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href="//tools.wmflabs.org/pageviews?project=ur.wikipedia.org&platform=all-access&agent=user&range=latest-20&pages=%D8%B5%D9%81%D8%AD%DB%82_%D8%A7%D9%88%D9%84"><span>تعداد ناظرین</span></a></li><li id="n-اپلوڈ-تصویر" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D8%A7%D9%BE%D9%84%D9%88%DA%88"><span>اپلوڈ تصویر</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%D8%B5%D9%81%D8%AD%DB%82_%D8%A7%D9%88%D9%84" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="ویکیپیڈیا" src="/static/images/mobile/copyright/wikipedia-wordmark-ur.svg" style="width: 5.4375em; height: 2em;"> <img class="mw-logo-tagline" alt="آزاد دائرۃ المعارف" src="/static/images/mobile/copyright/wikipedia-tagline-ur.svg" width="86" height="16" style="width: 5.375em; height: 1em;"> </span> </a> </div> <div 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class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%AF%D8%A7%D8%AE%D9%84_%DB%81%D9%88%DA%BA&returnto=%D8%AA%D9%82%D8%B3%DB%8C%D9%85+%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="کھاتے میں داخل ہونے کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں [o]" accesskey="o" class=""><span>داخل ہوں</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="مزید اختیارات" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="ذاتی آلات" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">ذاتی آلات</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ur.wikipedia.org&uselang=ur"><span>عطیہ دیجیے</span></a></li><li id="pt-contribute" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:Contribute"><span class="vector-icon mw-ui-icon-edit mw-ui-icon-wikimedia-edit"></span> <span>تعاون</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%AA%D8%AE%D9%84%DB%8C%D9%82_%DA%A9%DA%BE%D8%A7%D8%AA%DB%81&returnto=%D8%AA%D9%82%D8%B3%DB%8C%D9%85+%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="کھاتہ بنانے یا اس میں داخل ہونے کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>کھاتہ بنائیں</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%AF%D8%A7%D8%AE%D9%84_%DB%81%D9%88%DA%BA&returnto=%D8%AA%D9%82%D8%B3%DB%8C%D9%85+%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="کھاتے میں داخل ہونے کی سفارش کی جاتی ہے؛ تاہم یہ ضروری نہیں [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>داخل ہوں</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/%D9%85%D8%B9%D8%A7%D9%88%D9%86%D8%AA:%D8%AA%D8%B9%D8%A7%D8%B1%D9%81" aria-label="ترمیم کاری کے متعلق مزید تفصیلات"><span>مزید تفصیلات</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D8%AE%D8%A7%D8%B5:%D9%85%DB%8C%D8%B1%D8%A7_%D8%AA%D8%A8%D8%A7%D8%AF%D9%84%DB%81_%D8%AE%DB%8C%D8%A7%D9%84" title="اس آئی پی پتے کی ترامیم سے متعلق گفتگو [n]" accesskey="n"><span>تبادلۂ خیال</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="ویب گاہ"> <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="مندرجات" 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">مندرجات</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">بغلی پٹی پر جائیں</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">چھپائیں</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">دیباچہ</div> </a> </li> <li id="toc-متفرد" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#متفرد"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>متفرد</span> </div> </a> <button aria-controls="toc-متفرد-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>متفرد کے ذیلی قطعہ پر جائیں</span> </button> <ul id="toc-متفرد-sublist" class="vector-toc-list"> <li id="toc-مثال:_دو_رقمی_توزیعِ_احتمال" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#مثال:_دو_رقمی_توزیعِ_احتمال"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>مثال: دو رقمی توزیعِ احتمال</span> </div> </a> <ul id="toc-مثال:_دو_رقمی_توزیعِ_احتمال-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-متواصل" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#متواصل"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>متواصل</span> </div> </a> <button aria-controls="toc-متواصل-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>متواصل کے ذیلی قطعہ پر جائیں</span> </button> <ul id="toc-متواصل-sublist" class="vector-toc-list"> <li id="toc-مثال" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#مثال"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>مثال</span> </div> </a> <ul id="toc-مثال-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-مزید_دیکھیے" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#مزید_دیکھیے"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>مزید دیکھیے</span> </div> </a> <ul id="toc-مزید_دیکھیے-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="مندرجات" 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="فہرست عناوین دیکھیں" > <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">فہرست عناوین دیکھیں</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">تقسیم احتمال</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 56 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-56" 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">56 زبانیں</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="توزيع احتمال – عربی" lang="ar" hreflang="ar" data-title="توزيع احتمال" data-language-autonym="العربية" data-language-local-name="عربی" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84" title="توزیع احتمال – فارسی" lang="fa" hreflang="fa" data-title="توزیع احتمال" data-language-autonym="فارسی" data-language-local-name="فارسی" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Probability_distribution" title="Probability distribution – انگریزی" lang="en" hreflang="en" data-title="Probability distribution" data-language-autonym="English" data-language-local-name="انگریزی" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilid%C3%A1" title="Distribución de probabilidá – اسٹوریائی" lang="ast" hreflang="ast" data-title="Distribución de probabilidá" data-language-autonym="Asturianu" data-language-local-name="اسٹوریائی" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A7%8D%E0%A6%AD%E0%A6%BE%E0%A6%AC%E0%A6%A8%E0%A6%BE_%E0%A6%AC%E0%A6%BF%E0%A6%A8%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8" title="সম্ভাবনা বিন্যাস – بنگلہ" lang="bn" hreflang="bn" data-title="সম্ভাবনা বিন্যাস" data-language-autonym="বাংলা" data-language-local-name="بنگلہ" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5_%D1%96%D0%BC%D0%B0%D0%B2%D0%B5%D1%80%D0%BD%D0%B0%D1%81%D1%86%D0%B5%D0%B9" title="Размеркаванне імавернасцей – بیلاروسی" lang="be" hreflang="be" data-title="Размеркаванне імавернасцей" data-language-autonym="Беларуская" data-language-local-name="بیلاروسی" 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%A0%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8%D1%82%D0%B5" title="Разпределение на вероятностите – بلغاری" lang="bg" hreflang="bg" data-title="Разпределение на вероятностите" data-language-autonym="Български" data-language-local-name="بلغاری" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_de_probabilitat" title="Distribució de probabilitat – کیٹالان" lang="ca" hreflang="ca" data-title="Distribució de probabilitat" data-language-autonym="Català" data-language-local-name="کیٹالان" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%83%D0%BB%D0%B0%D1%8F%D1%81%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D0%B2%D0%B0%D0%BB%D0%B5%C3%A7%C4%95%D0%B2%C4%95" title="Пулаяслăхсен валеçĕвĕ – چوواش" lang="cv" hreflang="cv" data-title="Пулаяслăхсен валеçĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="چوواش" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rozd%C4%9Blen%C3%AD_pravd%C4%9Bpodobnosti" title="Rozdělení pravděpodobnosti – چیک" lang="cs" hreflang="cs" data-title="Rozdělení pravděpodobnosti" data-language-autonym="Čeština" data-language-local-name="چیک" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dosbarthiad_tebygolrwydd" title="Dosbarthiad tebygolrwydd – ویلش" lang="cy" hreflang="cy" data-title="Dosbarthiad tebygolrwydd" data-language-autonym="Cymraeg" data-language-local-name="ویلش" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Wahrscheinlichkeitsverteilung" title="Wahrscheinlichkeitsverteilung – جرمن" lang="de" hreflang="de" data-title="Wahrscheinlichkeitsverteilung" data-language-autonym="Deutsch" data-language-local-name="جرمن" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Jaotus_(matemaatika)" title="Jaotus (matemaatika) – اسٹونین" lang="et" hreflang="et" data-title="Jaotus (matemaatika)" data-language-autonym="Eesti" data-language-local-name="اسٹونین" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE_%CF%80%CE%B9%CE%B8%CE%B1%CE%BD%CF%8C%CF%84%CE%B7%CF%84%CE%B1%CF%82" title="Κατανομή πιθανότητας – یونانی" lang="el" hreflang="el" data-title="Κατανομή πιθανότητας" data-language-autonym="Ελληνικά" data-language-local-name="یونانی" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad" title="Distribución de probabilidad – ہسپانوی" lang="es" hreflang="es" data-title="Distribución de probabilidad" data-language-autonym="Español" data-language-local-name="ہسپانوی" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Probablodistribuo" title="Probablodistribuo – ایسپرانٹو" lang="eo" hreflang="eo" data-title="Probablodistribuo" data-language-autonym="Esperanto" data-language-local-name="ایسپرانٹو" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Probabilitate-banaketa" title="Probabilitate-banaketa – باسکی" lang="eu" hreflang="eu" data-title="Probabilitate-banaketa" data-language-autonym="Euskara" data-language-local-name="باسکی" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="بہترین مضمون"><a href="https://fr.wikipedia.org/wiki/Loi_de_probabilit%C3%A9" title="Loi de probabilité – فرانسیسی" lang="fr" hreflang="fr" data-title="Loi de probabilité" data-language-autonym="Français" data-language-local-name="فرانسیسی" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidade" title="Distribución de probabilidade – گالیشیائی" lang="gl" hreflang="gl" data-title="Distribución de probabilidade" data-language-autonym="Galego" data-language-local-name="گالیشیائی" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%95%EB%A5%A0_%EB%B6%84%ED%8F%AC" title="확률 분포 – کوریائی" lang="ko" hreflang="ko" data-title="확률 분포" data-language-autonym="한국어" data-language-local-name="کوریائی" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%BE%D5%A1%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%A2%D5%A1%D5%B7%D5%AD%D5%B8%D6%82%D5%B4" title="Հավանականության բաշխում – آرمینیائی" lang="hy" hreflang="hy" data-title="Հավանականության բաշխում" data-language-autonym="Հայերեն" data-language-local-name="آرمینیائی" class="interlanguage-link-target"><span>Հայերեն</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%BE%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%82%E0%A4%9F%E0%A4%A8" title="प्रायिकता बंटन – ہندی" lang="hi" hreflang="hi" data-title="प्रायिकता बंटन" data-language-autonym="हिन्दी" data-language-local-name="ہندی" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sebaran_probabilitas" title="Sebaran probabilitas – انڈونیثیائی" lang="id" hreflang="id" data-title="Sebaran probabilitas" data-language-autonym="Bahasa Indonesia" data-language-local-name="انڈونیثیائی" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variabile_casuale#Distribuzione_di_probabilità" title="Variabile casuale – اطالوی" lang="it" hreflang="it" data-title="Variabile casuale" data-language-autonym="Italiano" data-language-local-name="اطالوی" class="interlanguage-link-target"><span>Italiano</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" title="התפלגות – عبرانی" lang="he" hreflang="he" data-title="התפלגות" data-language-autonym="עברית" data-language-local-name="عبرانی" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%9A%E1%83%91%E1%83%90%E1%83%97%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="ალბათური განაწილება – جارجیائی" lang="ka" hreflang="ka" data-title="ალბათური განაწილება" data-language-autonym="ქართული" data-language-local-name="جارجیائی" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Distributio_probabilistica" title="Distributio probabilistica – لاطینی" lang="la" hreflang="la" data-title="Distributio probabilistica" data-language-autonym="Latina" data-language-local-name="لاطینی" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skirstinys" title="Skirstinys – لیتھوینین" lang="lt" hreflang="lt" data-title="Skirstinys" data-language-autonym="Lietuvių" data-language-local-name="لیتھوینین" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3sz%C3%ADn%C5%B1s%C3%A9g-eloszl%C3%A1s" title="Valószínűség-eloszlás – ہنگیرین" lang="hu" hreflang="hu" data-title="Valószínűség-eloszlás" data-language-autonym="Magyar" data-language-local-name="ہنگیرین" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D0%BE%D1%98%D0%B0%D1%82%D0%BD%D0%BE%D1%81%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="Веројатносна распределба – مقدونیائی" lang="mk" hreflang="mk" data-title="Веројатносна распределба" data-language-autonym="Македонски" data-language-local-name="مقدونیائی" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Taburan_kebarangkalian" title="Taburan kebarangkalian – مالے" lang="ms" hreflang="ms" data-title="Taburan kebarangkalian" data-language-autonym="Bahasa Melayu" data-language-local-name="مالے" 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/Kansverdeling" title="Kansverdeling – ڈچ" lang="nl" hreflang="nl" data-title="Kansverdeling" data-language-autonym="Nederlands" data-language-local-name="ڈچ" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E5%88%86%E5%B8%83" title="確率分布 – جاپانی" lang="ja" hreflang="ja" data-title="確率分布" data-language-autonym="日本語" data-language-local-name="جاپانی" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sannsynlighetsfordeling" title="Sannsynlighetsfordeling – نارویجین بوکمل" lang="nb" hreflang="nb" data-title="Sannsynlighetsfordeling" data-language-autonym="Norsk bokmål" data-language-local-name="نارویجین بوکمل" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sannsynsfordeling" title="Sannsynsfordeling – نارویجین نینورسک" lang="nn" hreflang="nn" data-title="Sannsynsfordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="نارویجین نینورسک" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_prawdopodobie%C5%84stwa" title="Rozkład prawdopodobieństwa – پولش" lang="pl" hreflang="pl" data-title="Rozkład prawdopodobieństwa" data-language-autonym="Polski" data-language-local-name="پولش" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="منتخب مضمون"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_de_probabilidade" title="Distribuição de probabilidade – پُرتگالی" lang="pt" hreflang="pt" data-title="Distribuição de probabilidade" data-language-autonym="Português" data-language-local-name="پُرتگالی" 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%9Bii_de_probabilitate" title="Distribuții de probabilitate – رومینین" lang="ro" hreflang="ro" data-title="Distribuții de probabilitate" data-language-autonym="Română" data-language-local-name="رومینین" 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%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Распределение вероятностей – روسی" lang="ru" hreflang="ru" data-title="Распределение вероятностей" data-language-autonym="Русский" data-language-local-name="روسی" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_e_probabilitetit" title="Shpërndarja e probabilitetit – البانی" lang="sq" hreflang="sq" data-title="Shpërndarja e probabilitetit" data-language-autonym="Shqip" data-language-local-name="البانی" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%B8%E0%B7%8A%E0%B6%B7%E0%B7%8F%E0%B7%80%E0%B7%92%E0%B6%AD%E0%B7%8F_%E0%B7%80%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%B4%E0%B7%8A%E0%B6%AD%E0%B7%92%E0%B6%BA" title="සම්භාවිතා ව්‍යාප්තිය – سنہالا" lang="si" hreflang="si" data-title="සම්භාවිතා ව්‍යාප්තිය" data-language-autonym="සිංහල" data-language-local-name="سنہالا" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Probability_distribution" title="Probability distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Probability 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Verjetnostna_porazdelitev" title="Verjetnostna porazdelitev – سلووینیائی" lang="sl" hreflang="sl" data-title="Verjetnostna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="سلووینیائی" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D0%BE%D0%B4%D0%B5%D0%BB%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D0%BD%D0%BE%D1%9B%D0%B5" title="Расподела вероватноће – سربین" lang="sr" hreflang="sr" data-title="Расподела вероватноће" data-language-autonym="Српски / srpski" data-language-local-name="سربین" 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_probabilitas" title="Sebaran probabilitas – سنڈانیز" lang="su" hreflang="su" data-title="Sebaran probabilitas" data-language-autonym="Sunda" data-language-local-name="سنڈانیز" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Todenn%C3%A4k%C3%B6isyysjakauma" title="Todennäköisyysjakauma – فینیش" lang="fi" hreflang="fi" data-title="Todennäköisyysjakauma" data-language-autonym="Suomi" data-language-local-name="فینیش" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sannolikhetsf%C3%B6rdelning" title="Sannolikhetsfördelning – سویڈش" lang="sv" hreflang="sv" data-title="Sannolikhetsfördelning" data-language-autonym="Svenska" data-language-local-name="سویڈش" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_probabilidad" title="Distribusyong probabilidad – ٹیگا لوگ" lang="tl" hreflang="tl" data-title="Distribusyong probabilidad" data-language-autonym="Tagalog" data-language-local-name="ٹیگا لوگ" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%95%E0%AE%B4%E0%AF%8D%E0%AE%A4%E0%AE%95%E0%AE%B5%E0%AF%81%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="நிகழ்தகவுப் பரவல் – تمل" lang="ta" hreflang="ta" data-title="நிகழ்தகவுப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="تمل" 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%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%99%E0%B9%88%E0%B8%B2%E0%B8%88%E0%B8%B0%E0%B9%80%E0%B8%9B%E0%B9%87%E0%B8%99" title="การแจกแจงความน่าจะเป็น – تھائی" lang="th" hreflang="th" data-title="การแจกแจงความน่าจะเป็น" data-language-autonym="ไทย" data-language-local-name="تھائی" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Olas%C4%B1l%C4%B1k_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Olasılık dağılımı – ترکی" lang="tr" hreflang="tr" data-title="Olasılık dağılımı" data-language-autonym="Türkçe" data-language-local-name="ترکی" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB_%D1%96%D0%BC%D0%BE%D0%B2%D1%96%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Розподіл імовірностей – یوکرینیائی" lang="uk" hreflang="uk" data-title="Розподіл імовірностей" data-language-autonym="Українська" data-language-local-name="یوکرینیائی" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ph%E1%BB%91i_x%C3%A1c_su%E1%BA%A5t" title="Phân phối xác suất – ویتنامی" lang="vi" hreflang="vi" data-title="Phân phối xác suất" data-language-autonym="Tiếng Việt" data-language-local-name="ویتنامی" 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%A6%82%E7%8E%87%E5%88%86%E5%B8%83" title="概率分布 – وو چائینیز" lang="wuu" hreflang="wuu" data-title="概率分布" data-language-autonym="吴语" data-language-local-name="وو چائینیز" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈 – کینٹونیز" lang="yue" hreflang="yue" data-title="概率分佈" data-language-autonym="粵語" data-language-local-name="کینٹونیز" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E5%B8%83" title="概率分布 – چینی" lang="zh" hreflang="zh" data-title="概率分布" data-language-autonym="中文" data-language-local-name="چینی" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q200726#sitelinks-wikipedia" title="ترمیم بین اللسانی روابط" 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</td></tr></tbody></table> <p>کسی <a href="/wiki/%D9%86%D9%85%D9%88%D9%86%DB%81_%D9%81%D8%B6%D8%A7" title="نمونہ فضا">نمونہ فضا</a> کے <a href="/w/index.php?title=Subset&action=edit&redlink=1" class="new" title="Subset (صفحہ موجود نہیں)">ذیلی مجموعات</a> کو <a href="/wiki/%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84_(%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84_%D9%86%D8%B8%D8%B1%DB%8C%DB%81)" title="احتمال (احتمال نظریہ)">احتمال</a> اس طرح سونپا جاتا ہے کہ احتمال کے <a href="/w/index.php?title=Axiom&action=edit&redlink=1" class="new" title="Axiom (صفحہ موجود نہیں)">مسلمات</a> پورے ہوتے ہوں۔ <a href="/wiki/%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%DB%8C_%D9%85%D8%AA%D8%BA%DB%8C%D8%B1" title="تصادفی متغیر">تصادفی متغیر</a> ایک <a href="/wiki/%D8%AF%D8%A7%D9%84%DB%81" class="mw-disambig" title="دالہ">دالہ</a> ہوتا ہے، جو نمونہ فضا کو اصل اعداد میں لے جاتا ہے۔ تصادفی متغیر <i>X</i> کے کسی عدد <i>x</i> سے کم ہونے کے احتمال کو بطور ایک فنکشن لکھا جاتا ہے، </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}(x)=\Pr(X\leq x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\Pr(X\leq x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572cdd5143d81980084f9dc0c43baa5986a73355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.076ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)=\Pr(X\leq x)}"></span></dd></dl> <p>اور اس فنکشن <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bc47af7d61f51df1b0c8e9a7c4328c1edbcc98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle \ F_{X}(x)}"></span> کو "تَراكُمی توزیعِ احتمال" کہا جاتا ہے۔ تَراكُمی توزیعِ احتمال مندرجہ ذیل خصوصیات کی حامل ہوتی ہے: </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}(x)\geq 0,\forall x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)\geq 0,\forall x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b524fc3f33986709056b4d0bce4fa689fdbeff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.183ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)\geq 0,\forall x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(-\infty )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(-\infty )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3372cb60fd86897f1009a784ef46f91c1984d66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:2.843ex;" alt="{\displaystyle F_{X}(-\infty )=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(\infty )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(\infty )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa1ea7dde5c7a9eb1389b754b9f6baa53e4705c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.521ex; height:2.843ex;" alt="{\displaystyle F_{X}(\infty )=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,x_{2}>x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,x_{2}>x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/614d7512e26969d3d574393461f6199cfadd3a83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.187ex; height:2.843ex;" alt="{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,x_{2}>x_{1}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="متفرد"><span id=".D9.85.D8.AA.D9.81.D8.B1.D8.AF"></span>متفرد</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%AA%D9%82%D8%B3%DB%8C%D9%85_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84&action=edit&section=1" title="تدوینِ قطعہ: متفرد"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D9%85%D8%AA%D9%81%D8%B1%D8%AF" title="متفرد">متفرد</a> تصافی متغیر <i>X</i> کی <i>احتمال کمیت تفاعل</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p_{X}(.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ p_{X}(.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235c66d7785c1c250050889475e3131b872c5326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.226ex; height:2.843ex;" alt="{\displaystyle \ p_{X}(.)}"></span> کسی بھی عدد <i>x</i> کے لیے یوں تعریف ہوتی ہے </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 p_{X}(x)=\Pr(X=x)=\Pr({\hbox{all }}s\in S:X(s)=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>all </mtext> </mstyle> </mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X}(x)=\Pr(X=x)=\Pr({\hbox{all }}s\in S:X(s)=x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5938e681b2daf2f17576c582b8f1c65a3be5687a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:46.955ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)=\Pr(X=x)=\Pr({\hbox{all }}s\in S:X(s)=x)}"></span></dd></dl> <p>یعنی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ p_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c978af7cab5d946ce61ad9507300e406ca8a288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.521ex; height:2.843ex;" alt="{\displaystyle \ p_{X}(x)}"></span> اس احتمال کے برابر ہے کہ تصادفی متغیر <i>X</i> کی قدر <i>x</i> بنے۔ یہاں <i>S</i> <a href="/wiki/%D9%86%D9%85%D9%88%D9%86%DB%81_%D9%81%D8%B6%D8%A7" title="نمونہ فضا">نمونہ فضا</a> ہے، جو تصادفی متغیر <i>X</i> کا <a href="/wiki/%D8%B3%D8%A7%D8%AD%DB%81_(%D8%B6%D8%AF%D8%A7%D8%A8%DB%81%D8%A7%D9%85)" class="mw-disambig" title="ساحہ (ضدابہام)">ساحہ</a> ہے۔ غور کرو کہ </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 p_{X}(x)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X}(x)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43deb83da99ae91cbf6ced3ee5f14fdac6004789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)\geq 0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}(x)\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X}(x)\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e3a10077aebc703c161aa39d424095287ac351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)\leq 1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sum _{x}p_{X}(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sum _{x}p_{X}(x)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9afa2a9533583a9b37dbdad33788d338551b319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.911ex; height:5.509ex;" alt="{\displaystyle \ \sum _{x}p_{X}(x)=1}"></span></dd></dl> <p>اب متفرد تصادفی متغیر <i>X</i> کی <i>تَراكُمی توزیع احتمال تفاعل</i> یوں تعریف ہو گی </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}(x)=\Pr(X\leq x)=\sum _{x:X(s)\leq x}p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\Pr(X\leq x)=\sum _{x:X(s)\leq x}p(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2babf609a16750fa4d21201e60ac1040bfe11313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.937ex; height:6.009ex;" alt="{\displaystyle F_{X}(x)=\Pr(X\leq x)=\sum _{x:X(s)\leq x}p(x)}"></span></dd></dl> <p>یعنی یہ احتمال کہ تصادفی متغیر <i>X</i> کی قدر <i>x</i> کے برابر یا اس سے کم ہو۔ غور کرو کہ </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}(x)\geq 0,\forall x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)\geq 0,\forall x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b524fc3f33986709056b4d0bce4fa689fdbeff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.183ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)\geq 0,\forall x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(-\infty )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(-\infty )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3372cb60fd86897f1009a784ef46f91c1984d66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:2.843ex;" alt="{\displaystyle F_{X}(-\infty )=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(\infty )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(\infty )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa1ea7dde5c7a9eb1389b754b9f6baa53e4705c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.521ex; height:2.843ex;" alt="{\displaystyle F_{X}(\infty )=1}"></span></dd></dl> <p>متفرد تصادفی متغیر <i>X</i> کے لیے اس کے "احتمال کمیت تفاعل" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ p_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c978af7cab5d946ce61ad9507300e406ca8a288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.521ex; height:2.843ex;" alt="{\displaystyle \ p_{X}(x)}"></span> اور "تَراكُمی توزیع احتمال تفاعل" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bc47af7d61f51df1b0c8e9a7c4328c1edbcc98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle \ F_{X}(x)}"></span>، دونوں کو "توزیعِ احتمال" پکارا جاتا ہے۔ </p> <figure typeof="mw:Error mw:File/Thumb"><a href="/w/index.php?title=%D8%AE%D8%A7%D8%B5:%D8%A7%D9%BE%D9%84%D9%88%DA%88&wpDestFile=Binomial_distribution_pmf.png" class="new" title="فائل:Binomial distribution pmf.png"><span class="mw-file-element mw-broken-media" data-width="280">فائل:Binomial distribution pmf.png</span></a><figcaption>تصویر 2: دو رقمی توزیع کی <i>احتمال کمیت تفاعل</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p_{X}(.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ p_{X}(.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235c66d7785c1c250050889475e3131b872c5326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.226ex; height:2.843ex;" alt="{\displaystyle \ p_{X}(.)}"></span></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D9%81%D8%A7%D8%A6%D9%84:Binomial_distribution_cdf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Binomial_distribution_cdf.png/280px-Binomial_distribution_cdf.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Binomial_distribution_cdf.png/420px-Binomial_distribution_cdf.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Binomial_distribution_cdf.png/560px-Binomial_distribution_cdf.png 2x" data-file-width="561" data-file-height="420" /></a><figcaption>تصویر 3: دو رقمی توزیع کی <i>تَراكُمی توزیع احتمال تفاعل</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F_{X}(.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{X}(.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed4be71dd975c20fc44bb270c0c7a07637ba97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.551ex; height:2.843ex;" alt="{\displaystyle \ F_{X}(.)}"></span></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="مثال:_دو_رقمی_توزیعِ_احتمال"><span id=".D9.85.D8.AB.D8.A7.D9.84:_.D8.AF.D9.88_.D8.B1.D9.82.D9.85.DB.8C_.D8.AA.D9.88.D8.B2.DB.8C.D8.B9.D9.90_.D8.A7.D8.AD.D8.AA.D9.85.D8.A7.D9.84"></span>مثال: دو رقمی توزیعِ احتمال</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%AA%D9%82%D8%B3%DB%8C%D9%85_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84&action=edit&section=2" title="تدوینِ قطعہ: مثال: دو رقمی توزیعِ احتمال"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>تفصیلی مضمون:</i> <a href="/wiki/%D8%AF%D9%88_%D8%B1%D9%82%D9%85%DB%8C_%D8%AA%D9%88%D8%B2%DB%8C%D8%B9" title="دو رقمی توزیع">دو رقمی توزیع احتمال</a> </p><p>بعض اوقات ایک ہی تجربہ کو متعدد بار دہرایا جاتا ہے (جیسے سکے کو بار بار فضا میں اچھالا جائے)۔ ایسے بار بار آزمائش میں فرض کرو کہ: </p> <ul><li>دو ممکنہ نتائج ہیں، "کامیابی" اور "ناکامی"</li> <li>ہر آزمائش پر "کامیابی" کا احتمال <i>p</i> ہے اور "ناکامی" کا احتمال <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1-p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237ce1b78f4fbddf0475ad53620f35ce473b68c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.753ex; height:2.509ex;" alt="{\displaystyle \ 1-p}"></span></li> <li>آزمائش کی تعداد <i>n</i> ہے</li> <li>ہر آزمائش دوسری آزمائشوں سے <a href="/wiki/%D8%A7%D8%AD%D8%B5%D8%A7%D8%A6%DB%8C_%D8%A2%D8%B2%D8%A7%D8%AF%DB%8C" title="احصائی آزادی">آزاد</a> ہے</li></ul> <p>فرض کرو کہ تصادفی متغیر <i>X</i> ہے، جو ان <i>n</i> آزمائشوں میں "کامیابی" کی تعداد ظاہر کرتا ہے۔ اس متفرد تصادفی متغیر کا حیطہ </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 \{0,1,2,\cdots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\cdots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f06ab2737ad7b28de87b3448b095c5216e852e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.453ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\cdots ,n\}}"></span></dd></dl> <p>ہے اور توزیعِ احتمال </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 p_{X}(x)={\frac {n!}{x!(n-x)!}}p^{x}(1-p)^{n-x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>x</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X}(x)={\frac {n!}{x!(n-x)!}}p^{x}(1-p)^{n-x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae75d54e25762c11d2dd4545b500db4fbb0dbe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:32.723ex; height:6.176ex;" alt="{\displaystyle p_{X}(x)={\frac {n!}{x!(n-x)!}}p^{x}(1-p)^{n-x}}"></span></dd></dl> <p>اس توزیع احتمال کو "دو رقمی توزیع" کے نام سے پکارا جاتا ہے۔ ( یہاں <i>!</i> کی علامت <a href="/wiki/%D8%B9%D8%A7%D9%85%D9%84%DB%8C%DB%81" title="عاملیہ">عامِلیہ</a> کو ظاہر کرتی ہے۔) </p> <div class="mw-heading mw-heading2"><h2 id="متواصل"><span id=".D9.85.D8.AA.D9.88.D8.A7.D8.B5.D9.84"></span>متواصل</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%AA%D9%82%D8%B3%DB%8C%D9%85_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84&action=edit&section=3" title="تدوینِ قطعہ: متواصل"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D9%85%D8%AA%D9%88%D8%A7%D8%B5%D9%84" title="متواصل">متواصل</a> تصافی متغیر <i>X</i> کی <i><a href="/w/index.php?title=Probability_density_function&action=edit&redlink=1" class="new" title="Probability density function (صفحہ موجود نہیں)">احتمال کثافت دالہ</a></i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f_{X}(.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f_{X}(.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2e10370a41b634053cd3df4768e23bf6a50b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.195ex; height:2.843ex;" alt="{\displaystyle \ f_{X}(.)}"></span> کی مدد سے تصادفی متغیر <i>X</i> کی قدر وقفہ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e68fabba679caa5f240a664bfc9110b6cf757e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.651ex; height:2.843ex;" alt="{\displaystyle \ (a,b)}"></span> میں ہونے کا احتمال یوں لکھا جا سکتا ہے: </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 \Pr(a<X\leq b)=\int _{a}^{b}f_{X}(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(a<X\leq b)=\int _{a}^{b}f_{X}(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2deee5bf7eee2930ce5b90432bce2c59b7322f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.052ex; height:6.343ex;" alt="{\displaystyle \Pr(a<X\leq b)=\int _{a}^{b}f_{X}(x)dx}"></span></dd></dl> <p>یعنی یہ احتمال <a href="/wiki/%D8%AF%D8%A7%D9%84%DB%81" class="mw-disambig" title="دالہ">دالہ</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 \ f_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f_{X}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eea88f60ffdeaae69f01d3f44d423495f4904ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.491ex; height:2.843ex;" alt="{\displaystyle \ f_{X}(x)}"></span> کے نیچے <i>a</i> سے <i>b</i> تک رقبہ کے برابر ہے۔ "احتمال کثافت تفاعل" مندرجہ ذیل خصوصیات کی حامل ہوتی ہے: </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}(x)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd69fc6fe04954cdfd622fccb3e1f337972a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.171ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)\geq 0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \int _{-\infty }^{\infty }f_{X}(x)dx=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <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}(x)dx=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9fe09bd84fa95add98dd833af00988dde9f3af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.518ex; height:6.009ex;" alt="{\displaystyle \ \int _{-\infty }^{\infty }f_{X}(x)dx=1}"></span></dd></dl> <p>اب متواصل تصادفی متغیر <i>X</i> کی <i>تَراكُمی توزیع احتمال تفاعل</i> یوں تعریف ہو گی </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}(x)=\Pr(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\Pr(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028f277d66361680b5d22be6172a7b755c9a57bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.464ex; height:6.009ex;" alt="{\displaystyle F_{X}(x)=\Pr(X\leq x)=\int _{-\infty }^{x}f_{X}(x)dx}"></span></dd></dl> <p>یعنی یہ احتمال کہ تصادفی متغیر <i>X</i> کی قدر <i>x</i> سے کم ہو۔ غور کرو کہ </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}(x)\geq 0,\forall x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)\geq 0,\forall x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b524fc3f33986709056b4d0bce4fa689fdbeff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.183ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)\geq 0,\forall x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(-\infty )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(-\infty )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3372cb60fd86897f1009a784ef46f91c1984d66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:2.843ex;" alt="{\displaystyle F_{X}(-\infty )=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(\infty )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(\infty )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa1ea7dde5c7a9eb1389b754b9f6baa53e4705c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.521ex; height:2.843ex;" alt="{\displaystyle F_{X}(\infty )=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,\,x_{2}>x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,\,x_{2}>x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea02e6c0285e28af1bdf38ac917464dfb6bc115b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.574ex; height:2.843ex;" alt="{\displaystyle F_{X}(x_{2})\geq F_{X}(x_{1})\,,\,\,\,\,x_{2}>x_{1}}"></span></dd></dl> <p>ان تعاریف سے پتہ چلتا ہے کہ </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 \Pr(a<X\leq b)=\Pr(X\leq b)-\Pr(X\leq a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(a<X\leq b)=\Pr(X\leq b)-\Pr(X\leq a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d31ea8e0a896af1128d1f883d998590a289dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.639ex; height:2.843ex;" alt="{\displaystyle \Pr(a<X\leq b)=\Pr(X\leq b)-\Pr(X\leq a)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(a<X\leq b)=F_{X}(b)-F_{X}(a)=\int _{a}^{b}f_{X}(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo><</mo> <mi>X</mi> <mo>≤</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(a<X\leq b)=F_{X}(b)-F_{X}(a)=\int _{a}^{b}f_{X}(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e5457d98844deb69d2da675737048b01263ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.09ex; height:6.343ex;" alt="{\displaystyle \Pr(a<X\leq b)=F_{X}(b)-F_{X}(a)=\int _{a}^{b}f_{X}(x)dx}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="مثال"><span id=".D9.85.D8.AB.D8.A7.D9.84"></span>مثال</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%AA%D9%82%D8%B3%DB%8C%D9%85_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84&action=edit&section=4" title="تدوینِ قطعہ: مثال"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D9%81%D8%A7%D8%A6%D9%84:Normal_distribution_pdf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/280px-Normal_distribution_pdf.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/420px-Normal_distribution_pdf.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/560px-Normal_distribution_pdf.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption>گاسین توزیع احتمال کی "احتمال کثافت تفاعل"۔ سبز رنگ میں معیاری گاسین توزیع احتمال ہے۔</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D9%81%D8%A7%D8%A6%D9%84:Normal_distribution_cdf.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Normal_distribution_cdf.png/280px-Normal_distribution_cdf.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Normal_distribution_cdf.png/420px-Normal_distribution_cdf.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Normal_distribution_cdf.png/560px-Normal_distribution_cdf.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption>گاسین توزیع احتمال کی "تراکمی توزیع تفاعل"۔ رنگ اوپر والی تصویر کے موافق ہیں۔</figcaption></figure> <p>سب سے مشہور متواصل توزیع احتمال، گاسین توزیع احتمال ہے، جس کی "احتمال کثافت تفاعل" تصادفی متغیر <i>X</i> کے لیے یوں لکھی جاتی ہے </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}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\;\exp \left(-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <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> <mspace width="thickmathspace" /> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\;\exp \left(-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c888ee84c54eae3e89a9077aa6f3e04704096380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.564ex; height:7.509ex;" alt="{\displaystyle f_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\;\exp \left(-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\!}"></span></dd></dl> <p>جبکہ اس کی <a href="/wiki/%D9%85%D8%AA%D9%88%D9%82%D8%B9_%D9%82%D8%AF%D8%B1" title="متوقع قدر">متوقع قدر</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 \ \mu =E(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu =E(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f2236f2805887d21076a9cd0c498fe08a75826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.646ex; height:2.843ex;" alt="{\displaystyle \ \mu =E(X)}"></span></dd></dl> <p>اور <a href="/wiki/%D8%AA%D9%81%D8%A7%D9%88%D8%AA" title="تفاوت">تفاوت</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 \ \sigma ^{2}=Var(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>V</mi> <mi>a</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ^{2}=Var(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aecb0efd7d774c6236669cfc4ee9dc5ab2785f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.919ex; height:3.176ex;" alt="{\displaystyle \ \sigma ^{2}=Var(X)}"></span></dd></dl> <p>ہیں۔ معیاری گاسین "احتمال کثافت تفاعل" (سبز) کے لیے <span class="mwe-math-element"><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"> <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/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}"></span> اور <span class="mwe-math-element"><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}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193637e200ca8b182a20186e31c4a195c7fd3555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.646ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}=1}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="مزید_دیکھیے"><span id=".D9.85.D8.B2.DB.8C.D8.AF_.D8.AF.DB.8C.DA.A9.DA.BE.DB.8C.DB.92"></span>مزید دیکھیے</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D8%AA%D9%82%D8%B3%DB%8C%D9%85_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84&action=edit&section=5" title="تدوینِ قطعہ: مزید دیکھیے"><span>ترمیم</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><div style="background-color: #f9f9f9; border: 1px dashed #aaaaaa; padding: .2em; text-align: center; margin: .5em 5em .5em 5em; font-size: large;"><small></small></div></li></ul><small></small><p><small><code>E=mc<sup>2</sup></code>     اردو ویکیپیڈیا پر ریاضی مساوات کو بائیں سے دائیں <small>LTR</small> پڑھیٔے     <a href="https://en.wikipedia.org/wiki/Table_of_mathematical_symbols" class="extiw" title="en:Table of mathematical symbols">ریاضی علامات </a></small> </p> <table class="mbox-small plainlinks sistersitebox" style="border:1px solid #a2a9b1;background-color:#f8f9fa"> <tbody><tr> <td class="mbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></td> <td class="mbox-text plainlist">ویکی ذخائر پر <b><a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">تقسیم احتمال </a></b> سے متعلق سمعی و بصری مواد ملاحظہ کریں۔</td></tr></tbody></table>    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title="زمرہ:احتمال و احصاء">احتمال و احصاء</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">پوشیدہ زمرہ جات: <ul><li><a href="/wiki/%D8%B2%D9%85%D8%B1%DB%81:%D9%81%D8%A7%D8%A6%D9%84%D9%88%DA%BA_%DA%A9%DB%92_%D8%B4%DA%A9%D8%B3%D8%AA%DB%81_%D8%B1%D9%88%D8%A7%D8%A8%D8%B7_%D9%BE%D8%B1_%D9%85%D8%B4%D8%AA%D9%85%D9%84_%D8%B5%D9%81%D8%AD%D8%A7%D8%AA" title="زمرہ:فائلوں کے شکستہ روابط پر مشتمل صفحات">فائلوں کے شکستہ روابط پر مشتمل صفحات</a></li><li><a href="/wiki/%D8%B2%D9%85%D8%B1%DB%81:%D9%85%D8%AA%D8%B6%D8%A7%D8%AF_%D8%A8%DB%8C%D9%86_%D8%A7%D9%84%D9%88%DB%8C%DA%A9%DB%8C_%D9%88%D8%A7%D9%84%DB%92_%D8%B5%D9%81%D8%AD%D8%A7%D8%AA" title="زمرہ:متضاد بین الویکی والے صفحات">متضاد بین الویکی والے صفحات</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"> اس صفحہ میں آخری بار مورخہ 1 جنوری 2024ء کو 12:50 بجے ترمیم کی گئی۔</li> <li 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