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[y]" accesskey="y"><span>Közreműködések</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Vit%C3%A1m" title="Az általad használt IP-címről végrehajtott szerkesztések megvitatása [n]" accesskey="n"><span>Vitalap</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Wiki"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Tartalomjegyzék" 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">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</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">Bevezető</div> </a> </li> <li id="toc-A_normális_eloszlást_jellemző_függvények" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_normális_eloszlást_jellemző_függvények"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>A normális eloszlást jellemző függvények</span> </div> </a> <ul id="toc-A_normális_eloszlást_jellemző_függvények-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sűrűségfüggvényének_tulajdonságai" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sűrűségfüggvényének_tulajdonságai"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Sűrűségfüggvényének tulajdonságai</span> </div> </a> <ul id="toc-Sűrűségfüggvényének_tulajdonságai-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_normális_eloszlást_jellemző_számok" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_normális_eloszlást_jellemző_számok"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>A normális eloszlást jellemző számok</span> </div> </a> <ul id="toc-A_normális_eloszlást_jellemző_számok-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normális_eloszlású_valószínűségi_változó_néhány_fontosabb_tulajdonsága" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Normális_eloszlású_valószínűségi_változó_néhány_fontosabb_tulajdonsága"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Normális eloszlású valószínűségi változó néhány fontosabb tulajdonsága</span> </div> </a> <ul id="toc-Normális_eloszlású_valószínűségi_változó_néhány_fontosabb_tulajdonsága-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Megjelenése_máshol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Megjelenése_máshol"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Megjelenése máshol</span> </div> </a> <ul id="toc-Megjelenése_máshol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Források</span> </div> </a> <ul id="toc-Források-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-További_információk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_információk"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>További információk</span> </div> </a> <ul id="toc-További_információk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kapcsolódó_szócikkek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kapcsolódó_szócikkek"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Kapcsolódó szócikkek</span> </div> </a> <ul id="toc-Kapcsolódó_szócikkek-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Normális eloszlás</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="Ugrás egy más nyelvű szócikkre. Elérhető 73 nyelven" > <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 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normal_distribution" title="Normal distribution – angol" lang="en" hreflang="en" data-title="Normal distribution" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung – svájci német" lang="gsw" hreflang="gsw" data-title="Normalverteilung" data-language-autonym="Alemannisch" data-language-local-name="svájci német" 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 – arab" lang="ar" hreflang="ar" data-title="توزيع احتمالي طبيعي" data-language-autonym="العربية" data-language-local-name="arab" 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 – asztúr" lang="ast" hreflang="ast" data-title="Distribución normal" data-language-autonym="Asturianu" data-language-local-name="asztúr" 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 – azerbajdzsáni" lang="az" hreflang="az" data-title="Normal paylanma" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" 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 – belarusz" lang="be" hreflang="be" data-title="Нармальнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="belarusz" 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 – bolgár" lang="bg" hreflang="bg" data-title="Нормално разпределение" data-language-autonym="Български" data-language-local-name="bolgár" 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 – bosnyák" lang="bs" hreflang="bs" data-title="Normalna raspodjela" data-language-autonym="Bosanski" data-language-local-name="bosnyák" 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 – katalán" lang="ca" hreflang="ca" data-title="Distribució normal" data-language-autonym="Català" data-language-local-name="katalán" 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 – cseh" lang="cs" hreflang="cs" data-title="Normální rozdělení" data-language-autonym="Čeština" data-language-local-name="cseh" 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 – csuvas" lang="cv" hreflang="cv" data-title="Гаусс валеçĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" 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 – walesi" lang="cy" hreflang="cy" data-title="Dosraniad normal" data-language-autonym="Cymraeg" data-language-local-name="walesi" 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 – dán" lang="da" hreflang="da" data-title="Normalfordeling" data-language-autonym="Dansk" data-language-local-name="dán" 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 – német" lang="de" hreflang="de" data-title="Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="német" 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 – görög" lang="el" hreflang="el" data-title="Κανονική κατανομή" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Normala_distribuo" title="Normala distribuo – eszperantó" lang="eo" hreflang="eo" data-title="Normala distribuo" data-language-autonym="Esperanto" data-language-local-name="eszperantó" 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 – spanyol" lang="es" hreflang="es" data-title="Distribución normal" data-language-autonym="Español" data-language-local-name="spanyol" 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 – észt" lang="et" hreflang="et" data-title="Normaaljaotus" data-language-autonym="Eesti" data-language-local-name="észt" 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 – baszk" lang="eu" hreflang="eu" data-title="Banaketa normal" data-language-autonym="Euskara" data-language-local-name="baszk" 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 – perzsa" lang="fa" hreflang="fa" data-title="توزیع نرمال" data-language-autonym="فارسی" data-language-local-name="perzsa" 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 – finn" lang="fi" hreflang="fi" data-title="Normaalijakauma" data-language-autonym="Suomi" data-language-local-name="finn" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="jó szócikk"><a href="https://fr.wikipedia.org/wiki/Loi_normale" title="Loi normale – francia" lang="fr" hreflang="fr" data-title="Loi normale" data-language-autonym="Français" data-language-local-name="francia" 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 – északi fríz" lang="frr" hreflang="frr" data-title="Normoolferdialang" data-language-autonym="Nordfriisk" data-language-local-name="északi fríz" 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 – ír" lang="ga" hreflang="ga" data-title="Dáileadh normalach" data-language-autonym="Gaeilge" data-language-local-name="ír" 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 – gallego" lang="gl" hreflang="gl" data-title="Distribución normal" data-language-autonym="Galego" data-language-local-name="gallego" 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 – héber" lang="he" hreflang="he" data-title="התפלגות נורמלית" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%E0%A4%AC%E0%A4%82%E0%A4%9F%E0%A4%A8" title="प्रसामान्य बंटन – hindi" lang="hi" hreflang="hi" data-title="प्रसामान्य बंटन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela – horvát" lang="hr" hreflang="hr" data-title="Normalna raspodjela" data-language-autonym="Hrvatski" data-language-local-name="horvát" class="interlanguage-link-target"><span>Hrvatski</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 – örmény" lang="hy" hreflang="hy" data-title="Նորմալ բաշխում" data-language-autonym="Հայերեն" data-language-local-name="örmény" 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 – indonéz" lang="id" hreflang="id" data-title="Distribusi normal" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" 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 – izlandi" lang="is" hreflang="is" data-title="Normaldreifing" data-language-autonym="Íslenska" data-language-local-name="izlandi" 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 – olasz" lang="it" hreflang="it" data-title="Distribuzione normale" data-language-autonym="Italiano" data-language-local-name="olasz" 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 – japán" lang="ja" hreflang="ja" data-title="正規分布" data-language-autonym="日本語" data-language-local-name="japán" 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 – grúz" lang="ka" hreflang="ka" data-title="ნორმალური განაწილება" data-language-autonym="ქართული" data-language-local-name="grúz" 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 – kazah" lang="kk" hreflang="kk" data-title="Қалыпты дисперсия" data-language-autonym="Қазақша" data-language-local-name="kazah" 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 – koreai" lang="ko" hreflang="ko" data-title="정규 분포" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Distributio_normalis" title="Distributio normalis – latin" lang="la" hreflang="la" data-title="Distributio normalis" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Distribuzzion_normala" title="Distribuzzion normala – 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 – litván" lang="lt" hreflang="lt" data-title="Normalusis skirstinys" data-language-autonym="Lietuvių" data-language-local-name="litván" 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 – lett" lang="lv" hreflang="lv" data-title="Normālais sadalījums" data-language-autonym="Latviešu" data-language-local-name="lett" 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 – macedón" lang="mk" hreflang="mk" data-title="Нормална распределба" data-language-autonym="Македонски" data-language-local-name="macedón" 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 – maráthi" lang="mr" hreflang="mr" data-title="सामान्य वितरण" data-language-autonym="मराठी" data-language-local-name="maráthi" 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 – maláj" lang="ms" hreflang="ms" data-title="Taburan normal" data-language-autonym="Bahasa Melayu" data-language-local-name="maláj" 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 – holland" lang="nl" hreflang="nl" data-title="Normale verdeling" data-language-autonym="Nederlands" data-language-local-name="holland" 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 – norvég (nynorsk)" lang="nn" hreflang="nn" data-title="Normalfordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="norvég (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 – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Normalfordeling" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (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 – lengyel" lang="pl" hreflang="pl" data-title="Rozkład normalny" data-language-autonym="Polski" data-language-local-name="lengyel" 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="kiemelt szócikk"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_normal" title="Distribuição normal – portugál" lang="pt" hreflang="pt" data-title="Distribuição normal" data-language-autonym="Português" data-language-local-name="portugál" 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 – román" lang="ro" hreflang="ro" data-title="Distribuția Gauss" data-language-autonym="Română" data-language-local-name="román" 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 – orosz" lang="ru" hreflang="ru" data-title="Нормальное распределение" data-language-autonym="Русский" data-language-local-name="orosz" 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 – szerbhorvát" lang="sh" hreflang="sh" data-title="Normalna raspodela" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="szerbhorvát" 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 – szlovák" lang="sk" hreflang="sk" data-title="Normálne rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="szlovák" 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 – szlovén" lang="sl" hreflang="sl" data-title="Normalna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="szlovén" 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 – albán" lang="sq" hreflang="sq" data-title="Shpërndarja normale" data-language-autonym="Shqip" data-language-local-name="albán" 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 – szerb" lang="sr" hreflang="sr" data-title="Нормална расподела" data-language-autonym="Српски / srpski" data-language-local-name="szerb" 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 – szundanéz" lang="su" hreflang="su" data-title="Sebaran normal" data-language-autonym="Sunda" data-language-local-name="szundanéz" 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 – svéd" lang="sv" hreflang="sv" data-title="Normalfördelning" data-language-autonym="Svenska" data-language-local-name="svéd" 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 – tamil" lang="ta" hreflang="ta" data-title="இயல்நிலைப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="tamil" 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 – török" lang="tr" hreflang="tr" data-title="Normal dağılım" data-language-autonym="Türkçe" data-language-local-name="török" 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 – tatár" lang="tt" hreflang="tt" data-title="Гаусс бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="tatár" 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 – ukrán" lang="uk" hreflang="uk" data-title="Нормальний розподіл" data-language-autonym="Українська" data-language-local-name="ukrán" 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 – vietnámi" lang="vi" hreflang="vi" data-title="Phân phối chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" 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 kínai" lang="wuu" hreflang="wuu" data-title="正态分布" data-language-autonym="吴语" data-language-local-name="wu kínai" 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 – jiddis" lang="yi" hreflang="yi" data-title="נארמאלע פארטיילונג" data-language-autonym="ייִדיש" data-language-local-name="jiddis" 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 – kínai" lang="zh" hreflang="zh" data-title="正态分布" data-language-autonym="中文" data-language-local-name="kínai" 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 – min nan kínai" lang="nan" hreflang="nan" data-title="Siông-thài hun-pò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan kínai" 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 – kantoni" lang="yue" hreflang="yue" data-title="常態分佈" data-language-autonym="粵語" data-language-local-name="kantoni" 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							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Speci%C3%A1lis:Badtitle/Message">ellenőrizve</a>: <i>2023. június 1.</i> <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Badtitle/Message&oldid=26165537&diff=cur">3 változtatás</a> vár ellenőrzésre.<p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Normal_distribution_pdf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Normal_distribution_pdf.svg/300px-Normal_distribution_pdf.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Normal_distribution_pdf.svg/450px-Normal_distribution_pdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Normal_distribution_pdf.svg/600px-Normal_distribution_pdf.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>A normális eloszlás sűrűségfüggvénye, ha <div class="legend"><span class="legend-color" style="display:inline-block; width:1.5em; height:1.5em; margin:1px 0; border:1px solid black; background-color: #FF7E7E; color:black; font-size:90%; text-align:center;"> </span> m = 0 és σ² = 0,2</div> <div class="legend"><span class="legend-color" style="display:inline-block; width:1.5em; height:1.5em; margin:1px 0; border:1px solid black; background-color: #70B770; color:black; font-size:90%; text-align:center;"> </span> m = 0 és σ² = 1 (standard normális eloszlás)</div> <div class="legend"><span class="legend-color" style="display:inline-block; width:1.5em; height:1.5em; margin:1px 0; border:1px solid black; background-color: #7D7DFF; color:black; font-size:90%; text-align:center;"> </span> m = 0 és σ² = 5</div> <div class="legend"><span class="legend-color" style="display:inline-block; width:1.5em; height:1.5em; margin:1px 0; border:1px solid black; background-color: #7CFFFF; color:black; font-size:90%; text-align:center;"> </span> m = –2 és σ² = 0,5</div></figcaption></figure> <p>Az <i>X</i> <a href="/wiki/Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gi_v%C3%A1ltoz%C3%B3" title="Valószínűségi változó">valószínűségi változó</a> <b>normális eloszlás</b>t követ – vagy rövidebben: normális eloszlású – pontosan akkor, ha <a href="/wiki/S%C5%B1r%C5%B1s%C3%A9gf%C3%BCggv%C3%A9ny" title="Sűrűségfüggvény">sűrűségfüggvénye</a> </p> <center><span class="mwe-math-element"><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 }}}}\cdot e^{-{\frac {(x-m)^{2}}{2\sigma ^{2}}}},}"> <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> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot e^{-{\frac {(x-m)^{2}}{2\sigma ^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6630d6dc545504a89748833809f7d0aa79825ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.681ex; height:7.176ex;" alt="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot e^{-{\frac {(x-m)^{2}}{2\sigma ^{2}}}},}"></span></center> <p>ahol a két paraméter, <i>m</i> és <i><a href="/wiki/Szigma" title="Szigma">σ</a></i> ∈ <b><a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">R</a></b>, valamint <i><a href="/wiki/Szigma" title="Szigma">σ</a></i> > 0. A normális eloszlást szokták <b>Gauss-eloszlás</b>nak vagy néha <b>normál eloszlás</b>nak is nevezni. </p><p>Azt, hogy az <i>X</i> valószínűségi változó normális eloszlást követ, a következő módon szoktuk jelölni: </p> <center><span class="mwe-math-element"><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}}(m,\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>m</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}}(m,\sigma ^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c6b860bf1fc497f64c179bf8c8199d96249ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.269ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(m,\sigma ^{2}).}"></span></center> <p>Speciálisan, ha <i>X</i> ~ <i>N</i>(0, 1), akkor <i>X</i>-et <a href="/wiki/Standard_norm%C3%A1lis_eloszl%C3%A1s" class="mw-redirect" title="Standard normális eloszlás">standard normális eloszlásúnak</a> (vagy sztenderd normális eloszlásúnak) nevezzük. </p><p>A fenti sűrűségfüggvény grafikonját alakja miatt szokás <b>haranggörbé</b>nek nevezni. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="A_normális_eloszlást_jellemző_függvények"><span id="A_norm.C3.A1lis_eloszl.C3.A1st_jellemz.C5.91_f.C3.BCggv.C3.A9nyek"></span>A normális eloszlást jellemző függvények</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=1" title="Szakasz szerkesztése: A normális eloszlást jellemző függvények"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Eloszl%C3%A1sf%C3%BCggv%C3%A9ny" title="Eloszlásfüggvény">Eloszlásfüggvénye</a> </p> <center><span class="mwe-math-element"><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)=\int \limits _{-\infty }^{x}{\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot e^{-{\frac {(t-m)^{2}}{2\sigma ^{2}}}}dt\left(=\int \limits _{-\infty }^{x}f(t)dt\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> <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> <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> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <mi>d</mi> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <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>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\int \limits _{-\infty }^{x}{\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot e^{-{\frac {(t-m)^{2}}{2\sigma ^{2}}}}dt\left(=\int \limits _{-\infty }^{x}f(t)dt\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680c0397c4e2c090f2c6ec86d65ac8f587bfcb90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.05ex; height:9.176ex;" alt="{\displaystyle F(x)=\int \limits _{-\infty }^{x}{\frac {1}{\sigma {\sqrt {2\pi }}}}\cdot e^{-{\frac {(t-m)^{2}}{2\sigma ^{2}}}}dt\left(=\int \limits _{-\infty }^{x}f(t)dt\right)}"></span></center> <p><a href="/wiki/Karakterisztikus_f%C3%BCggv%C3%A9ny_(val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gsz%C3%A1m%C3%ADt%C3%A1s)" title="Karakterisztikus függvény (valószínűségszámítás)">Karakterisztikus függvénye</a> </p> <center><span class="mwe-math-element"><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 (t)=e^{itm-{\frac {\sigma ^{2}t^{2}}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)=e^{itm-{\frac {\sigma ^{2}t^{2}}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1867de16c950d07dab5d73aacc0b91ee2d5e899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.149ex; height:4.509ex;" alt="{\displaystyle \varphi (t)=e^{itm-{\frac {\sigma ^{2}t^{2}}{2}}}}"></span></center> <div class="mw-heading mw-heading2"><h2 id="Sűrűségfüggvényének_tulajdonságai"><span id="S.C5.B1r.C5.B1s.C3.A9gf.C3.BCggv.C3.A9ny.C3.A9nek_tulajdons.C3.A1gai"></span>Sűrűségfüggvényének tulajdonságai</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=2" title="Szakasz szerkesztése: Sűrűségfüggvényének tulajdonságai"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Maximumhelye <i>m</i> (de nem emiatt lesz az eloszlás <a href="/wiki/V%C3%A1rhat%C3%B3_%C3%A9rt%C3%A9k" title="Várható érték">várható értéke</a> is <i>m</i>, az egybeesés a szimmetriának köszönhető).</li> <li><a href="/wiki/Szimmetria" title="Szimmetria">Szimmetrikus</a> a maximumhelyére vonatkozóan.</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 (\forall x\in \mathbb {R} )f(x)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall x\in \mathbb {R} )f(x)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7ed2ab6c2e25af8135da14b65a434b3c8edb4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.629ex; height:2.843ex;" alt="{\displaystyle (\forall x\in \mathbb {R} )f(x)>0}"></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 \lim _{x\to -\infty }f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1d79067550b15d37a608056218f3d7a760710a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.571ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to -\infty }f(x)=0}"></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 \lim _{x\to \infty }f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca224044bfa1b0bcdd732c49816668da7fb96db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.292ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)=0}"></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 \int _{-\infty }^{\infty }f(x)dx=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>d</mi> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(x)dx=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3476049f123f74d83f499899f459eef56a34cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.058ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)dx=1}"></span></li> <li><a href="/wiki/Folytonos_f%C3%BCggv%C3%A9ny" title="Folytonos függvény">Folytonos függvény</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="A_normális_eloszlást_jellemző_számok"><span id="A_norm.C3.A1lis_eloszl.C3.A1st_jellemz.C5.91_sz.C3.A1mok"></span>A normális eloszlást jellemző számok</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=3" title="Szakasz szerkesztése: A normális eloszlást jellemző számok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/V%C3%A1rhat%C3%B3_%C3%A9rt%C3%A9k" title="Várható érték">Várható értéke</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (X)=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (X)=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c89a1bd98525109bf7312995564415d896b61ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.685ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} (X)=m}"></span></dd></dl> <p><a href="/wiki/Sz%C3%B3r%C3%A1s_(val%C3%B3sz%C3%ADn%C5%B1s%C3%A9gsz%C3%A1m%C3%ADt%C3%A1s)" title="Szórás (valószínűségszámítás)">Szórása</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} (X)=\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} (X)=\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f56d94b959889d5ae9db01f3c617ecb4e53369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.267ex; height:2.843ex;" alt="{\displaystyle \mathbf {D} (X)=\sigma }"></span></dd></dl> <p><a href="/wiki/Momentum_(matematika)" title="Momentum (matematika)">Momentumai</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} \left(X^{p}\right)={\begin{cases}0&{\text{ha }}p{\text{ páratlan,}}\\\sigma ^{p}\,(p-1)!!&{\text{ha }}p{\text{ páros.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ha </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> páratlan,</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ha </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> páros.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} \left(X^{p}\right)={\begin{cases}0&{\text{ha }}p{\text{ páratlan,}}\\\sigma ^{p}\,(p-1)!!&{\text{ha }}p{\text{ páros.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a800a4ea6702bf4893e687bedc6eda85c49b0ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.155ex; height:7.509ex;" alt="{\displaystyle \mathrm {E} \left(X^{p}\right)={\begin{cases}0&{\text{ha }}p{\text{ páratlan,}}\\\sigma ^{p}\,(p-1)!!&{\text{ha }}p{\text{ páros.}}\end{cases}}}"></span></dd></dl> <p><a href="/wiki/Abszol%C3%BAt_momentum" title="Abszolút momentum">Abszolút momentumai</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left(|X|^{p}\right)=\sigma ^{p}\,(p-1)!!\cdot \left.{\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{ha }}p{\text{ páratlan}}\\1&{\text{ha }}p{\text{ páros}}\end{cases}}\right\}=\sigma ^{p}\cdot {\frac {2^{\frac {p}{2}}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>⋅</mo> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </msqrt> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ha </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> páratlan</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>ha </mtext> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> páros</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(|X|^{p}\right)=\sigma ^{p}\,(p-1)!!\cdot \left.{\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{ha }}p{\text{ páratlan}}\\1&{\text{ha }}p{\text{ páros}}\end{cases}}\right\}=\sigma ^{p}\cdot {\frac {2^{\frac {p}{2}}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3b9096a1289012e49aab5a7405899c1eeda46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:67.722ex; height:9.343ex;" alt="{\displaystyle \operatorname {E} \left(|X|^{p}\right)=\sigma ^{p}\,(p-1)!!\cdot \left.{\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{ha }}p{\text{ páratlan}}\\1&{\text{ha }}p{\text{ páros}}\end{cases}}\right\}=\sigma ^{p}\cdot {\frac {2^{\frac {p}{2}}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}}"></span></dd></dl> <p><a href="/wiki/Ferdes%C3%A9g" title="Ferdeség">Ferdesége</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}(X)=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}(X)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8d41cc52895f1d438c54d3ca7429b07ac3ea51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.807ex; height:2.843ex;" alt="{\displaystyle \beta _{1}(X)=0\,}"></span></dd></dl> <p><a href="/wiki/Lapults%C3%A1g" title="Lapultság">Lapultsága</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}(X)=0\,}"> <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 stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}(X)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3854b2394533ed342ace6e1f86f6d41956cc3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.807ex; height:2.843ex;" alt="{\displaystyle \beta _{2}(X)=0\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Normális_eloszlású_valószínűségi_változó_néhány_fontosabb_tulajdonsága"><span id="Norm.C3.A1lis_eloszl.C3.A1s.C3.BA_val.C3.B3sz.C3.ADn.C5.B1s.C3.A9gi_v.C3.A1ltoz.C3.B3_n.C3.A9h.C3.A1ny_fontosabb_tulajdons.C3.A1ga"></span>Normális eloszlású valószínűségi változó néhány fontosabb tulajdonsága</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=4" title="Szakasz szerkesztése: Normális eloszlású valószínűségi változó néhány fontosabb tulajdonsága"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Ha <i>X</i> ~ <i>N</i>(<i>m</i>, σ²), akkor bármilyen nullától különböző <a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">valós</a> <i>a</i> és bármilyen valós <i>b</i> szám esetén az <i>Y</i> = <i>aX</i> + <i>b</i> valószínűségi változó is normális eloszlást követ, pontosabban <i>Y</i> ~ <i>N</i>(<i>am</i> + <i>b</i>, <i>a</i>²σ²).<br /> Az eloszlás eme tulajdonságán alapul a <b>standardizálás</b> módszere: ha <i>X</i> ~ <i>N</i>(<i>m</i>, σ²), akkor (<i>X</i>–<i>m</i>)/σ ~ <i>N</i>(0, 1).</li> <li>Normális eloszlású <a href="/w/index.php?title=F%C3%BCggetlens%C3%A9g_(matematika)&action=edit&redlink=1" class="new" title="Függetlenség (matematika) (a lap nem létezik)">független</a> valószínűségi változók összege is normális eloszlású. Pontosabban ha <i>X</i><sub>1</sub> ~ <i>N</i>(<i>m</i><sub>1</sub>, σ<sub>1</sub>²) és <i>X</i><sub>2</sub> ~ <i>N</i>(<i>m</i><sub>2</sub>, σ<sub>2</sub>²) független valószínűségi változók, akkor <i>X</i><sub>1</sub> + <i>X</i><sub>2</sub> ~ <i>N</i>(<i>m</i><sub>1</sub> + <i>m</i><sub>2</sub>, σ<sub>1</sub>² + σ<sub>2</sub>²).</li></ul> <ul><li>Fordítva: ha <i>X</i><sub>1</sub> és <i>X</i><sub>2</sub> független valószínűségi változó, és <i>X</i><sub>1</sub> + <i>X</i><sub>2</sub> normális eloszlású, akkor <i>X</i><sub>1</sub> is és <i>X</i><sub>2</sub> is normális eloszlású.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Megjelenése_máshol"><span id="Megjelen.C3.A9se_m.C3.A1shol"></span>Megjelenése máshol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=5" title="Szakasz szerkesztése: Megjelenése máshol"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:Error mw:File/Thumb"><a href="/w/index.php?title=Speci%C3%A1lis:Felt%C3%B6lt%C3%A9s&wpDestFile=10_DM_Serie4_Vorderseite.jpg" class="new" title="Fájl:10 DM Serie4 Vorderseite.jpg"><span class="mw-file-element mw-broken-media" data-width="220">Fájl:10 DM Serie4 Vorderseite.jpg</span></a><figcaption>A korábbi 10 <a href="/wiki/N%C3%A9met_m%C3%A1rka" title="Német márka">márkás</a> <a href="/wiki/Bankjegy" title="Bankjegy">bankjegy</a> <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> portréjával</figcaption></figure> <p><a href="/wiki/1989" title="1989">1989</a>-ben a <a href="/wiki/Deutsche_Bundesbank" title="Deutsche Bundesbank">Német Szövetségi Bank</a> olyan 10 <a href="/wiki/N%C3%A9met_m%C3%A1rka" title="Német márka">márkás</a> <a href="/wiki/Bankjegy" title="Bankjegy">bankjegyet</a> bocsátott ki, melyen <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> képe mellett a standard normális eloszlás sűrűségfüggvényének grafikonja és képlete is látható.<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> Ez a bankjegy <a href="/wiki/2001" title="2001">2001</a>-ig volt forgalomban, amikor is <a href="/wiki/N%C3%A9metorsz%C3%A1g" title="Németország">Németország</a> áttért az <a href="/wiki/Eur%C3%B3" title="Euró">euróra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=6" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation"><a rel="nofollow" class="external text" href="https://old.nationalcurvebank.org/gaussdist/gaussdist.htm"><i>The Gaussian Distribution</i></a>. The National Curve Bank. (Hozzáférés: 2022. január 26.)</span></span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=7" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Fazekas István (szerk.): <i>Bevezetés a matematikai statisztikába</i> (Kossuth Egyetemi Kiadó, Debrecen, <a href="/wiki/2000" title="2000">2000</a>)</li> <li>Lukács Ottó: <i>Matematikai statisztika</i> (Műszaki, <a href="/wiki/2002" title="2002">2002</a>) <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9631630366" title="Speciális:Könyvforrások/9631630366">ISBN 963-16-3036-6</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="További_információk"><span id="Tov.C3.A1bbi_inform.C3.A1ci.C3.B3k"></span>További információk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=8" title="Szakasz szerkesztése: További információk"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="tarsprojekt-alap noviewer"> <style data-mw-deduplicate="TemplateStyles:r27252708">.mw-parser-output .tarsprojekt-alap{float:right;clear:right;width:272px;display:flex;column-gap:10px;margin:0 0 1em 1em;border:1px solid #a2a9b1;padding:4px;background-color:var(--background-color-neutral-subtle,#f8f9fa);color:var(--color-base,#202122);font-size:95%}@media screen and (max-width:719px){.mw-parser-output .tarsprojekt-alap{float:none;width:auto;margin:0 0 1em}}@media print{.mw-parser-output .tarsprojekt-alap{display:none}}</style> <div class="tarsprojekt-alap-ikon"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/Category:Normal_distribution" title="commons:Category:Normal distribution"><img alt="Commons:Category:Normal distribution" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/50px-Commons-logo.svg.png" decoding="async" width="50" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/75px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/100px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="tarsprojekt-alap-szoveg">A <a href="/wiki/Wikim%C3%A9dia_Commons" title="Wikimédia Commons">Wikimédia Commons</a> tartalmaz <i><b><a href="https://commons.wikimedia.org/wiki/Category:Normal_distribution" class="extiw" title="commons:Category:Normal distribution">Normális eloszlás</a></b></i> témájú médiaállományokat.</div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20151208122113/http://siva.bgk.uni-obuda.hu/jegyzetek/Matematika/Gepesz/Ngepesz/masodik_felev/standard_normalis_eloszlas.pdf">A standard normális eloszlású változó eloszlásfüggvényének táblázata</a></li> <li><a rel="nofollow" class="external text" href="http://nagysandor.eu/AsimovTeka/RandVarExp/index.html">Interaktív Java szimuláció a normális (és további 10 folytonos) eloszlás tanulmányozásához</a>. 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Szerzők: Kyle Siegrist & Dawn Duehring</li> <li><a rel="nofollow" class="external text" href="http://nagysandor.eu/AsimovTeka/N01/index.html">Interaktív Flash szimuláció a standard normális eloszlásértékekről (magyarított)</a>. Szerző: Jim Reed</li> <li><a rel="nofollow" class="external text" href="http://www.elektro-energetika.cz/calculations/no.php?language=magyar">Online kalkulátor Normális eloszlás</a>. Szerző: René Vápeník</li></ul> <div class="mw-heading mw-heading2"><h2 id="Kapcsolódó_szócikkek"><span id="Kapcsol.C3.B3d.C3.B3_sz.C3.B3cikkek"></span>Kapcsolódó szócikkek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lis_eloszl%C3%A1s&action=edit&section=9" title="Szakasz szerkesztése: Kapcsolódó szócikkek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/68%E2%80%9395%E2%80%9399,7_szab%C3%A1ly" title="68–95–99,7 szabály">68–95–99,7 szabály</a></li> <li><a href="/wiki/Kh%C3%AD-n%C3%A9gyzet_eloszl%C3%A1s" title="Khí-négyzet eloszlás">Khí-négyzet eloszlás</a></li> <li><a href="/wiki/Centr%C3%A1lis_hat%C3%A1reloszl%C3%A1s-t%C3%A9tel" title="Centrális határeloszlás-tétel">Centrális határeloszlás-tétel</a></li> <li><a href="/wiki/Log-norm%C3%A1lis_eloszl%C3%A1s" title="Log-normális eloszlás">Log-normális eloszlás</a></li></ul> <div class="noprint noviewer" style="overflow: hidden; clear: both;"><div style="margin-left:0; margin-right:2px;"><ul style="display:block; list-style-image:none; list-style-type:none; width:100%; vertical-align:middle; margin:0; padding:0; min-height: 27px;"><li style="float:left; min-height: 27px; line-height:25px; width:100%; margin:0; margin-top:.5em; margin-left:0; margin-right:0; padding:0; border:1px solid #CCF; background-color:#F0EEFF"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:P_cartesian_graph.svg" class="mw-file-description" title="Matematika"><img alt="Matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/25px-P_cartesian_graph.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/38px-P_cartesian_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/50px-P_cartesian_graph.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Matematikaportál</a></b> • összefoglaló, színes tartalomajánló lap</li></ul></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Normális_eloszlás&oldid=26165537">https://hu.wikipedia.org/w/index.php?title=Normális_eloszlás&oldid=26165537</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9g-eloszl%C3%A1sok" title="Kategória:Valószínűség-eloszlások">Valószínűség-eloszlások</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Rejtett kategória: <ul><li><a href="/wiki/Kateg%C3%B3ria:Hib%C3%A1s_f%C3%A1jlhivatkoz%C3%A1sokat_tartalmaz%C3%B3_sz%C3%B3cikkek" title="Kategória:Hibás fájlhivatkozásokat tartalmazó szócikkek">Hibás fájlhivatkozásokat tartalmazó szócikkek</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A lap utolsó módosítása: 2023. június 1., 19:50</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. 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