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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theorema_limitis_centralis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Theorema limitis centralis</span> </div> </a> <ul id="toc-Theorema_limitis_centralis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietates_parametrorum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietates_parametrorum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Proprietates parametrorum</span> </div> </a> <ul id="toc-Proprietates_parametrorum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ad_parametra_aestimanda" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ad_parametra_aestimanda"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ad parametra aestimanda</span> </div> </a> <ul id="toc-Ad_parametra_aestimanda-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functio_momenta_generans_et_momenta_centralia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Functio_momenta_generans_et_momenta_centralia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Functio momenta generans et momenta centralia</span> </div> </a> <ul id="toc-Functio_momenta_generans_et_momenta_centralia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mensurae_momentis_centralibus_fundatae" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mensurae_momentis_centralibus_fundatae"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Mensurae momentis centralibus fundatae</span> </div> </a> <ul id="toc-Mensurae_momentis_centralibus_fundatae-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nexus_interni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nexus_interni"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Nexus interni</span> </div> </a> <ul id="toc-Nexus_interni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notae" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notae"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notae</span> </div> </a> <ul id="toc-Notae-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="Index" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Distributio normalis</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 73 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-73" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">73 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung – Alemannic" lang="gsw" hreflang="gsw" data-title="Normalverteilung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D9%8A_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="توزيع احتمالي طبيعي – Arabica" lang="ar" hreflang="ar" data-title="توزيع احتمالي طبيعي" data-language-autonym="العربية" data-language-local-name="Arabica" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – Asturian" lang="ast" hreflang="ast" data-title="Distribución normal" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Normal_paylanma" title="Normal paylanma – Atropatenica" lang="az" hreflang="az" data-title="Normal paylanma" data-language-autonym="Azərbaycanca" data-language-local-name="Atropatenica" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%D9%88%D8%B1%D9%85%D8%A7%D9%84_%D8%AF%D8%A7%D8%BA%DB%8C%D9%84%DB%8C%D9%85" title="نورمال داغیلیم – South Azerbaijani" lang="azb" hreflang="azb" data-title="نورمال داغیلیم" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Нармальнае размеркаванне – Ruthenica Alba" lang="be" hreflang="be" data-title="Нармальнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="Ruthenica Alba" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормално разпределение – Bulgarica" lang="bg" hreflang="bg" data-title="Нормално разпределение" data-language-autonym="Български" data-language-local-name="Bulgarica" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela – Bosnica" lang="bs" hreflang="bs" data-title="Normalna raspodjela" data-language-autonym="Bosanski" data-language-local-name="Bosnica" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_normal" title="Distribució normal – Catalana" lang="ca" hreflang="ca" data-title="Distribució normal" data-language-autonym="Català" data-language-local-name="Catalana" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD" title="Normální rozdělení – Bohemica" lang="cs" hreflang="cs" data-title="Normální rozdělení" data-language-autonym="Čeština" data-language-local-name="Bohemica" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B2%D0%B0%D0%BB%D0%B5%C3%A7%C4%95%D0%B2%C4%95" title="Гаусс валеçĕвĕ – Chuvassica" lang="cv" hreflang="cv" data-title="Гаусс валеçĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvassica" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dosraniad_normal" title="Dosraniad normal – Cambrica" lang="cy" hreflang="cy" data-title="Dosraniad normal" data-language-autonym="Cymraeg" data-language-local-name="Cambrica" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – Danica" lang="da" hreflang="da" data-title="Normalfordeling" data-language-autonym="Dansk" data-language-local-name="Danica" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung – Germanica" lang="de" hreflang="de" data-title="Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="Germanica" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CE%BD%CE%BF%CE%BD%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE" title="Κανονική κατανομή – Graeca" lang="el" hreflang="el" data-title="Κανονική κατανομή" data-language-autonym="Ελληνικά" data-language-local-name="Graeca" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normal_distribution" title="Normal distribution – Anglica" lang="en" hreflang="en" data-title="Normal distribution" data-language-autonym="English" data-language-local-name="Anglica" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Normala_distribuo" title="Normala distribuo – Esperantica" lang="eo" hreflang="eo" data-title="Normala distribuo" data-language-autonym="Esperanto" data-language-local-name="Esperantica" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – Hispanica" lang="es" hreflang="es" data-title="Distribución normal" data-language-autonym="Español" data-language-local-name="Hispanica" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Normaaljaotus" title="Normaaljaotus – Estonica" lang="et" hreflang="et" data-title="Normaaljaotus" data-language-autonym="Eesti" data-language-local-name="Estonica" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Banaketa_normal" title="Banaketa normal – Vasconica" lang="eu" hreflang="eu" data-title="Banaketa normal" data-language-autonym="Euskara" data-language-local-name="Vasconica" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="توزیع نرمال – Persica" lang="fa" hreflang="fa" data-title="توزیع نرمال" data-language-autonym="فارسی" data-language-local-name="Persica" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Normaalijakauma" title="Normaalijakauma – Finnica" lang="fi" hreflang="fi" data-title="Normaalijakauma" data-language-autonym="Suomi" data-language-local-name="Finnica" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fr.wikipedia.org/wiki/Loi_normale" title="Loi normale – Francogallica" lang="fr" hreflang="fr" data-title="Loi normale" data-language-autonym="Français" data-language-local-name="Francogallica" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Normoolferdialang" title="Normoolferdialang – Northern Frisian" lang="frr" hreflang="frr" data-title="Normoolferdialang" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%A1ileadh_normalach" title="Dáileadh normalach – Hibernica" lang="ga" hreflang="ga" data-title="Dáileadh normalach" data-language-autonym="Gaeilge" data-language-local-name="Hibernica" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal – Gallaica" lang="gl" hreflang="gl" data-title="Distribución normal" data-language-autonym="Galego" data-language-local-name="Gallaica" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="התפלגות נורמלית – Hebrew" lang="he" hreflang="he" data-title="התפלגות נורמלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%AC%E0%A4%82%E0%A4%9F%E0%A4%A8" title="प्रसामान्य बंटन – Hindica" lang="hi" hreflang="hi" data-title="प्रसामान्य बंटन" data-language-autonym="हिन्दी" data-language-local-name="Hindica" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela – Croatica" lang="hr" hreflang="hr" data-title="Normalna raspodjela" data-language-autonym="Hrvatski" data-language-local-name="Croatica" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Norm%C3%A1lis_eloszl%C3%A1s" title="Normális eloszlás – Hungarica" lang="hu" hreflang="hu" data-title="Normális eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarica" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B8%D6%80%D5%B4%D5%A1%D5%AC_%D5%A2%D5%A1%D5%B7%D5%AD%D5%B8%D6%82%D5%B4" title="Նորմալ բաշխում – Armenica" lang="hy" hreflang="hy" data-title="Նորմալ բաշխում" data-language-autonym="Հայերեն" data-language-local-name="Armenica" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_normal" title="Distribusi normal – Indonesian" lang="id" hreflang="id" data-title="Distribusi normal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Normaldreifing" title="Normaldreifing – Islandica" lang="is" hreflang="is" data-title="Normaldreifing" data-language-autonym="Íslenska" data-language-local-name="Islandica" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_normale" title="Distribuzione normale – Italiana" lang="it" hreflang="it" data-title="Distribuzione normale" data-language-autonym="Italiano" data-language-local-name="Italiana" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83" title="正規分布 – Iaponica" lang="ja" hreflang="ja" data-title="正規分布" data-language-autonym="日本語" data-language-local-name="Iaponica" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%9D%E1%83%A0%E1%83%9B%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%9C%E1%83%90%E1%83%AC%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="ნორმალური განაწილება – Georgiana" lang="ka" hreflang="ka" data-title="ნორმალური განაწილება" data-language-autonym="ქართული" data-language-local-name="Georgiana" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%B0%D0%BB%D1%8B%D0%BF%D1%82%D1%8B_%D0%B4%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F" title="Қалыпты дисперсия – Cazachica" lang="kk" hreflang="kk" data-title="Қалыпты дисперсия" data-language-autonym="Қазақша" data-language-local-name="Cazachica" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EA%B7%9C_%EB%B6%84%ED%8F%AC" title="정규 분포 – Coreana" lang="ko" hreflang="ko" data-title="정규 분포" data-language-autonym="한국어" data-language-local-name="Coreana" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Distribuzzion_normala" title="Distribuzzion normala – Lombard" lang="lmo" hreflang="lmo" data-title="Distribuzzion normala" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Normalusis_skirstinys" title="Normalusis skirstinys – Lithuanica" lang="lt" hreflang="lt" data-title="Normalusis skirstinys" data-language-autonym="Lietuvių" data-language-local-name="Lithuanica" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Norm%C4%81lais_sadal%C4%ABjums" title="Normālais sadalījums – Lettonica" lang="lv" hreflang="lv" data-title="Normālais sadalījums" data-language-autonym="Latviešu" data-language-local-name="Lettonica" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B1%D0%B0" title="Нормална распределба – Macedonica" lang="mk" hreflang="mk" data-title="Нормална распределба" data-language-autonym="Македонски" data-language-local-name="Macedonica" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A4%B0%E0%A4%A3" title="सामान्य वितरण – Marathica" lang="mr" hreflang="mr" data-title="सामान्य वितरण" data-language-autonym="मराठी" data-language-local-name="Marathica" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Taburan_normal" title="Taburan normal – Malayana" lang="ms" hreflang="ms" data-title="Taburan normal" data-language-autonym="Bahasa Melayu" data-language-local-name="Malayana" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Normale_verdeling" title="Normale verdeling – Batava" lang="nl" hreflang="nl" data-title="Normale verdeling" data-language-autonym="Nederlands" data-language-local-name="Batava" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Normalfordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Normalfordeling" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_normalny" title="Rozkład normalny – Polonica" lang="pl" hreflang="pl" data-title="Rozkład normalny" data-language-autonym="Polski" data-language-local-name="Polonica" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Distribussion_%C3%ABd_Gauss" title="Distribussion ëd Gauss – Piedmontese" lang="pms" hreflang="pms" data-title="Distribussion ëd Gauss" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="pagina mensis"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_normal" title="Distribuição normal – Lusitana" lang="pt" hreflang="pt" data-title="Distribuição normal" data-language-autonym="Português" data-language-local-name="Lusitana" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Distribu%C8%9Bia_Gauss" title="Distribuția Gauss – Dacoromanica" lang="ro" hreflang="ro" data-title="Distribuția Gauss" data-language-autonym="Română" data-language-local-name="Dacoromanica" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормальное распределение – Russica" lang="ru" hreflang="ru" data-title="Нормальное распределение" data-language-autonym="Русский" data-language-local-name="Russica" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Normalna_raspodela" title="Normalna raspodela – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Normalna raspodela" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Normal_distribution" title="Normal distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Normal distribution" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Norm%C3%A1lne_rozdelenie" title="Normálne rozdelenie – Slovaca" lang="sk" hreflang="sk" data-title="Normálne rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="Slovaca" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Normalna_porazdelitev" title="Normalna porazdelitev – Slovena" lang="sl" hreflang="sl" data-title="Normalna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovena" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_normale" title="Shpërndarja normale – Albanica" lang="sq" hreflang="sq" data-title="Shpërndarja normale" data-language-autonym="Shqip" data-language-local-name="Albanica" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D1%81%D0%BF%D0%BE%D0%B4%D0%B5%D0%BB%D0%B0" title="Нормална расподела – Serbica" lang="sr" hreflang="sr" data-title="Нормална расподела" data-language-autonym="Српски / srpski" data-language-local-name="Serbica" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_normal" title="Sebaran normal – Sundanese" lang="su" hreflang="su" data-title="Sebaran normal" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Normalf%C3%B6rdelning" title="Normalfördelning – Suecica" lang="sv" hreflang="sv" data-title="Normalfördelning" data-language-autonym="Svenska" data-language-local-name="Suecica" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B2%E0%AF%8D%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="இயல்நிலைப் பரவல் – Tamulica" lang="ta" hreflang="ta" data-title="இயல்நிலைப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="Tamulica" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%88%E0%B8%81%E0%B9%81%E0%B8%88%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%81%E0%B8%95%E0%B8%B4" title="การแจกแจงปรกติ – Thai" lang="th" hreflang="th" data-title="การแจกแจงปรกติ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_normal" title="Distribusyong normal – Tagalog" lang="tl" hreflang="tl" data-title="Distribusyong normal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Normal_da%C4%9F%C4%B1l%C4%B1m" title="Normal dağılım – Turcica" lang="tr" hreflang="tr" data-title="Normal dağılım" data-language-autonym="Türkçe" data-language-local-name="Turcica" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B1%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D1%88%D0%B5" title="Гаусс бүленеше – Tatarica" lang="tt" hreflang="tt" data-title="Гаусс бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarica" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D1%80%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB" title="Нормальний розподіл – Ucrainica" lang="uk" hreflang="uk" data-title="Нормальний розподіл" data-language-autonym="Українська" data-language-local-name="Ucrainica" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%A7%D8%B1%D9%85%D9%84_%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="نارمل تقسیم – Urdu" lang="ur" hreflang="ur" data-title="نارمل تقسیم" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ph%E1%BB%91i_chu%E1%BA%A9n" title="Phân phối chuẩn – Vietnamica" lang="vi" hreflang="vi" data-title="Phân phối chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamica" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 – Wu" lang="wuu" hreflang="wuu" data-title="正态分布" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%90%D7%A8%D7%9E%D7%90%D7%9C%D7%A2_%D7%A4%D7%90%D7%A8%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A0%D7%92" title="נארמאלע פארטיילונג – Yiddish" lang="yi" hreflang="yi" data-title="נארמאלע פארטיילונג" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83" title="正态分布 – Sinica" lang="zh" hreflang="zh" data-title="正态分布" data-language-autonym="中文" data-language-local-name="Sinica" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%C3%B4ng-th%C3%A0i_hun-p%C3%B2%CD%98" title="Siông-thài hun-pò͘ – Minnan" lang="nan" hreflang="nan" data-title="Siông-thài hun-pò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B8%B8%E6%85%8B%E5%88%86%E4%BD%88" title="常態分佈 – Cantonese" lang="yue" hreflang="yue" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/ButtonOrange.svg/12px-ButtonOrange.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/ButtonOrange.svg/16px-ButtonOrange.svg.png 2x" data-file-width="480" data-file-height="480" /></a></span></div></div> </div> <div id="siteSub" class="noprint">E Vicipaedia</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="la" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fasciculus:Standard_deviation_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/350px-Standard_deviation_diagram.svg.png" decoding="async" width="350" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/525px-Standard_deviation_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/700px-Standard_deviation_diagram.svg.png 2x" data-file-width="400" data-file-height="200" /></a><figcaption><b>Densitas distributionis normalis (curva campanae forma Gaussiana) et deviatio canonica.</b> Color <a href="/wiki/Caeruleus" title="Caeruleus">caeruleus</a> opacus congruit erroribus a <a href="/wiki/Valor_medius_exspectatus" class="mw-redirect" title="Valor medius exspectatus">valore medio exspectato</a> μ intra unam <a href="/wiki/Deviatio_canonica" title="Deviatio canonica">deviationem canonicam</a> σ, <a href="/wiki/Probabilitas" title="Probabilitas">probabilitate</a> 68.3 %; caeruleus opacus plus caeruleus medianus, erroribus intra duas deviationes canonicas, probabilitate 95.4%; caeruleus opacus plus medianus plus clarus, erroribus intra tres deviationes, probabilitate 99.7 %.</figcaption></figure> <p><b>Distributio normalis</b> sive <b>distributio Gaussiana</b> est <a href="/wiki/Distributio_probabilistica" title="Distributio probabilistica">distributio probabilistica</a> continua et <a href="/wiki/Symmetria" title="Symmetria">symmetros</a>, cui congruit densitas probabilistica <span class="mwe-math-element"><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\colon \mathbb {R} \to \mathbb {R} _{0}^{+},\ x\mapsto f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} _{0}^{+},\ x\mapsto f(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb06e5edeb51eb231d2ed8e0cb58eff60c5f7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.416ex; height:3.176ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} _{0}^{+},\ x\mapsto f(x),}"></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 f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left[{\frac {x-\mu }{\sigma }}\right]^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left[{\frac {x-\mu }{\sigma }}\right]^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456bea210557fc2ce6162146007a0c96a2a2ec2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.104ex; height:7.509ex;" alt="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left[{\frac {x-\mu }{\sigma }}\right]^{2}\right),}"></span></dd></dl> <p>ubi <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <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/745951ff5642eef22ccff7952a964ec0be349202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \,\mu }"></span> est <a href="/wiki/Valor_medius_exspectatus" class="mw-redirect" title="Valor medius exspectatus">valor medius exspectatus</a> huius distributionis et <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7b69856011606cb91141da00d75255e94db0f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle \,\sigma }"></span> <a href="/wiki/Deviatio_canonica" title="Deviatio canonica">deviatio canonica</a>. Haec densitas, ob suam <a href="/wiki/Campana" title="Campana">campanae</a> similem formam, saepe etiam dicitur <i>curva campanae forma Gaussiana</i>. </p><p>Cum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8697a3864c2ce3049efc6fde36a7915e474ee206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.772ex; height:2.676ex;" alt="{\displaystyle \,\sigma ^{2}}"></span> deviationem canonicam quadratam designat, quae <a href="/w/index.php?title=Variantia_theoretica&action=edit&redlink=1" class="new" title="Variantia theoretica (non est haec pagina)">variantia</a> quoque appellatur; modo mathematico haec <a href="/w/index.php?title=Distributio&action=edit&redlink=1" class="new" title="Distributio (non est haec pagina)">distributio</a> etiam symbolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}"></span> describitur. <a href="/w/index.php?title=Variabile_fortuitum&action=edit&redlink=1" class="new" title="Variabile fortuitum (non est haec pagina)">Variabile fortuitum</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 \,X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <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/77c9459aca0b4981e1f3933b8354b37ea3049dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.367ex; height:2.176ex;" alt="{\displaystyle \,X}"></span> sic distributum deinde repraesentatur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aeea1216143061c89f6a1944928a0aeee1b9cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.63ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Distributio_normalis_canonica_et_usus_eius_ad_functionem_distributivam_calculandam">Distributio normalis canonica et usus eius ad functionem distributivam calculandam</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=1" title="Recensere partem: Distributio normalis canonica et usus eius ad functionem distributivam calculandam" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=1" title="Edit section's source code: Distributio normalis canonica et usus eius ad functionem distributivam calculandam"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Distributio normalis canonica</i> dicitur distributio cuius valor medius exspectatus est 0 cuiusque deviatio canonica est 1. <a href="/w/index.php?title=Densitas_probabilistica&action=edit&redlink=1" class="new" title="Densitas probabilistica (non est haec pagina)">Densitas probabilistica</a> huius distributionis est </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 \varphi (x)={\frac {1}{\sqrt {2\pi }}}\mathrm {e} ^{-x^{2}/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}\mathrm {e} ^{-x^{2}/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa61052e4f1aa8c1d0fe2efe5d2f726de69d06fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.63ex; height:6.176ex;" alt="{\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}\mathrm {e} ^{-x^{2}/2}.}"></span></dd></dl> <p>Cui congruit <a href="/w/index.php?title=Functio_distributiva&action=edit&redlink=1" class="new" title="Functio distributiva (non est haec pagina)">functio distributiva</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 \Phi \colon \mathbb {R} \to [0,1],\ x\mapsto \Phi (x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">↦</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi \colon \mathbb {R} \to [0,1],\ x\mapsto \Phi (x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be62a4ffb83426154afce6a00dd45a26498e0150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.679ex; height:2.843ex;" alt="{\displaystyle \Phi \colon \mathbb {R} \to [0,1],\ x\mapsto \Phi (x),}"></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 \Phi (x)=\int \limits _{-\infty }^{x}\varphi (t)\,\mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munderover> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x)=\int \limits _{-\infty }^{x}\varphi (t)\,\mathrm {d} t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71cdae2e2185d3558ce8b9cf7621090b95fb6231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.195ex; height:8.843ex;" alt="{\displaystyle \Phi (x)=\int \limits _{-\infty }^{x}\varphi (t)\,\mathrm {d} t,}"></span></dd></dl> <p>cuius valores in multis tabellis statisticis inspici possunt. </p><p>In parametrorum <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <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/745951ff5642eef22ccff7952a964ec0be349202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \,\mu }"></span> et <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7b69856011606cb91141da00d75255e94db0f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle \,\sigma }"></span> quorumlibet casu, quaequam <a href="/w/index.php?title=Functio_distributiva&action=edit&redlink=1" class="new" title="Functio distributiva (non est haec pagina)">functio distributiva</a> normalis <span class="mwe-math-element"><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\colon \mathbb {R} \to [0,1],\ x\mapsto F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon \mathbb {R} \to [0,1],\ x\mapsto F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ffaa025caf34f2ac574d84313a243f592e82d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.158ex; height:2.843ex;" alt="{\displaystyle F\colon \mathbb {R} \to [0,1],\ x\mapsto F(x)}"></span> ad eam canonicam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2da7bb82c61e971aa33b017426d9c8e4ded327e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.065ex; height:2.176ex;" alt="{\displaystyle \,\Phi }"></span> reduci potest per formulam </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)=P(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=P(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d531737424cc4466595d0b347c677bd044b5d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.968ex; height:6.176ex;" alt="{\displaystyle F(x)=P(X\leq x)=\Phi \left({\frac {x-\mu }{\sigma }}\right).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Theorema_limitis_centralis">Theorema limitis centralis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=2" title="Recensere partem: Theorema limitis centralis" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=2" title="Edit section's source code: Theorema limitis centralis"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Distributio normalis est maximi momenti in <a href="/wiki/Mathematica" title="Mathematica">mathematica</a> ob <a href="/w/index.php?title=Theorema_limitis_centralis&action=edit&redlink=1" class="new" title="Theorema limitis centralis (non est haec pagina)">Theorema limitis centralis</a>, quod dicit ad distributionem normalem tendere omnes distributiones de rebus quae quantitates independentes fortuitas summant vel quae earum valores medios aestimant, si conditiones certae, quae variantias pertinent, valent. Cum numeri elementi conlati in summa sunt maiores, distributio normalis accuratius approximat. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietates_parametrorum">Proprietates parametrorum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=3" title="Recensere partem: Proprietates parametrorum" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=3" title="Edit section's source code: Proprietates parametrorum"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parametrum <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <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/745951ff5642eef22ccff7952a964ec0be349202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \,\mu }"></span> designat centrum distributionis normalis. Densitas distributionis probabilistica duo puncta inflexionis ad <span class="mwe-math-element"><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 \pm \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>±</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \pm \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e204c02117a4f9075f3ee4fd83758cd771da083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.572ex; height:2.676ex;" alt="{\displaystyle \mu \pm \sigma }"></span> habet. </p> <div class="mw-heading mw-heading2"><h2 id="Ad_parametra_aestimanda">Ad parametra aestimanda</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=4" title="Recensere partem: Ad parametra aestimanda" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=4" title="Edit section's source code: Ad parametra aestimanda"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parametra <span class="mwe-math-element"><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"> <mspace width="thinmathspace" /> <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/745951ff5642eef22ccff7952a964ec0be349202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \,\mu }"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8697a3864c2ce3049efc6fde36a7915e474ee206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.772ex; height:2.676ex;" alt="{\displaystyle \,\sigma ^{2}}"></span> <a href="/w/index.php?title=Valor_medius&action=edit&redlink=1" class="new" title="Valor medius (non est haec pagina)">valore medio empirico</a> et <a href="/w/index.php?title=Variantia_empirica&action=edit&redlink=1" class="new" title="Variantia empirica (non est haec pagina)">variantia empirica</a> efficienter aestimantur. Hi valores empirici ex datis selectionis fortuitae calculari solent. </p><p><b>Exemplum historicum:</b> <a href="/wiki/Carolus_Fridericus_Gauss" title="Carolus Fridericus Gauss">Carolus Fridericus Gauss</a> in <a href="/wiki/Angulus" title="Angulus">angulos</a> metiendo versatus est, quod in pristina scida pecuniae pretio decem <a href="/w/index.php?title=Marca_Germanica&action=edit&redlink=1" class="new" title="Marca Germanica (non est haec pagina)">marcarum Germanicarum</a> videri potest. Mensiones erroribus subiectae erant, ut Gauss valorem medium plurium mensionum eiusdem anguli calcularet. Ad excusandum valore medio usum errores normaliter distributos esse obtinuit, quia sub hac distributione valorem medium esse aestimatorem efficientem scivit. Haec adsumptio etiam hodie in multis analysibus statisticis facta est. </p><p><a href="/w/index.php?title=Curva&action=edit&redlink=1" class="new" title="Curva (non est haec pagina)">Curva</a> quae distributionem normalem describit etiam <a href="/wiki/Curva_errorum" title="Curva errorum">curva errorum</a> vocatur. </p> <div class="mw-heading mw-heading2"><h2 id="Functio_momenta_generans_et_momenta_centralia">Functio momenta generans et momenta centralia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=5" title="Recensere partem: Functio momenta generans et momenta centralia" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=5" title="Edit section's source code: Functio momenta generans et momenta centralia"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Functio_momenta_generans" title="Functio momenta generans">Functio momenta generans</a> distributionis normalis est <span class="mwe-math-element"><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}\colon \mathbb {R} \to \mathbb {R} ^{+},\ t\mapsto M_{X}(t),}"> <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>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo stretchy="false">↦</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{X}\colon \mathbb {R} \to \mathbb {R} ^{+},\ t\mapsto M_{X}(t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8d8f66c3daedbf1aea215c433f53784c981120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.652ex; height:3.009ex;" alt="{\displaystyle M_{X}\colon \mathbb {R} \to \mathbb {R} ^{+},\ t\mapsto M_{X}(t),}"></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 M_{X}(t)=\operatorname {E} (\operatorname {e} ^{tX})=\operatorname {e} ^{\mu t+\sigma ^{2}t^{2}/2},}"> <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 mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi mathvariant="normal">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{X}(t)=\operatorname {E} (\operatorname {e} ^{tX})=\operatorname {e} ^{\mu t+\sigma ^{2}t^{2}/2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6511bafc0dad20347888e2df8f876e1619d31c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.999ex; height:3.509ex;" alt="{\displaystyle M_{X}(t)=\operatorname {E} (\operatorname {e} ^{tX})=\operatorname {e} ^{\mu t+\sigma ^{2}t^{2}/2},}"></span></dd></dl> <p>unde parametro <span class="mwe-math-element"><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 =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7362c5ba78891b1d3168efc4de7082a8c76dccb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.676ex;" alt="{\displaystyle \,\mu =0}"></span> haec <a href="/w/index.php?title=Momentum_(statistica)&action=edit&redlink=1" class="new" title="Momentum (statistica) (non est haec pagina)">momenta centralia</a> ordinis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b71acd0632140110d57428fc8e822d6f608734b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.117ex; height:2.176ex;" alt="{\displaystyle \,k\in \mathbb {N} }"></span> consequuntur: </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 _{k}=0,~~\mathrm {si~} k\mathrm {~est~impar} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mtext> </mtext> </mrow> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mtext> </mtext> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}=0,~~\mathrm {si~} k\mathrm {~est~impar} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/882b49101d8e88fdb012f95741f3fc32f791a3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.912ex; height:2.676ex;" alt="{\displaystyle \mu _{k}=0,~~\mathrm {si~} k\mathrm {~est~impar} ,}"></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 \mu _{k}={\frac {k!\cdot \sigma ^{k}}{(k/2)!\cdot 2^{k/2}}},~~\mathrm {si~} k\mathrm {~est~par} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> <mo>⋅</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mtext> </mtext> </mrow> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}={\frac {k!\cdot \sigma ^{k}}{(k/2)!\cdot 2^{k/2}}},~~\mathrm {si~} k\mathrm {~est~par} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1e54656085600fd3cfa733fdc940faf596b780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.57ex; height:6.843ex;" alt="{\displaystyle \mu _{k}={\frac {k!\cdot \sigma ^{k}}{(k/2)!\cdot 2^{k/2}}},~~\mathrm {si~} k\mathrm {~est~par} .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Mensurae_momentis_centralibus_fundatae">Mensurae momentis centralibus fundatae</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=6" title="Recensere partem: Mensurae momentis centralibus fundatae" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=6" title="Edit section's source code: Mensurae momentis centralibus fundatae"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Valor_medius_exspectatus" class="mw-redirect" title="Valor medius exspectatus">Valor medius exspectatus</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 \,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <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/745951ff5642eef22ccff7952a964ec0be349202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \,\mu }"></span> est primum momentum: <span class="mwe-math-element"><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_{1}=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,m_{1}=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d590ed1af1604a0a41e19a4a9c42b406bcc1b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.982ex; height:2.176ex;" alt="{\displaystyle \,m_{1}=\mu }"></span>; hoc non est momentum centrale; secundum momentum centrale est <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\mu _{2}=\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\mu _{2}=\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91da407dbfbb7c17ad616ffdec3fa38e9f42ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle \,\mu _{2}=\sigma ^{2}}"></span>, i.e. <a href="/w/index.php?title=Variantia_theoretica&action=edit&redlink=1" class="new" title="Variantia theoretica (non est haec pagina)">variantia</a>. </p><p><a href="/w/index.php?title=Obliquitas_(statistica)&action=edit&redlink=1" class="new" title="Obliquitas (statistica) (non est haec pagina)">Obliquitas</a> distributionis normalis est </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 \gamma _{1}={\frac {\mu _{3}}{\mu _{2}^{3/2}}}={\frac {\mu _{3}}{\sigma ^{3}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}={\frac {\mu _{3}}{\mu _{2}^{3/2}}}={\frac {\mu _{3}}{\sigma ^{3}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d595213ad458048a432d29ef70798eeda7cd42c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.944ex; height:6.509ex;" alt="{\displaystyle \gamma _{1}={\frac {\mu _{3}}{\mu _{2}^{3/2}}}={\frac {\mu _{3}}{\sigma ^{3}}}=0}"></span></dd></dl> <p>propter <a href="/wiki/Symmetria" title="Symmetria">symmetriam</a> et exsistentiam tertii momenti. </p><p>Distributionis normalis mensurae canonicae ad fornicem et excessum describendum sunt: </p> <ul><li>Fractio momentorum centralium (antea Graece <i><a href="/w/index.php?title=Kurtosis&action=edit&redlink=1" class="new" title="Kurtosis (non est haec pagina)">kurtosis</a></i> appellata): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}}=3;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}}=3;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dabc49690f3cbef1d5cc8697dc44d8f9e69eda88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.059ex; height:5.843ex;" alt="{\displaystyle \beta _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}}=3;}"></span></li> <li><a href="/w/index.php?title=Excessus_(statistica)&action=edit&redlink=1" class="new" title="Excessus (statistica) (non est haec pagina)">Excessus</a> (nunc Graece etiam <i>kurtosis</i> appellatur): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}={\frac {\mu _{4}}{\sigma _{\,}^{4}}}-3=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}={\frac {\mu _{4}}{\sigma _{\,}^{4}}}-3=0;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60dc82ef75c989330b30cdcc505ca6c5c06afbc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.56ex; height:5.509ex;" alt="{\displaystyle \gamma _{2}={\frac {\mu _{4}}{\sigma _{\,}^{4}}}-3=0;}"></span></li> <li><a href="/w/index.php?title=Fornix_(statistica)&action=edit&redlink=1" class="new" title="Fornix (statistica) (non est haec pagina)">Fornix</a> (Anglice: <i>arch</i><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>): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{S}={\frac {1}{2}}\left({\frac {\mu _{4}}{\sigma ^{4}}}-1\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{S}={\frac {1}{2}}\left({\frac {\mu _{4}}{\sigma ^{4}}}-1\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5541bb1a6e0c59dc4e5825f12b296a6f9fe2f76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.888ex; height:6.176ex;" alt="{\displaystyle \alpha _{S}={\frac {1}{2}}\left({\frac {\mu _{4}}{\sigma ^{4}}}-1\right)=1.}"></span></li></ul> <p>Postremae duae mensurae distributioni normali sic aptatae sunt, ut valores canonicos 0 et 1 habeant. Postrema mensura, fornix, rationabilior est, quia potentiae in mensura quarti momenti pares sunt, ut valores negativos adsumere non possint. Valores excessus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\gamma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\gamma _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f18ca3fa9e88765cb77316d197864e1e463bb84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.646ex; height:2.176ex;" alt="{\displaystyle \,\gamma _{2}}"></span>, iacent in <span class="mwe-math-element"><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,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-2,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e2607c8438791a3e661071ec5d2dd16be61633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle [-2,\infty )}"></span>, contra valores fornicis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\alpha _{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\alpha _{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332c6ef89698ec1ea469c513badb4e05bd5c87c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.167ex; height:2.009ex;" alt="{\displaystyle \,\alpha _{S}}"></span>, in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}"></span> iacent. </p> <div class="mw-heading mw-heading2"><h2 id="Nexus_interni">Nexus interni</h2></div> <ul><li><a href="/wiki/Carolus_Fridericus_Gauss" title="Carolus Fridericus Gauss">Carolus Fridericus Gauss</a></li> <li><a href="/w/index.php?title=Distributio_symmetros&action=edit&redlink=1" class="new" title="Distributio symmetros (non est haec pagina)">Distributio symmetros</a></li> <li><a href="/w/index.php?title=Fornix_(statistica)&action=edit&redlink=1" class="new" title="Fornix (statistica) (non est haec pagina)">Fornix (statistica)</a></li> <li><a href="/wiki/Methodus_quadratorum_minimorum" title="Methodus quadratorum minimorum">Methodus quadratorum minimorum</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notae">Notae</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributio_normalis&veaction=edit&section=7" title="Recensere partem: Notae" class="mw-editsection-visualeditor"><span>recensere</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributio_normalis&action=edit&section=7" title="Edit section's source code: Notae"><span>fontem recensere</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">M. Bachmaier et V. Guiard, "An alternative and generalized measure for the kurtosis and its advantages," <i>Statistical Papers</i> 41 (2000), pp. 37–52. <a href="/wiki/Berolinum" title="Berolinum">Berolini</a>, <a href="/wiki/Heidelberga" title="Heidelberga">Heidelbergae</a>, <a href="/wiki/Urbs_Novum_Eboracum" class="mw-redirect" title="Urbs Novum Eboracum">Novi Eboraci</a>: Springer.</span> </li> </ol></div></div> <p><br /> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Receptum de "<a dir="ltr" href="https://la.wikipedia.org/w/index.php?title=Distributio_normalis&oldid=3611341">https://la.wikipedia.org/w/index.php?title=Distributio_normalis&oldid=3611341</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Specialis:Categoriae" title="Specialis:Categoriae">Categoria</a>: <ul><li><a href="/wiki/Categoria:Res_statisticae" title="Categoria:Res statisticae">Res statisticae</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoriae celatae: <ul><li><a href="/wiki/Categoria:L_%3F%3F" title="Categoria:L ??">L ??</a></li><li><a href="/wiki/Categoria:Latinitas_%3F%3F_(nondum_censa)" title="Categoria:Latinitas ?? 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