gösteren fakat bağımsız olmayan rassal değişken için örneğin <a href="/w/index.php?title=Normal_da%C4%9F%C4%B1l%C4%B1m_g%C3%B6sterip_hi%C3%A7_korelasyon_g%C3%B6stermemek_ba%C4%9F%C4%B1ms%C4%B1z_olmak_demek_de%C4%9Fildir&action=edit&redlink=1" class="new" title="Normal dağılım gösterip hiç korelasyon göstermemek bağımsız olmak demek değildir (sayfa mevcut değil)">normal dağılım gösterip hiç korelasyon göstermemek bağımsız olmak demek değildir</a> maddesine bakınız. </p> <div class="mw-heading mw-heading2"><h2 id="Daha_yüksek_momentler"><span id="Daha_y.C3.BCksek_momentler"></span>Daha yüksek momentler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=9" title="Değiştirilen bölüm: Daha yüksek momentler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=9" title="Bölümün kaynak kodunu değiştir: Daha yüksek momentler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Genel olarak <i>X</i> için <i>k</i>inci derecede momentler şöyle tanımlanmaktadır: </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 \mu _{1,\dots ,N}(X)\ {\stackrel {\mathrm {def} }{=}}\ \mu _{r_{1},\dots ,r_{N}}(X)\ {\stackrel {\mathrm {def} }{=}}\ E\left[\prod \limits _{j=1}^{N}X_{j}^{r_{j}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1,\dots ,N}(X)\ {\stackrel {\mathrm {def} }{=}}\ \mu _{r_{1},\dots ,r_{N}}(X)\ {\stackrel {\mathrm {def} }{=}}\ E\left[\prod \limits _{j=1}^{N}X_{j}^{r_{j}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57ed43c60b7444a6b0219eb6a9cef2c825bc607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.221ex; height:7.676ex;" alt="{\displaystyle \mu _{1,\dots ,N}(X)\ {\stackrel {\mathrm {def} }{=}}\ \mu _{r_{1},\dots ,r_{N}}(X)\ {\stackrel {\mathrm {def} }{=}}\ E\left[\prod \limits _{j=1}^{N}X_{j}^{r_{j}}\right]}"></span></dd></dl> <p>Burada <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_{1}+r_{2}+\cdots +r_{N}=k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}+r_{2}+\cdots +r_{N}=k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4555de8e2da57a0c65b427303cadb84c45bc48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.147ex; height:2.509ex;" alt="{\displaystyle r_{1}+r_{2}+\cdots +r_{N}=k.}"></span> </p><p>Merkezsel <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>inci derecede momentler şöyle verilir: </p><p>(a)Eger <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> tek ise <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 _{1,\dots ,N}(X-\mu )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1,\dots ,N}(X-\mu )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5975871b02043d62f0d866c897693517de4cc87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.048ex; height:3.009ex;" alt="{\displaystyle \mu _{1,\dots ,N}(X-\mu )=0}"></span> olur. (b)Eger <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> cift ise ve <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=2\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efb2d6b9b27f1f62c9fcf4e5a5d90c2f992e178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.827ex; height:2.176ex;" alt="{\displaystyle k=2\lambda }"></span>, o halde </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 \mu _{1,\dots ,2\lambda }(X-\mu )=\sum \left(\sigma _{ij}\sigma _{kl}\cdots \sigma _{XZ}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>2</mn> <mi>λ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>⋯</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1,\dots ,2\lambda }(X-\mu )=\sum \left(\sigma _{ij}\sigma _{kl}\cdots \sigma _{XZ}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34283d7556263a5eebb6d27ddf226066eb577d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:37.114ex; height:3.843ex;" alt="{\displaystyle \mu _{1,\dots ,2\lambda }(X-\mu )=\sum \left(\sigma _{ij}\sigma _{kl}\cdots \sigma _{XZ}\right)}"></span></dd></dl> <p>Burada toplam <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 \left\{1,\dots ,2\lambda \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>2</mn> <mi>λ</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{1,\dots ,2\lambda \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4c2f4e6213fe7147ed5dcfefdf9a1572604414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.183ex; height:2.843ex;" alt="{\displaystyle \left\{1,\dots ,2\lambda \right\}}"></span> setinin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> (sıralanmamış) çiftler üzerine tahsis edilmelerinin hepsi birlikte alınmasıdır. Bu işlem sonucunda toplam içinde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2\lambda -1)!/(2^{\lambda -1}(\lambda -1)!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>λ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2\lambda -1)!/(2^{\lambda -1}(\lambda -1)!)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570bb8738874f62b0cf007a09dd04e2109e381e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.216ex; height:3.176ex;" alt="{\displaystyle (2\lambda -1)!/(2^{\lambda -1}(\lambda -1)!)}"></span> sayıda terim bulunur, Her bir terim <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tane kovaryansın çarpımıdır. </p><p><br /> Özellikle, 4-üncü derecedeki momentler şöyle verilirler: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[X_{i}^{4}\right]=3(\sigma _{ii})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>]</mo> </mrow> <mo>=</mo> <mn>3</mn> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[X_{i}^{4}\right]=3(\sigma _{ii})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ea9402dceae9478358cb7fb1a12b0b21c08268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.972ex; height:3.343ex;" alt="{\displaystyle E\left[X_{i}^{4}\right]=3(\sigma _{ii})^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[X_{i}^{3}X_{j}\right]=3\sigma _{ii}\sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>3</mn> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[X_{i}^{3}X_{j}\right]=3\sigma _{ii}\sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0967165c68e8faaf2fb9b05289d9bac27e48dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.747ex; height:3.176ex;" alt="{\displaystyle E\left[X_{i}^{3}X_{j}\right]=3\sigma _{ii}\sigma _{ij}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[X_{i}^{2}X_{j}^{2}\right]=\sigma _{ii}\sigma _{jj}+2\left(\sigma _{ij}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[X_{i}^{2}X_{j}^{2}\right]=\sigma _{ii}\sigma _{jj}+2\left(\sigma _{ij}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510b29da3930184935a609b9a518477b141f70de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.838ex; height:4.843ex;" alt="{\displaystyle E\left[X_{i}^{2}X_{j}^{2}\right]=\sigma _{ii}\sigma _{jj}+2\left(\sigma _{ij}\right)^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[X_{i}^{2}X_{j}X_{k}\right]=\sigma _{ii}\sigma _{jk}+2\sigma _{ij}\sigma _{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[X_{i}^{2}X_{j}X_{k}\right]=\sigma _{ii}\sigma _{jk}+2\sigma _{ij}\sigma _{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0905759cb975f2e75a267d4e467c28973289f671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.677ex; height:3.176ex;" alt="{\displaystyle E\left[X_{i}^{2}X_{j}X_{k}\right]=\sigma _{ii}\sigma _{jk}+2\sigma _{ij}\sigma _{ik}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[X_{i}X_{j}X_{k}X_{n}\right]=\sigma _{ij}\sigma _{kn}+\sigma _{ik}\sigma _{jn}+\sigma _{in}\sigma _{jk}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[X_{i}X_{j}X_{k}X_{n}\right]=\sigma _{ij}\sigma _{kn}+\sigma _{ik}\sigma _{jn}+\sigma _{in}\sigma _{jk}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf0a7d73ff2e7923b591abde8aeb5026f047ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.217ex; height:3.009ex;" alt="{\displaystyle E\left[X_{i}X_{j}X_{k}X_{n}\right]=\sigma _{ij}\sigma _{kn}+\sigma _{ik}\sigma _{jn}+\sigma _{in}\sigma _{jk}.}"></span></dd></dl> <p>Dört değişken halindeki dördüncü derece moment içinde üç tane terim bulunur. </p><p>Altıncı-derecede moment içinde (3 × 5 =) 15 terim; sekizinci derecede momentler arasında (3 × 5 × 7) = 105 terim bulunur. Altıncı-derecedeki moment için ifade şöyle genişletilebilir: </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 {\begin{aligned}&{}E[X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\\&{}=E[X_{1}X_{2}]E[X_{3}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{6}]E[X_{4}X_{5}]\\&{}+E[X_{1}X_{3}]E[X_{2}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{6}]E[X_{4}X_{5}]\\&+E[X_{1}X_{4}]E[X_{2}X_{3}]E[X_{5}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{5}]E[X_{3}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{6}]E[X_{3}X_{5}]\\&+E[X_{1}X_{5}]E[X_{2}X_{3}]E[X_{4}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{4}]E[X_{3}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{6}]E[X_{3}X_{4}]\\&+E[X_{1}X_{6}]E[X_{2}X_{3}]E[X_{4}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{4}]E[X_{3}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{5}]E[X_{3}X_{4}].\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> 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class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> 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<mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> 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class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{}E[X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\\&{}=E[X_{1}X_{2}]E[X_{3}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{6}]E[X_{4}X_{5}]\\&{}+E[X_{1}X_{3}]E[X_{2}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{6}]E[X_{4}X_{5}]\\&+E[X_{1}X_{4}]E[X_{2}X_{3}]E[X_{5}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{5}]E[X_{3}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{6}]E[X_{3}X_{5}]\\&+E[X_{1}X_{5}]E[X_{2}X_{3}]E[X_{4}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{4}]E[X_{3}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{6}]E[X_{3}X_{4}]\\&+E[X_{1}X_{6}]E[X_{2}X_{3}]E[X_{4}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{4}]E[X_{3}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{5}]E[X_{3}X_{4}].\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7ea4f5aa3c42b15afa0ee0504e29aad74b1d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:91.156ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}&{}E[X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\\&{}=E[X_{1}X_{2}]E[X_{3}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{2}]E[X_{3}X_{6}]E[X_{4}X_{5}]\\&{}+E[X_{1}X_{3}]E[X_{2}X_{4}]E[X_{5}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{5}]E[X_{4}X_{6}]+E[X_{1}X_{3}]E[X_{2}X_{6}]E[X_{4}X_{5}]\\&+E[X_{1}X_{4}]E[X_{2}X_{3}]E[X_{5}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{5}]E[X_{3}X_{6}]+E[X_{1}X_{4}]E[X_{2}X_{6}]E[X_{3}X_{5}]\\&+E[X_{1}X_{5}]E[X_{2}X_{3}]E[X_{4}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{4}]E[X_{3}X_{6}]+E[X_{1}X_{5}]E[X_{2}X_{6}]E[X_{3}X_{4}]\\&+E[X_{1}X_{6}]E[X_{2}X_{3}]E[X_{4}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{4}]E[X_{3}X_{5}]+E[X_{1}X_{6}]E[X_{2}X_{5}]E[X_{3}X_{4}].\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Koşullu_dağılımlar"><span id="Ko.C5.9Fullu_da.C4.9F.C4.B1l.C4.B1mlar"></span>Koşullu dağılımlar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=10" title="Değiştirilen bölüm: Koşullu dağılımlar" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=10" title="Bölümün kaynak kodunu değiştir: Koşullu dağılımlar"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eğer <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> ve <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 mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> şu şekilde kısımlara ayrılırlarsa: </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 \mu ={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d711bc792479b43d43c412fe8e749f797fcc713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.485ex; height:6.176ex;" alt="{\displaystyle \mu ={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}\quad }"></span> Büyüklüğü şu olur; <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{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}}"> <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> <mi>q</mi> <mo>×</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24969ed92deafd811bca33c78df4bddb5b01490a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.992ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}}"></span></dd></dl> <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 \Sigma ={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma ={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843a3fdf2c4afff317b1c5ce9bfd0cf66dfbda58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.737ex; height:6.176ex;" alt="{\displaystyle \Sigma ={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}\quad }"></span> Büyüklüğü şu olur: <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{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}}"> <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> <mi>q</mi> <mo>×</mo> <mi>q</mi> </mtd> <mtd> <mi>q</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>q</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7a3c208427b1a9c2ce76807972faf11125be76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.628ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}}"></span></dd></dl> <p>Bu halde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d5db8c46e4fe0053e49e9e263f0baae32edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.712ex; height:2.009ex;" alt="{\displaystyle x_{2}=a}"></span> ifadesiyle koşullu olan <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> şöyle özetlenen çokdeğişirli normal dağılım gösterir: </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_{1}|X_{2}=a)\sim N({\bar {\mu }},{\overline {\Sigma }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1}|X_{2}=a)\sim N({\bar {\mu }},{\overline {\Sigma }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0de68328df13ebd59410f801c6926cae106621dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.941ex; height:3.509ex;" alt="{\displaystyle (X_{1}|X_{2}=a)\sim N({\bar {\mu }},{\overline {\Sigma }})}"></span></dd></dl> <p>Burada </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 {\bar {\mu }}=\mu _{1}+\Sigma _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mu }}=\mu _{1}+\Sigma _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99068188ce2ba9155a48b719ee8b7be8362fc72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.084ex; height:3.343ex;" alt="{\displaystyle {\bar {\mu }}=\mu _{1}+\Sigma _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}"></span></dd></dl> <p>olur ve covaryans matrisi şöyle verilir: </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 {\overline {\Sigma }}=\Sigma _{11}-\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Σ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\Sigma }}=\Sigma _{11}-\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1acdf4c420f1190f4ab0f40a579ad36410b146fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.052ex; height:3.676ex;" alt="{\displaystyle {\overline {\Sigma }}=\Sigma _{11}-\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21}.}"></span></dd></dl> <p>Bu matris <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 {\mathbf {\Sigma } }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {\Sigma } }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d79435492d63d29ce2506eab5d7934b1414bf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\mathbf {\Sigma } }}"></span> içinde <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 {\mathbf {\Sigma } _{22}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {\Sigma } _{22}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43da912519bf4cd48d0598c23d66c0096a5e91c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.807ex; height:2.509ex;" alt="{\displaystyle {\mathbf {\Sigma } _{22}}}"></span> ifadesinin <a href="/w/index.php?title=Schur_tamamlay%C4%B1c%C4%B1s%C4%B1&action=edit&redlink=1" class="new" title="Schur tamamlayıcısı (sayfa mevcut değil)">Schur tamamlayıcısı</a> olur. </p><p>Bundan dikkati çekmesi gereken şu sonuçlar çıkartılır: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span> değerinin <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> olduğunu bilmek varyansı değiştirir. Daha şaşırtıcı olarak, ortalama değeri <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 _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07c8b8367dd3ba8476cba3655c162feecde5c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.288ex; height:3.343ex;" alt="{\displaystyle \Sigma _{12}\Sigma _{22}^{-1}\left(a-\mu _{2}\right)}"></span> ile kayma gösterir. Eğer <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> bilinmese idi, <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> nin göstereceği dağılım <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_{q}\left(\mu _{1},\Sigma _{11}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{q}\left(\mu _{1},\Sigma _{11}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1342b7f5f2f8306fdfef8dedcc9c466e7b6a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.095ex; height:3.009ex;" alt="{\displaystyle N_{q}\left(\mu _{1},\Sigma _{11}\right)}"></span> olurdu. </p><p><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 _{12}\Sigma _{22}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{12}\Sigma _{22}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46204ea60669f5d8fc055b499e606bbe29820889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.565ex; height:3.343ex;" alt="{\displaystyle \Sigma _{12}\Sigma _{22}^{-1}}"></span> matrisi <a href="/wiki/Regresyon_analizi" title="Regresyon analizi">regresyon</a> katsayıları olarak da bilinirler. </p> <div class="mw-heading mw-heading2"><h2 id="Fisher'in_enformasyon_matrisi"><span id="Fisher.27in_enformasyon_matrisi"></span>Fisher'in enformasyon matrisi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=11" title="Değiştirilen bölüm: Fisher'in enformasyon matrisi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=11" title="Bölümün kaynak kodunu değiştir: Fisher'in enformasyon matrisi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bir normal dağılım için <a href="/w/index.php?title=Fisher%27in_enformasyon_matrisi&action=edit&redlink=1" class="new" title="Fisher'in enformasyon matrisi (sayfa mevcut değil)">Fisher'in enformasyon matrisi</a> bir ozel sekil alir. <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 N(\mu (\theta ),\Sigma (\theta ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim N(\mu (\theta ),\Sigma (\theta ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b5e5ef2162455dbfc91a37c6e4812b9b6227a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.865ex; height:2.843ex;" alt="{\displaystyle X\sim N(\mu (\theta ),\Sigma (\theta ))}"></span> için Fisher'in enformasyon matrisinin <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,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274d4857135a7d28a94ba9ee8135779615084d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}"></span> elemanı su olur: </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 {\mathcal {I}}_{m,n}={\frac {\partial \mu }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \mu ^{\top }}{\partial \theta _{n}}}+{\frac {1}{2}}\mathrm {tr} \left(\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{n}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Σ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Σ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}_{m,n}={\frac {\partial \mu }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \mu ^{\top }}{\partial \theta _{n}}}+{\frac {1}{2}}\mathrm {tr} \left(\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{n}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4502e9eaa2471aca2b4b3b257c0b90de11a229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.069ex; width:49.418ex; height:6.343ex;" alt="{\displaystyle {\mathcal {I}}_{m,n}={\frac {\partial \mu }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \mu ^{\top }}{\partial \theta _{n}}}+{\frac {1}{2}}\mathrm {tr} \left(\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{n}}}\right)}"></span></dd></dl> <p>Burada </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mu }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}&{\frac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \mu _{N}}{\partial \theta _{m}}}&\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd /> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mu }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}&{\frac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \mu _{N}}{\partial \theta _{m}}}&\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de94483d44c3b9d8815db5336042e20765d242a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.836ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \mu }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}&{\frac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \mu _{N}}{\partial \theta _{m}}}&\end{bmatrix}}}"></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 {\frac {\partial \mu ^{\top }}{\partial \theta _{m}}}=\left({\frac {\partial \mu }{\partial \theta _{m}}}\right)^{\top }={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}\\\\{\frac {\partial \mu _{2}}{\partial \theta _{m}}}\\\\\vdots \\\\{\frac {\partial \mu _{N}}{\partial \theta _{m}}}\\\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mu ^{\top }}{\partial \theta _{m}}}=\left({\frac {\partial \mu }{\partial \theta _{m}}}\right)^{\top }={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}\\\\{\frac {\partial \mu _{2}}{\partial \theta _{m}}}\\\\\vdots \\\\{\frac {\partial \mu _{N}}{\partial \theta _{m}}}\\\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5101248a48676e871919df2a73b7d4ade93175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.171ex; width:29.076ex; height:31.509ex;" alt="{\displaystyle {\frac {\partial \mu ^{\top }}{\partial \theta _{m}}}=\left({\frac {\partial \mu }{\partial \theta _{m}}}\right)^{\top }={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \theta _{m}}}\\\\{\frac {\partial \mu _{2}}{\partial \theta _{m}}}\\\\\vdots \\\\{\frac {\partial \mu _{N}}{\partial \theta _{m}}}\\\\\end{bmatrix}}}"></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 {\frac {\partial \Sigma }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \Sigma _{1,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{1,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{1,N}}{\partial \theta _{m}}}\\\\{\frac {\partial \Sigma _{2,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{2,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{2,N}}{\partial \theta _{m}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial \Sigma _{N,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{N,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{N,N}}{\partial \theta _{m}}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Σ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Sigma }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \Sigma _{1,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{1,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{1,N}}{\partial \theta _{m}}}\\\\{\frac {\partial \Sigma _{2,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{2,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{2,N}}{\partial \theta _{m}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial \Sigma _{N,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{N,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{N,N}}{\partial \theta _{m}}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6818713a0923a306c96ae25d4798fe3d3bb358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:38.361ex; height:29.509ex;" alt="{\displaystyle {\frac {\partial \Sigma }{\partial \theta _{m}}}={\begin{bmatrix}{\frac {\partial \Sigma _{1,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{1,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{1,N}}{\partial \theta _{m}}}\\\\{\frac {\partial \Sigma _{2,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{2,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{2,N}}{\partial \theta _{m}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial \Sigma _{N,1}}{\partial \theta _{m}}}&{\frac {\partial \Sigma _{N,2}}{\partial \theta _{m}}}&\cdots &{\frac {\partial \Sigma _{N,N}}{\partial \theta _{m}}}\end{bmatrix}}}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812681cabb4e0ad561bc0e91a24e09d9ae15d6f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \mathrm {tr} }"></span> <a href="/w/index.php?title=Trace_(matris)&action=edit&redlink=1" class="new" title="Trace (matris) (sayfa mevcut değil)">trace</a> fonksiyonu olur.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Kullback-Leibler_ayrılımı"><span id="Kullback-Leibler_ayr.C4.B1l.C4.B1m.C4.B1"></span>Kullback-Leibler ayrılımı</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=12" title="Değiştirilen bölüm: Kullback-Leibler ayrılımı" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=12" title="Bölümün kaynak kodunu değiştir: Kullback-Leibler ayrılımı"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><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 N0_{N}(\mu _{0},\Sigma _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N0_{N}(\mu _{0},\Sigma _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e217df513b2d0b537bf97a6c9d781aabbda0d80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.949ex; height:2.843ex;" alt="{\displaystyle N0_{N}(\mu _{0},\Sigma _{0})}"></span> den <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 N1_{N}(\mu _{1},\Sigma _{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N1_{N}(\mu _{1},\Sigma _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd841fb1758da6d16b956024c7f04a92ea277034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.949ex; height:2.843ex;" alt="{\displaystyle N1_{N}(\mu _{1},\Sigma _{1})}"></span> dağılımına <a href="/w/index.php?title=Kullback-Leibler_ayr%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Kullback-Leibler ayrılımı (sayfa mevcut değil)">Kullback-Leibler ayrılımı</a> şöyle verilir: </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 D_{\text{KL}}(N0\|N1)={1 \over 2}\left(\log _{e}\left({\det \Sigma _{1} \over \det \Sigma _{0}}\right)+\mathrm {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)+\left(\mu _{1}-\mu _{0}\right)^{\top }\Sigma _{1}^{-1}(\mu _{1}-\mu _{0})-N\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>N</mi> <mn>0</mn> <mo fence="false" stretchy="false">‖</mo> <mi>N</mi> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo movablelimits="true" form="prefix">det</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>N</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{KL}}(N0\|N1)={1 \over 2}\left(\log _{e}\left({\det \Sigma _{1} \over \det \Sigma _{0}}\right)+\mathrm {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)+\left(\mu _{1}-\mu _{0}\right)^{\top }\Sigma _{1}^{-1}(\mu _{1}-\mu _{0})-N\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2957bfe9351e557f4f04e52a2bd131d3503cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:84.708ex; height:6.176ex;" alt="{\displaystyle D_{\text{KL}}(N0\|N1)={1 \over 2}\left(\log _{e}\left({\det \Sigma _{1} \over \det \Sigma _{0}}\right)+\mathrm {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)+\left(\mu _{1}-\mu _{0}\right)^{\top }\Sigma _{1}^{-1}(\mu _{1}-\mu _{0})-N\right).}"></span></dd></dl> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Parametrelerin_kestrimi">Parametrelerin kestrimi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=13" title="Değiştirilen bölüm: Parametrelerin kestrimi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=13" title="Bölümün kaynak kodunu değiştir: Parametrelerin kestrimi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cokdegisirli normal dağılımın kovaryansinin <a href="/wiki/Maksimum_olabilirlik" class="mw-redirect" title="Maksimum olabilirlik">maksimum olabilirlik</a> kestiriminin elde edilmesi şaşırtıcı şekilde düzenli ve zekice yapılmıştır. <a href="/w/index.php?title=Kovaryans_matrislerin_kestirimi&action=edit&redlink=1" class="new" title="Kovaryans matrislerin kestirimi (sayfa mevcut değil)">Kovaryans matrislerin kestirimi</a> maddesine bakın. Bir <i>N</i>-boyutlu cokludegisirli normal dağılımın olasılık yoğunluk fonksiyonu şöyle verilir: </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)=(2\pi )^{-N/2}\det(\Sigma )^{-1/2}\exp \left(-{1 \over 2}(x-\mu )^{T}\Sigma ^{-1}(x-\mu )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(2\pi )^{-N/2}\det(\Sigma )^{-1/2}\exp \left(-{1 \over 2}(x-\mu )^{T}\Sigma ^{-1}(x-\mu )\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c24a584dc789da49670db1b0b2f9b302663cf2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.844ex; height:6.176ex;" alt="{\displaystyle f(x)=(2\pi )^{-N/2}\det(\Sigma )^{-1/2}\exp \left(-{1 \over 2}(x-\mu )^{T}\Sigma ^{-1}(x-\mu )\right)}"></span></dd></dl> <p>ve kovaryans matrisinin maksimum olabilirlik kestirimi söyle yazılır: </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 {\widehat {\Sigma }}={1 \over n}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\Sigma }}={1 \over n}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4824f6e90473f638eb4bf6331fe5e2f39dd619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.149ex; height:6.843ex;" alt="{\displaystyle {\widehat {\Sigma }}={1 \over n}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}}"></span></dd></dl> <p>Bu basit olarak bir <i>n</i> büyüklüğünde bir örneklem için örneklem kovaryans matrisidir. Bu bir <a href="/w/index.php?title=Yanli_kestirim&action=edit&redlink=1" class="new" title="Yanli kestirim (sayfa mevcut değil)">yanli kestirim</a> olup beklenen değeri </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[{\widehat {\Sigma }}]={n-1 \over n}\Sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mi mathvariant="normal">Σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[{\widehat {\Sigma }}]={n-1 \over n}\Sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6792e687b17e3dcc425f941fe9271344cd472b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.404ex; height:5.176ex;" alt="{\displaystyle E[{\widehat {\Sigma }}]={n-1 \over n}\Sigma .}"></span></dd></dl> <p>Oliur. Bir yansız örneklem kovaryansi kestirmi sudur: </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 {\widehat {\Sigma }}={1 \over n-1}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\Sigma }}={1 \over n-1}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70d2a6bda16861ecbe5b22e0e5fb6b304e26264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.798ex; height:6.843ex;" alt="{\displaystyle {\widehat {\Sigma }}={1 \over n-1}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(X_{i}-{\overline {X}})^{T}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Entropi">Entropi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=14" title="Değiştirilen bölüm: Entropi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=14" title="Bölümün kaynak kodunu değiştir: Entropi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Çokdeğişirli normal dağılım için <a href="/w/index.php?title=Diferansiyel_entropi&action=edit&redlink=1" class="new" title="Diferansiyel entropi (sayfa mevcut değil)">diferansiyel entropi</a> ifadesi şöyle verilir:<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><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 h\left(f\right)=-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</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> <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> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\left(f\right)=-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1264a90035e8b43c0b485769c3e5c19188307ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.2ex; height:6.009ex;" alt="{\displaystyle h\left(f\right)=-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{2}}\left(N+N\ln \left(2\pi \right)+\ln \left|\Sigma \right|\right)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>+</mo> <mi>N</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mi mathvariant="normal">Σ</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}\left(N+N\ln \left(2\pi \right)+\ln \left|\Sigma \right|\right)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec5cdf922b34d0e6c9197f91ca135ce9309b55e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.166ex; width:27.777ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}\left(N+N\ln \left(2\pi \right)+\ln \left|\Sigma \right|\right)\!}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{2}}\ln\{(2\pi e)^{N}\left|\Sigma \right|\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>e</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mrow> <mo>|</mo> <mi mathvariant="normal">Σ</mi> <mo>|</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}\ln\{(2\pi e)^{N}\left|\Sigma \right|\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578b5a15a1ea7b04c9c79d818f55868cc1b54662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.541ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}\ln\{(2\pi e)^{N}\left|\Sigma \right|\}}"></span></dd></dl> <p>Burada <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 \left|\Sigma \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi mathvariant="normal">Σ</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\Sigma \right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71353694c7c8d764de548151cd633e4ebfeb539a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left|\Sigma \right|}"></span> covaryans matrisi olan <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 mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span>nın <a href="/w/index.php?title=Determan&action=edit&redlink=1" class="new" title="Determan (sayfa mevcut değil)">determani</a> olur: </p> <div class="mw-heading mw-heading2"><h2 id="Çokdeğişirli_normallik_sınamaları"><span id=".C3.87okde.C4.9Fi.C5.9Firli_normallik_s.C4.B1namalar.C4.B1"></span>Çokdeğişirli normallik sınamaları</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=15" title="Değiştirilen bölüm: Çokdeğişirli normallik sınamaları" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=15" title="Bölümün kaynak kodunu değiştir: Çokdeğişirli normallik sınamaları"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Çokdeğişirli <a href="/wiki/Normallik_s%C4%B1namalar%C4%B1" title="Normallik sınamaları">normallik sınamaları</a> bir verilmiş veri seti için bir teorik çokdeğişirli normal dağılıma benzerlik olup olmadığını sınamak için hazırlanmıştır. Bu sınamalarda <a href="/wiki/S%C4%B1f%C4%B1r_hipotez" class="mw-redirect" title="Sıfır hipotez">sıfır hipotez</a> veri setinin çokdeğişirli normal dağılıma benzerlik gösterdiğidir. Eğer sınama ile bulunan <a href="/wiki/P-de%C4%9Feri" class="mw-redirect" title="P-değeri">p-değeri</a> yeter derece küçük ise (yani genellikle 0,05 veya 0,01den daha küçük ise), sıfır hipotez reddedilir ve verinin çokludeğişirli normal dağılım göstermediği kabul edilir. Bu çokludeğişirli normallik sınamaları arasında popüler olan Cox-Small sınamasıdır:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Smith ve Jain'in Friedman-Rafsky testini adaptasyonu için şu referansa bakın: <sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dağılımdan_değerlerin_bulunması"><span id="Da.C4.9F.C4.B1l.C4.B1mdan_de.C4.9Ferlerin_bulunmas.C4.B1"></span>Dağılımdan değerlerin bulunması</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=16" title="Değiştirilen bölüm: Dağılımdan değerlerin bulunması" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=16" title="Bölümün kaynak kodunu değiştir: Dağılımdan değerlerin bulunması"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><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> ortalama vektörü ve (simetrik ve pozitif kesin olması gereken) <a href="/wiki/Kovaryans_matrisi" title="Kovaryans matrisi">kovaryans matrisi</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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> olan bir <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>-boyutlu çokdeğişirli normal dağılımdan bir rastgele vektör <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"> <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> çekmek için çok kullanılan bir yöntem şöyle uygulanır: </p> <ol><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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> için (matris kare kökü olan) <a href="/w/index.php?title=%C3%87oleski_dekompozisyonu&action=edit&redlink=1" class="new" title="Çoleski dekompozisyonu (sayfa mevcut değil)">Çoleski dekompozisyonu</a> hesap edilir. Yani <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\,A^{T}=\Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,A^{T}=\Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0702a80d59f36cdba335f9961ba7f31f3fe6897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.039ex; height:2.676ex;" alt="{\displaystyle A\,A^{T}=\Sigma }"></span> koşuluna uyan tek bir alt üçgensel matris olan <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> bulunur.</li> <li>Örneğin <a href="/w/index.php?title=Box-M%C3%BCller_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC&action=edit&redlink=1" class="new" title="Box-Müller dönüşümü (sayfa mevcut değil)">Box-Müller dönüşümü</a> ile üretilip elde edilebilen <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> tane birebirine <a href="/w/index.php?title=Ba%C4%9F%C4%B1msiz&action=edit&redlink=1" class="new" title="Bağımsiz (sayfa mevcut değil)">bağımsiz</a> <a href="/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım">normal dağılım</a> gösteren değişebilir parçalarından oluşan bir vektör <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=(z_{1},\dots ,z_{N})^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=(z_{1},\dots ,z_{N})^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a477c02b8eb90e18dc77e6bc12d24052687ef7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.064ex; height:3.176ex;" alt="{\displaystyle Z=(z_{1},\dots ,z_{N})^{T}}"></span> bulunur.</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, <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 +AZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>A</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +AZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2422f2a952bee6d8e26ffddd38475f0f3d10787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.666ex; height:2.676ex;" alt="{\displaystyle \mu +AZ}"></span> ifadesine eşit olarak bulunur.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Kaynakça"><span id="Kaynak.C3.A7a"></span>Kaynakça</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&veaction=edit&section=17" title="Değiştirilen bölüm: Kaynakça" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&section=17" title="Bölümün kaynak kodunu değiştir: Kaynakça"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r32805677">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-count:2}.mw-parser-output .reflist-columns-3{column-count:3}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><strong><a href="#cite_ref-1">^</a></strong> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="http://www.math.wsu.edu/faculty/genz/software/software.html">[1]</a> 15 Nisan 2008 tarihinde <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080415030646/http://www.math.wsu.edu/faculty/genz/software/software.html">arşivlendi</a>. (<a href="/wiki/FORTRAN" class="mw-redirect" title="FORTRAN">FORTRAN</a> yazılımlı <a href="/wiki/Kaynak_kodu" title="Kaynak kodu">kodu</a> kapsar.)</span> </li> <li id="cite_note-2"><strong><a href="#cite_ref-2">^</a></strong> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="http://alex.strashny.org/a/Multivariate-normal-cumulative-distribution-function-(cdf)-in-MATLAB.html">[2]</a> 13 Mayıs 2008 tarihinde <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080513222841/http://alex.strashny.org/a/Multivariate-normal-cumulative-distribution-function-(cdf)-in-MATLAB.html">arşivlendi</a>. ( <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> yazılımlı <a href="/wiki/Kaynak_kodu" title="Kaynak kodu">koduda</a> kapsar )</span> </li> <li id="cite_note-3"><strong><a href="#cite_ref-3">^</a></strong> <span class="reference-text"><cite class="kaynak web">Nikolaus Hansen. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110927081250/http://www.bionik.tu-berlin.de/user/niko/cmatutorial.pdf">"The CMA Evolution Strategy: A Tutorial"</a> <span style="font-size:85%;">(PDF)</span>. 27 Eylül 2011 tarihinde <a rel="nofollow" class="external text" href="http://www.bionik.tu-berlin.de/user/niko/cmatutorial.pdf">kaynağından</a> <span style="font-size:85%;">(PDF)</span> arşivlendi.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+CMA+Evolution+Strategy%3A+A+Tutorial&rft.au=Nikolaus+Hansen&rft_id=http%3A%2F%2Fwww.bionik.tu-berlin.de%2Fuser%2Fniko%2Fcmatutorial.pdf&rfr_id=info%3Asid%2Ftr.wikipedia.org%3A%C3%87okde%C4%9Fi%C5%9Firli+normal+da%C4%9F%C4%B1l%C4%B1m" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><strong><a href="#cite_ref-4">^</a></strong> <span class="reference-text"><cite class="kaynak dergi">Gokhale, DV (Mayıs 1989). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109/18.30996">"Entropy Expressions and Their Estimators for Multivariate Distributions"</a>. <i>Information Theory, IEEE Transactions on</i>. <b>35</b> (3). ss. 688-692.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Entropy+Expressions+and+Their+Estimators+for+Multivariate+Distributions&rft.pages=688-692&rft.date=1989-05&rft.aulast=Gokhale&rft.aufirst=DV&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1109%2F18.30996&rfr_id=info%3Asid%2Ftr.wikipedia.org%3A%C3%87okde%C4%9Fi%C5%9Firli+normal+da%C4%9F%C4%B1l%C4%B1m" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><strong><a href="#cite_ref-5">^</a></strong> <span class="reference-text"><cite class="kaynak dergi">Cox, D. R. (Ağustos 1978). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_biometrika_1978-08_65_2/page/263">"Testing multivariate normality (Çokdeğişirli normallik testi)"</a>. <i>Biometrika</i>. <b>65</b> (2). ss. 263-272.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Testing+multivariate+normality+%28%C3%87okde%C4%9Fi%C5%9Firli+normallik+testi%29&rft.pages=263-272&rft.date=1978-08&rft.aulast=Cox&rft.aufirst=D.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_biometrika_1978-08_65_2%2Fpage%2F263&rfr_id=info%3Asid%2Ftr.wikipedia.org%3A%C3%87okde%C4%9Fi%C5%9Firli+normal+da%C4%9F%C4%B1l%C4%B1m" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><strong><a href="#cite_ref-6">^</a></strong> <span class="reference-text"><cite class="kaynak dergi">Smith, Stephen P. (Eylül 1988). "A test to determine the multivariate normality of a dataset (Bir veri setinin çokdeğişirli normallik gösterip göstermediği için bir sınama)". <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>. <b>10</b> (5). ss. 757-761. <a href="/wiki/Say%C4%B1sal_nesne_tan%C4%B1mlay%C4%B1c%C4%B1s%C4%B1" title="Sayısal nesne tanımlayıcısı">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109/34.6789">10.1109/34.6789</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+test+to+determine+the+multivariate+normality+of+a+dataset+%28Bir+veri+setinin+%C3%A7okde%C4%9Fi%C5%9Firli+normallik+g%C3%B6sterip+g%C3%B6stermedi%C4%9Fi+i%C3%A7in+bir+s%C4%B1nama%29&rft.pages=757-761&rft.date=1988-09&rft.aulast=Smith&rft.aufirst=Stephen+P.&rfr_id=info%3Asid%2Ftr.wikipedia.org%3A%C3%87okde%C4%9Fi%C5%9Firli+normal+da%C4%9F%C4%B1l%C4%B1m" class="Z3988"><span style="display:none;"> </span></span></span> 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style="white-space:nowrap"> <a href="/w/index.php?title=Ters-gamma_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Ters-gamma dağılım (sayfa mevcut değil)">Ters gamma</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=L%C3%A9vy_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Lévy dağılımı (sayfa mevcut değil)">Lévy</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Log-normal_da%C4%9F%C4%B1l%C4%B1m" title="Log-normal dağılım">Log-normal</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Log-logistik_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Log-logistik dağılım (sayfa mevcut değil)">Log-logistik</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Maxwell%E2%80%93Boltzmann_da%C4%9F%C4%B1l%C4%B1m%C4%B1" class="mw-redirect" title="Maxwell–Boltzmann dağılımı">Maxwell-Boltzmann</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Maxwell_h%C4%B1z_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Maxwell hız dağılımı (sayfa mevcut değil)">Maxwell hız</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Nakagami_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Nakagami dağılımı (sayfa mevcut değil)">Nakagami</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Merkezsel_olmayan_ki-kare_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Merkezsel olmayan ki-kare dağılımı (sayfa mevcut değil)">Merkezsel olmayan ki-kare</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Pareto_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Pareto dağılımı">Pareto</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Faz-tipi_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Faz-tipi dağılımı (sayfa mevcut değil)">Faz-tipi</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Rayleigh_sa%C3%A7%C4%B1l%C4%B1m%C4%B1" class="mw-redirect" title="Rayleigh saçılımı">Rayleigh</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Relativistik_Breit%E2%80%93Wigner_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Relativistik Breit–Wigner dağılımı (sayfa mevcut değil)">Relativistik Breit–Wigner</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Rice_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Rice dağılımı">Rice</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Rosin%E2%80%93Rammler_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Rosin–Rammler dağılımı (sayfa mevcut değil)">Rosin–Rammler</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Kayd%C4%B1r%C4%B1lm%C4%B1%C5%9F_Gompertz_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Kaydırılmış Gompertz dağılımı (sayfa mevcut değil)">Kaydırılmış Gompertz</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Kesilmi%C5%9F_normal_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Kesilmiş normal dağılımı (sayfa mevcut değil)">Kesilmiş normal</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=2.tip_Gumbel_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="2.tip Gumbel dağılımı (sayfa mevcut değil)">2.tip Gumbel</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Weibull_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Weibull dağılımı">Weibull</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Wilks%27in_lambda_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Wilks'in lambda dağılımı (sayfa mevcut değil)">Wilks'in lambda</a> </span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1m%C4%B1#Tüm_reel_çizgi_üzerinde_desteklenenler" title="Olasılık dağılımı">Sürekli tek değişkenli ve <br /> (-∞,∞) arasındaki tüm reel doğru<br /> üzerinde destekli</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"><div> <p><span style="white-space:nowrap"> <a href="/wiki/Cauchy_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Cauchy dağılımı">Cauchy</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=U%C3%A7sal_de%C4%9Fer_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Uçsal değer dağılımı (sayfa mevcut değil)">Uçsal değer</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=%C3%9Cstel_g%C3%BC%C3%A7_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Üstel güç dağılımı (sayfa mevcut değil)">Üstel güç</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Fisher%27in_z-da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Fisher'in z-dağılımı (sayfa mevcut değil)">Fisher'in z</a>  <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Genelle%C5%9Ftirilmi%C5%9F_hiperbolik_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Genelleştirilmiş hiperbolik dağılım (sayfa mevcut değil)">Genelleştirilmiş hiperbolik </a>  <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Gumbel_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Gumbel dağılımı (sayfa mevcut değil)">Gumbel</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Hiperbolik_sekant_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Hiperbolik sekant dağılımı (sayfa mevcut değil)">Hiperbolik sekant</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Landau_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Landau dağılımı (sayfa mevcut değil)">Landau</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Laplace_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Laplace dağılımı">Laplace</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=L%C3%A9vy_%C3%A7arp%C4%B1k_alfa-dura%C4%9Fan_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Lévy çarpık alfa-durağan dağılım (sayfa mevcut değil)">Lévy çarpık alfa-durağan</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Logistik_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Logistik dağılımı (sayfa mevcut değil)">Logistik</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım">Normal (Gauss tipi)</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Normal-ters_Gauss-tipi_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Normal-ters Gauss-tipi dağılım (sayfa mevcut değil)">Normal ters Gauss-tipi</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=%C3%87arp%C4%B1k_normal_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Çarpık normal dağılım (sayfa mevcut değil)">Çarpık normal</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Student%27in_t_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Student'in t dağılımı">Student'in <i>t</i></a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=1.tip_Gumbel_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="1.tip Gumbel dağılımı (sayfa mevcut değil)">1.tip Gumbel</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Varyans-gamma_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Varyans-gamma dağılımı (sayfa mevcut değil)">Varyans-Gamma</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Voigt_profili&action=edit&redlink=1" class="new" title="Voigt profili (sayfa mevcut değil)">Voigt</a> </span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1m%C4%B1#Birleşik_dağılımlar" title="Olasılık dağılımı">Çok değişkenli (birleşik)</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"><div> <p><span style="white-space:nowrap"> <b>Ayrık:</b> <a href="/w/index.php?title=Ewens_%C3%B6rneklem_form%C3%BCl%C3%BC&action=edit&redlink=1" class="new" title="Ewens örneklem formülü (sayfa mevcut değil)">Ewens</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Beta-binom_modeli&action=edit&redlink=1" class="new" title="Beta-binom modeli (sayfa mevcut değil)">Beta-binom</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Multinom_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Multinom dağılım (sayfa mevcut değil)">Multinom</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_Polya_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Çokdeğişirli Polya dağılımı (sayfa mevcut değil)">Çokdeğişirli Polya</a></span><br /> <span style="white-space:nowrap"> <b>Sürekli:</b> <a href="/w/index.php?title=Dirichlet_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Dirichlet dağılımı (sayfa mevcut değil)">Dirichlet</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Genelle%C5%9Ftirilmi%C5%9F_Dirichlet_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Genelleştirilmiş Dirichlet dağılımı (sayfa mevcut değil)">Genelleştirilmiş Dirichlet</a> <b>·</b></span> <span style="white-space:nowrap"> <a class="mw-selflink selflink">Çokdeğişirli normal</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=%C3%87okde%C4%9Fi%C5%9Firli_Student_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Çokdeğişirli Student dağılım (sayfa mevcut değil)">Çokdeğişirli Student</a>  <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Normal-%C3%B6l%C3%A7eklenmi%C5%9F_ters_gamma_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Normal-ölçeklenmiş ters gamma dağılımı (sayfa mevcut değil)">normal-ölçeklenmiş ters gamma</a>  <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Normal-gamma_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Normal-gamma dağılım (sayfa mevcut değil)">Normal-gamma</a> </span><br /> <span style="white-space:nowrap"> <b><a href="/wiki/Olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1m%C4%B1#Matris-değerli_dağılımlar" title="Olasılık dağılımı">Matris-değerli</a>:</b> <a href="/w/index.php?title=Ters-Wishart_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Ters-Wishart dağılımı (sayfa mevcut değil)">Ters-Wishart</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Matris_normal_da%C4%9F%C4%B1l%C4%B1m" title="Matris normal dağılım">Matris normal</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Wishart_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Wishart dağılımı (sayfa mevcut değil)">Wishart</a> </span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Y%C3%B6nsel_istatistikler&action=edit&redlink=1" class="new" title="Yönsel istatistikler (sayfa mevcut değil)">Yönsel</a>, <a href="/wiki/Bozulmu%C5%9F_da%C4%9F%C4%B1l%C4%B1m" title="Bozulmuş dağılım">Bozulmuş</a> ve <a href="/w/index.php?title=Singuler_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Singuler dağılım (sayfa mevcut değil)">singuler</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"><div> <p><span style="white-space:nowrap"> <b><a href="/w/index.php?title=Y%C3%B6nsel_istatistikler&action=edit&redlink=1" class="new" title="Yönsel istatistikler (sayfa mevcut değil)">Yönsel</a>:</b> <a href="/w/index.php?title=Kent_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Kent dağılımı (sayfa mevcut değil)">Kent</a>  <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Von_Mises_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Von Mises dağılımı">von Mises</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Von_Mises%E2%80%93Fisher_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Von Mises–Fisher dağılımı (sayfa mevcut değil)">von Mises–Fisher</a> <br /> <b><a href="/wiki/Bozulmu%C5%9F_da%C4%9F%C4%B1l%C4%B1m" title="Bozulmuş dağılım">Bozulmuş</a>:</b> <a href="/wiki/Bozulmu%C5%9F_da%C4%9F%C4%B1l%C4%B1m" title="Bozulmuş dağılım">Ayrık bozulmuş</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Dirac_delta_fonksiyonu" title="Dirac delta fonksiyonu">Dirac delta fonksiyonu</a> <br /> <b><a href="/w/index.php?title=Singuler_da%C4%9F%C4%B1l%C4%B1m&action=edit&redlink=1" class="new" title="Singuler dağılım (sayfa mevcut değil)">Singuler</a>:</b> <a href="/w/index.php?title=Cantor_da%C4%9F%C4%B1l%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Cantor dağılımı (sayfa mevcut değil)">Cantor</a> <b>·</b></span> <span style="white-space:nowrap"> </span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Aileler</th><td class="navbox-list navbox-even hlist" 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