Binomial distribution

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</button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Expected_value_and_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expected_value_and_variance"> <div class="vector-toc-text"> 2.1 Expected value and variance </div> </a> <ul id="toc-Expected_value_and_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_moments"> <div class="vector-toc-text"> 2.2 Higher moments </div> </a> <ul id="toc-Higher_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mode" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mode"> <div class="vector-toc-text"> 2.3 Mode </div> </a> <ul id="toc-Mode-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Median" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Median"> <div class="vector-toc-text"> 2.4 Median </div> </a> <ul id="toc-Median-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tail_bounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tail_bounds"> <div class="vector-toc-text"> 2.5 Tail bounds </div> </a> <ul id="toc-Tail_bounds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Statistical_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Statistical_inference"> <div class="vector-toc-text"> 3 Statistical inference </div> </a> <button aria-controls="toc-Statistical_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Statistical inference subsection </button> <ul id="toc-Statistical_inference-sublist" class="vector-toc-list"> <li id="toc-Estimation_of_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation_of_parameters"> <div class="vector-toc-text"> 3.1 Estimation of parameters </div> </a> <ul id="toc-Estimation_of_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Confidence_intervals_for_the_parameter_p" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Confidence_intervals_for_the_parameter_p"> <div class="vector-toc-text"> 3.2 Confidence intervals for the parameter p </div> </a> <ul id="toc-Confidence_intervals_for_the_parameter_p-sublist" class="vector-toc-list"> <li id="toc-Wald_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Wald_method"> <div class="vector-toc-text"> 3.2.1 Wald method </div> </a> <ul id="toc-Wald_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Agresti–Coull_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Agresti–Coull_method"> <div class="vector-toc-text"> 3.2.2 Agresti–Coull method </div> </a> <ul id="toc-Agresti–Coull_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arcsine_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Arcsine_method"> <div class="vector-toc-text"> 3.2.3 Arcsine method </div> </a> <ul id="toc-Arcsine_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wilson_(score)_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Wilson_(score)_method"> <div class="vector-toc-text"> 3.2.4 Wilson (score) method </div> </a> <ul id="toc-Wilson_(score)_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Comparison"> <div class="vector-toc-text"> 3.2.5 Comparison </div> </a> <ul id="toc-Comparison-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> 4 Related distributions </div> </a> <button aria-controls="toc-Related_distributions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related distributions subsection </button> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> <li id="toc-Sums_of_binomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sums_of_binomials"> <div class="vector-toc-text"> 4.1 Sums of binomials </div> </a> <ul id="toc-Sums_of_binomials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_binomial_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_binomial_distribution"> <div class="vector-toc-text"> 4.2 Poisson binomial distribution </div> </a> <ul id="toc-Poisson_binomial_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ratio_of_two_binomial_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ratio_of_two_binomial_distributions"> <div class="vector-toc-text"> 4.3 Ratio of two binomial distributions </div> </a> <ul id="toc-Ratio_of_two_binomial_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_binomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditional_binomials"> <div class="vector-toc-text"> 4.4 Conditional binomials </div> </a> <ul id="toc-Conditional_binomials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bernoulli_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bernoulli_distribution"> <div class="vector-toc-text"> 4.5 Bernoulli distribution </div> </a> <ul id="toc-Bernoulli_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_approximation"> <div class="vector-toc-text"> 4.6 Normal approximation </div> </a> <ul id="toc-Normal_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_approximation"> <div class="vector-toc-text"> 4.7 Poisson approximation </div> </a> <ul id="toc-Poisson_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limiting_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limiting_distributions"> <div class="vector-toc-text"> 4.8 Limiting distributions </div> </a> <ul id="toc-Limiting_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beta_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beta_distribution"> <div class="vector-toc-text"> 4.9 Beta distribution </div> </a> <ul id="toc-Beta_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computational_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computational_methods"> <div class="vector-toc-text"> 5 Computational methods </div> </a> <button aria-controls="toc-Computational_methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Computational methods subsection </button> <ul id="toc-Computational_methods-sublist" class="vector-toc-list"> <li id="toc-Random_number_generation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_number_generation"> <div class="vector-toc-text"> 5.1 Random number generation </div> </a> <ul id="toc-Random_number_generation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 6 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 9 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Binomial distribution</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 52 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-52" aria-hidden="true" > 52 languages </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%AB%D9%86%D8%A7%D8%A6%D9%8A_%D8%A7%D9%84%D8%AD%D8%AF%D9%8A%D9%86" title="توزيع ثنائي الحدين – Arabic" lang="ar" hreflang="ar" data-title="توزيع ثنائي الحدين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_bin%C3%B3mica" title="Distribución binómica – Asturian" lang="ast" hreflang="ast" data-title="Distribución binómica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%AA%E0%A6%A6%E0%A7%80_%E0%A6%AC%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%A8" title="দ্বিপদী বন্টন – Bangla" lang="bn" hreflang="bn" data-title="দ্বিপদী বন্টন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%BE%D0%BC%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="Біномнае размеркаванне – Belarusian" lang="be" hreflang="be" data-title="Біномнае размеркаванне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_binomial" title="Distribució binomial – Catalan" lang="ca" hreflang="ca" data-title="Distribució binomial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Binomick%C3%A9_rozd%C4%9Blen%C3%AD" title="Binomické rozdělení – Czech" lang="cs" hreflang="cs" data-title="Binomické rozdělení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Binomialfordelingen" title="Binomialfordelingen – Danish" lang="da" hreflang="da" data-title="Binomialfordelingen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Binomialverteilung" title="Binomialverteilung – German" lang="de" hreflang="de" data-title="Binomialverteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Binoomjaotus" title="Binoomjaotus – Estonian" lang="et" hreflang="et" data-title="Binoomjaotus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CF%89%CE%BD%CF%85%CE%BC%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE" title="Διωνυμική κατανομή – Greek" lang="el" hreflang="el" data-title="Διωνυμική κατανομή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_binomial" title="Distribución binomial – Spanish" lang="es" hreflang="es" data-title="Distribución binomial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Banaketa_binomial" title="Banaketa binomial – Basque" lang="eu" hreflang="eu" data-title="Banaketa binomial" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</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%AF%D9%88%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="توزیع دوجمله‌ای – Persian" lang="fa" hreflang="fa" data-title="توزیع دوجمله‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fr.wikipedia.org/wiki/Loi_binomiale" title="Loi binomiale – French" lang="fr" hreflang="fr" data-title="Loi binomiale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_binomial" title="Distribución binomial – Galician" lang="gl" hreflang="gl" data-title="Distribución binomial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%ED%95%AD_%EB%B6%84%ED%8F%AC" title="이항 분포 – Korean" lang="ko" hreflang="ko" data-title="이항 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AA%E0%A4%A6_%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A4%B0%E0%A4%A3" title="द्विपद वितरण – Hindi" lang="hi" hreflang="hi" data-title="द्विपद वितरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_binomial" title="Distribusi binomial – Indonesian" lang="id" hreflang="id" data-title="Distribusi binomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_binomiale" title="Distribuzione binomiale – Italian" lang="it" hreflang="it" data-title="Distribuzione binomiale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</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%91%D7%99%D7%A0%D7%95%D7%9E%D7%99%D7%AA" title="התפלגות בינומית – Hebrew" lang="he" hreflang="he" data-title="התפלגות בינומית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%B4%D1%8B%D2%9B_%D2%AF%D0%BB%D0%B5%D1%81%D1%82%D1%96%D1%80%D1%96%D0%BB%D1%83" title="Биномдық үлестірілу – Kazakh" lang="kk" hreflang="kk" data-title="Биномдық үлестірілу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Binomi%C4%81lais_sadal%C4%ABjums" title="Binomiālais sadalījums – Latvian" lang="lv" hreflang="lv" data-title="Binomiālais sadalījums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Binominis_skirstinys" title="Binominis skirstinys – Lithuanian" lang="lt" hreflang="lt" data-title="Binominis skirstinys" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Binomi%C3%A1lis_eloszl%C3%A1s" title="Binomiális eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Binomiális eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%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="Биномна распределба – Macedonian" lang="mk" hreflang="mk" data-title="Биномна распределба" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Taburan_binomial" title="Taburan binomial – Malay" lang="ms" hreflang="ms" data-title="Taburan binomial" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Binomiale_verdeling" title="Binomiale verdeling – Dutch" lang="nl" hreflang="nl" data-title="Binomiale verdeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%88%86%E5%B8%83" title="二項分布 – Japanese" lang="ja" hreflang="ja" data-title="二項分布" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Binomisk_fordeling" title="Binomisk fordeling – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Binomisk fordeling" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Binomisk_fordeling" title="Binomisk fordeling – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Binomisk fordeling" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Binomial_taqsimot" title="Binomial taqsimot – Uzbek" lang="uz" hreflang="uz" data-title="Binomial taqsimot" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_dwumianowy" title="Rozkład dwumianowy – Polish" lang="pl" hreflang="pl" data-title="Rozkład dwumianowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_binomial" title="Distribuição binomial – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição binomial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Distribu%C8%9Bia_binomial%C4%83" title="Distribuția binomială – Romanian" lang="ro" hreflang="ro" data-title="Distribuția binomială" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%BE%D0%BC%D0%B8%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="Биномиальное распределение – Russian" lang="ru" hreflang="ru" data-title="Биномиальное распределение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_binomiale" title="Shpërndarja binomiale – Albanian" lang="sq" hreflang="sq" data-title="Shpërndarja binomiale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binomial_distribution" title="Binomial distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binomial distribution" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Binomick%C3%A9_rozdelenie" title="Binomické rozdelenie – Slovak" lang="sk" hreflang="sk" data-title="Binomické rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Binomska_porazdelitev" title="Binomska porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Binomska porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Binomna_raspodela" title="Binomna raspodela – Serbian" lang="sr" hreflang="sr" data-title="Binomna raspodela" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_binomial" title="Sebaran binomial – Sundanese" lang="su" hreflang="su" data-title="Sebaran binomial" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Binomijakauma" title="Binomijakauma – Finnish" lang="fi" hreflang="fi" data-title="Binomijakauma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Binomialf%C3%B6rdelning" title="Binomialfördelning – Swedish" lang="sv" hreflang="sv" data-title="Binomialfördelning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distribusyong_binomial" title="Distribusyong binomial – Tagalog" lang="tl" hreflang="tl" data-title="Distribusyong binomial" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%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="ஈருறுப்புப் பரவல் – Tamil" lang="ta" hreflang="ta" data-title="ஈருறுப்புப் பரவல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</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%97%E0%B8%A7%E0%B8%B4%E0%B8%99%E0%B8%B2%E0%B8%A1" title="การแจกแจงทวินาม – Thai" lang="th" hreflang="th" data-title="การแจกแจงทวินาม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Binom_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Binom dağılımı – Turkish" lang="tr" hreflang="tr" data-title="Binom dağılımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%BE%D0%BC%D1%96%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="Біноміальний розподіл – Ukrainian" lang="uk" hreflang="uk" data-title="Біноміальний розподіл" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Binomial model" redirects here. For the binomial model in options pricing, see <a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1259743904">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1);padding:0.3em 0.4em}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Binomial distribution</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability mass function</div><a href="/wiki/File:Binomial_distribution_pmf.svg" class="mw-file-description" title="Probability mass function for the binomial distribution"><img alt="Probability mass function for the binomial distribution" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Binomial_distribution_pmf.svg/300px-Binomial_distribution_pmf.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Binomial_distribution_pmf.svg/450px-Binomial_distribution_pmf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Binomial_distribution_pmf.svg/600px-Binomial_distribution_pmf.svg.png 2x" data-file-width="434" data-file-height="289" /></a></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><a href="/wiki/File:Binomial_distribution_cdf.svg" class="mw-file-description" title="Cumulative distribution function for the binomial distribution"><img alt="Cumulative distribution function for the binomial distribution" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/300px-Binomial_distribution_cdf.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/450px-Binomial_distribution_cdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/600px-Binomial_distribution_cdf.svg.png 2x" data-file-width="640" data-file-height="480" /></a></td></tr><tr><th scope="row" class="infobox-label">Notation</th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(n,p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle B(n,p)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \{0,1,2,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \{0,1,2,\ldots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc276297ae0fc59977699b50d120092fb616bb00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.873ex; height:2.843ex;" alt="{\displaystyle n\in \{0,1,2,\ldots \}}"> – number of trials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c3a52aa7b2d00227e85c641cca67e85583c43c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.752ex; height:2.843ex;" alt="{\displaystyle p\in [0,1]}"> – success probability for each trial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1-p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e387fe24ba3da5f9a0dc424923cdfc2c08990c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.34ex; height:2.509ex;" alt="{\displaystyle q=1-p}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \{0,1,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \{0,1,\ldots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b6be671ec370d41685c66622c8b541116866c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.309ex; height:2.843ex;" alt="{\displaystyle k\in \{0,1,\ldots ,n\}}"> – number of successes</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20edfc22372742d64909cf7c7f97593bade88338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.507ex; height:6.176ex;" alt="{\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb1d2742d9450195d6f8155a44eae165eb40ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.645ex; height:3.009ex;" alt="{\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )}"> (the <a href="/wiki/Beta_function#Incomplete_beta_function" title="Beta function">regularized incomplete beta function</a>)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.009ex;" alt="{\displaystyle np}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor np\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor np\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0273d137830eeaf408348219fe159f8f97d53ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle \lfloor np\rfloor }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil np\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mi>p</mi> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil np\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ade49121f3f129664d81676c95c51302a43653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle \lceil np\rceil }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor (n+1)p\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor (n+1)p\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcda6d9602cb8abc4e83e4a1b19ed23bc5062932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle \lfloor (n+1)p\rfloor }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil (n+1)p\rceil -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo fence="false" stretchy="false">⌉</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil (n+1)p\rceil -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d92b493838f6bfe891a53e1629bdf8f85fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.444ex; height:2.843ex;" alt="{\displaystyle \lceil (n+1)p\rceil -1}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle npq=np(1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle npq=np(1-p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d284719a2524ed948d98054608bdeac05d344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.278ex; height:2.843ex;" alt="{\displaystyle npq=np(1-p)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q-p}{\sqrt {npq}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>p</mi> </mrow> <msqrt> <mi>n</mi> <mi>p</mi> <mi>q</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q-p}{\sqrt {npq}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d444fad2af6ac72538c9a3f57a7b0bab6b2ca6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:6.406ex; height:6.176ex;" alt="{\displaystyle {\frac {q-p}{\sqrt {npq}}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Excess kurtosis</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-6pq}{npq}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>6</mn> <mi>p</mi> <mi>q</mi> </mrow> <mrow> <mi>n</mi> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-6pq}{npq}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebfa3016450291a45a550007269491b841263ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.24ex; height:5.843ex;" alt="{\displaystyle {\frac {1-6pq}{npq}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>e</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80dd2bb6dc89a676674079f3d70d78285ce7174c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.085ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}"> in <a href="/wiki/Shannon_(unit)" title="Shannon (unit)">shannons</a>. For <a href="/wiki/Nat_(unit)" title="Nat (unit)">nats</a>, use the natural log in the log.</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q+pe^{t})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mi>p</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q+pe^{t})^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba4e2f1a111de13a9aab0be599278330c7bcd60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.017ex; height:3.009ex;" alt="{\displaystyle (q+pe^{t})^{n}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q+pe^{it})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>q</mi> <mo>+</mo> <mi>p</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q+pe^{it})^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98ad235a77669a9b8b3426fa73427bc0ce935c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.584ex; height:3.176ex;" alt="{\displaystyle (q+pe^{it})^{n}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability-generating_function" title="Probability-generating function">PGF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=[q+pz]^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>q</mi> <mo>+</mo> <mi>p</mi> <mi>z</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(z)=[q+pz]^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40494c697ce2f88ebb396ac0191946285cadcbdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.502ex; height:2.843ex;" alt="{\displaystyle G(z)=[q+pz]^{n}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{n}(p)={\frac {n}{pq}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{n}(p)={\frac {n}{pq}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c465cbbf5c0c8c8e201ae4da50823e4f61f25d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.48ex; height:5.176ex;" alt="{\displaystyle g_{n}(p)={\frac {n}{pq}}}"> (for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">)</td></tr></tbody></table> <style 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.sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a class="mw-selflink selflink">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output 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href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pascal%27s_triangle;_binomial_distribution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Pascal%27s_triangle%3B_binomial_distribution.svg/280px-Pascal%27s_triangle%3B_binomial_distribution.svg.png" decoding="async" width="280" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Pascal%27s_triangle%3B_binomial_distribution.svg/420px-Pascal%27s_triangle%3B_binomial_distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Pascal%27s_triangle%3B_binomial_distribution.svg/560px-Pascal%27s_triangle%3B_binomial_distribution.svg.png 2x" data-file-width="794" data-file-height="688" /></a><figcaption>Binomial distribution for p = 0.5 with n and k as in <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> The probability that a ball in a <a href="/wiki/Bean_machine" class="mw-redirect" title="Bean machine">Galton box</a> with 8 layers (n = 8) ends up in the central bin (k = 4) is 70/256.</figcaption></figure> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the binomial distribution with parameters n and p is the <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a> of the number of successes in a sequence of n <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> <a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">experiments</a>, each asking a <a href="/wiki/Yes%E2%80%93no_question" title="Yes–no question">yes–no question</a>, and each with its own <a href="/wiki/Boolean-valued_function" title="Boolean-valued function">Boolean</a>-valued <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcome</a>: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a> or Bernoulli experiment, and a sequence of outcomes is called a <a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a>; for a single trial, i.e., n = 1, the binomial distribution is a <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a>. The binomial distribution is the basis for the <a href="/wiki/Binomial_test" title="Binomial test">binomial test</a> of <a href="/wiki/Statistical_significance" title="Statistical significance">statistical significance</a>.<a href="#cite_note-1">[1]</a> The binomial distribution is frequently used to model the number of successes in a sample of size n drawn <a href="/wiki/With_replacement" class="mw-redirect" title="With replacement">with replacement</a> from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a <a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">hypergeometric distribution</a>, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=1" title="Edit section: Definitions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Probability_mass_function">Probability mass function</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=2" title="Edit section: Probability mass function">edit</a>]</div> If the <a href="/wiki/Random_variable" title="Random variable">random variable</a> X follows the binomial distribution with parameters n ∈ <a href="/wiki/Natural_number" title="Natural number"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></a> and p ∈ [0, 1], we write X ~ B(n, p). The probability of getting exactly k successes in n independent Bernoulli trials (with the same rate p) is given by the <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k,n,p)=\Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k,n,p)=\Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad3f324ca5778bd1cae96fed07ce09063a71a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.131ex; height:6.176ex;" alt="{\displaystyle f(k,n,p)=\Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}"></dd></dl> for k = 0, 1, 2, ..., n, where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33401621fb832dd2f9783e80a906d562f669008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.511ex; height:6.343ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}"></dd></dl> is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>. The formula can be understood as follows: pk qn−k is the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are "successes" and the remaining n − k trials result in "failure". Since the trials are independent with probabilities remaining constant between them, any sequence of n trials with k successes (and n − k failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\binom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4f99a2b24b9135bb87a91746d846a3f84833c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\textstyle {\binom {n}{k}}}"> such sequences, since the binomial coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\binom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4f99a2b24b9135bb87a91746d846a3f84833c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\textstyle {\binom {n}{k}}}"> counts the number of ways to choose the positions of the k successes among the n trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them (pk qn−k) must be added <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\binom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4f99a2b24b9135bb87a91746d846a3f84833c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\textstyle {\binom {n}{k}}}"> times, hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f238b0228e33bf8e82bf61dce5f262b54acf64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.401ex; height:3.176ex;" alt="{\textstyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}">. In creating reference tables for binomial distribution probability, usually, the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k,n,p)=f(n-k,n,1-p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k,n,p)=f(n-k,n,1-p).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c0fe9d29eeb85a1937b53dde044616752b1011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.846ex; height:2.843ex;" alt="{\displaystyle f(k,n,p)=f(n-k,n,1-p).}"></dd></dl> Looking at the expression f(k, n, p) as a function of k, there is a k value that maximizes it. This k value can be found by calculating <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d211bc315502f007bd8fb7f9d8c244d59b39ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.71ex; height:6.509ex;" alt="{\displaystyle {\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}}"></dd></dl> and comparing it to 1. There is always an integer M that satisfies<a href="#cite_note-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)p-1\leq M<(n+1)p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>M</mi> <mo><</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)p-1\leq M<(n+1)p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70d7a3d60734136e1f05651e856e4fdb28274ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.041ex; height:2.843ex;" alt="{\displaystyle (n+1)p-1\leq M<(n+1)p.}"></dd></dl> f(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which f is maximal: (n + 1) p and (n + 1) p − 1. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>. Equivalently, M − p < np ≤ M + 1 − p. Taking the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor function</a>, we obtain M = floor(np).<a href="#cite_note-3">[note 1]</a> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=3" title="Edit section: Example">edit</a>]</div> Suppose a <a href="/wiki/Fair_coin" title="Fair coin">biased coin</a> comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(4,6,0.3)={\binom {6}{4}}0.3^{4}(1-0.3)^{6-4}=0.059535.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>0.3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mn>0.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>0.3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>0.059535.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(4,6,0.3)={\binom {6}{4}}0.3^{4}(1-0.3)^{6-4}=0.059535.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/341ee6e07355e681839edfb828fbcacb2c0f7912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.629ex; height:6.176ex;" alt="{\displaystyle f(4,6,0.3)={\binom {6}{4}}0.3^{4}(1-0.3)^{6-4}=0.059535.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Cumulative_distribution_function">Cumulative distribution function</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=4" title="Edit section: Cumulative distribution function">edit</a>]</div> The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> can be expressed as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8689ba703ee51a5f66f605ea1f293fc74fa380b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.404ex; height:7.676ex;" alt="{\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor k\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor k\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a64054ec38bec048113dd328cf867e9e62b581d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.276ex; height:2.843ex;" alt="{\displaystyle \lfloor k\rfloor }"> is the "floor" under k, i.e. the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">greatest integer</a> less than or equal to k. It can also be represented in terms of the <a href="/wiki/Regularized_incomplete_beta_function" class="mw-redirect" title="Regularized incomplete beta function">regularized incomplete beta function</a>, as follows:<a href="#cite_note-4">[3]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F(k;n,p)&=\Pr(X\leq k)\\&=I_{1-p}(n-k,k+1)\\&=(n-k){n \choose k}\int _{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(k;n,p)&=\Pr(X\leq k)\\&=I_{1-p}(n-k,k+1)\\&=(n-k){n \choose k}\int _{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f874721853e1614ba99af968f826aefd075e143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:48.836ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}F(k;n,p)&=\Pr(X\leq k)\\&=I_{1-p}(n-k,k+1)\\&=(n-k){n \choose k}\int _{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt,\end{aligned}}}"></dd></dl> which is equivalent to the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> of the <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> and of the <a href="/wiki/F-distribution" title="F-distribution">F-distribution</a>:<a href="#cite_note-5">[4]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)=F_{\text{beta-distribution}}\left(x=1-p;\alpha =n-k,\beta =k+1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>beta-distribution</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>;</mo> <mi>α</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)=F_{\text{beta-distribution}}\left(x=1-p;\alpha =n-k,\beta =k+1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f92250fe71290d61de0333a7ddd9b568942e59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.835ex; height:2.843ex;" alt="{\displaystyle F(k;n,p)=F_{\text{beta-distribution}}\left(x=1-p;\alpha =n-k,\beta =k+1\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)=F_{F{\text{-distribution}}}\left(x={\frac {1-p}{p}}{\frac {k+1}{n-k}};d_{1}=2(n-k),d_{2}=2(k+1)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-distribution</mtext> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>;</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)=F_{F{\text{-distribution}}}\left(x={\frac {1-p}{p}}{\frac {k+1}{n-k}};d_{1}=2(n-k),d_{2}=2(k+1)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cdde2f8cc02dcf7f7c8a128bf830215faaaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.375ex; height:6.176ex;" alt="{\displaystyle F(k;n,p)=F_{F{\text{-distribution}}}\left(x={\frac {1-p}{p}}{\frac {k+1}{n-k}};d_{1}=2(n-k),d_{2}=2(k+1)\right).}"></dd></dl> Some closed-form bounds for the cumulative distribution function are given <a href="#Tail_bounds">below</a>. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=5" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Expected_value_and_variance">Expected value and variance</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=6" title="Edit section: Expected value and variance">edit</a>]</div> If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of X is:<a href="#cite_note-6">[5]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]=np.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=np.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f16b365410a1b23b5592c53d3ae6354f1a79aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.166ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]=np.}"></dd></dl> This follows from the linearity of the expected value along with the fact that X is the sum of n identical Bernoulli random variables, each with expected value p. In other words, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}"> are identical (and independent) Bernoulli random variables with parameter p, then X = X1 + ... + Xn and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]=\operatorname {E} [X_{1}+\cdots +X_{n}]=\operatorname {E} [X_{1}]+\cdots +\operatorname {E} [X_{n}]=p+\cdots +p=np.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>p</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=\operatorname {E} [X_{1}+\cdots +X_{n}]=\operatorname {E} [X_{1}]+\cdots +\operatorname {E} [X_{n}]=p+\cdots +p=np.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f238d520c68a1d1b9b318492ddda39f4cc45bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.884ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]=\operatorname {E} [X_{1}+\cdots +X_{n}]=\operatorname {E} [X_{1}]+\cdots +\operatorname {E} [X_{n}]=p+\cdots +p=np.}"></dd></dl> The <a href="/wiki/Variance" title="Variance">variance</a> is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (X)=npq=np(1-p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X)=npq=np(1-p).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca514f76e21fa0f7b2d2fd8193c4cf4c2cf866e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.63ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} (X)=npq=np(1-p).}"></dd></dl> This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances. <div class="mw-heading mw-heading3"><h3 id="Higher_moments">Higher moments</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=7" title="Edit section: Higher moments">edit</a>]</div> The first 6 <a href="/wiki/Central_moment" title="Central moment">central moments</a>, defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5ea3e05b674668550675c3c4593c725a1ec86b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.751ex; height:2.843ex;" alt="{\displaystyle \mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right]}">, are given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=np(1-p),\\\mu _{3}&=np(1-p)(1-2p),\\\mu _{4}&=np(1-p)(1+(3n-6)p(1-p)),\\\mu _{5}&=np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\\mu _{6}&=np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^{2}p^{2}(1-p)^{2}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mi>n</mi> <mo>−</mo> <mn>12</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>30</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5</mn> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>−</mo> <mn>26</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mn>15</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=np(1-p),\\\mu _{3}&=np(1-p)(1-2p),\\\mu _{4}&=np(1-p)(1+(3n-6)p(1-p)),\\\mu _{5}&=np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\\mu _{6}&=np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^{2}p^{2}(1-p)^{2}).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c1eb8cb4c395baf1beb5ba3054cd0dab131e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:95.622ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=np(1-p),\\\mu _{3}&=np(1-p)(1-2p),\\\mu _{4}&=np(1-p)(1+(3n-6)p(1-p)),\\\mu _{5}&=np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\\mu _{6}&=np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^{2}p^{2}(1-p)^{2}).\end{aligned}}}"></dd></dl> The non-central moments satisfy <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [X]&=np,\\\operatorname {E} [X^{2}]&=np(1-p)+n^{2}p^{2},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [X]&=np,\\\operatorname {E} [X^{2}]&=np(1-p)+n^{2}p^{2},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f2b3a9af52fc1ea633476400290069fc3ae7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.483ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [X]&=np,\\\operatorname {E} [X^{2}]&=np(1-p)+n^{2}p^{2},\end{aligned}}}"></dd></dl> and in general <a href="#cite_note-Andreas2008-7">[6]</a><a href="#cite_note-Nguyen2021-8">[7]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{c}]=\sum _{k=0}^{c}\left\{{c \atop k}\right\}n^{\underline {k}}p^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>c</mi> <mi>k</mi> </mfrac> </mrow> <mo>}</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>k</mi> <mo>_</mo> </munder> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{c}]=\sum _{k=0}^{c}\left\{{c \atop k}\right\}n^{\underline {k}}p^{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db435ced7af59fa481fe26a023a1429d18a6a83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.305ex; height:7.009ex;" alt="{\displaystyle \operatorname {E} [X^{c}]=\sum _{k=0}^{c}\left\{{c \atop k}\right\}n^{\underline {k}}p^{k},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \left\{{c \atop k}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>c</mi> <mi>k</mi> </mfrac> </mrow> <mo>}</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \left\{{c \atop k}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bac7c5417d94b0e4afe33363f1f1b6abc06428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.124ex; height:3.176ex;" alt="{\displaystyle \textstyle \left\{{c \atop k}\right\}}"> are the <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>k</mi> <mo>_</mo> </munder> </mrow> </msup> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f07c553938ed18bdb5fee981a95a9b50af9b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.941ex; height:3.176ex;" alt="{\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)}"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th <a href="/wiki/Falling_and_rising_factorials" title="Falling and rising factorials">falling power</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. A simple bound <a href="#cite_note-9">[8]</a> follows by bounding the Binomial moments via the <a href="/wiki/Poisson_distribution#Higher_moments" title="Poisson distribution">higher Poisson moments</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{c}]\leq \left({\frac {c}{\ln(c/(np)+1)}}\right)^{c}\leq (np)^{c}\exp \left({\frac {c^{2}}{2np}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{c}]\leq \left({\frac {c}{\ln(c/(np)+1)}}\right)^{c}\leq (np)^{c}\exp \left({\frac {c^{2}}{2np}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4150b3bc8d3ada7391088c4493928df6798680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.398ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} [X^{c}]\leq \left({\frac {c}{\ln(c/(np)+1)}}\right)^{c}\leq (np)^{c}\exp \left({\frac {c^{2}}{2np}}\right).}"></dd></dl> This shows that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=O({\sqrt {np}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=O({\sqrt {np}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e92fb16bfa3a153ae951523ff7f44f6ddd8881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.188ex; height:3.176ex;" alt="{\displaystyle c=O({\sqrt {np}})}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{c}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{c}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5989f9c9f5202059ad1c0a4026d267ee2a975761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.818ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X^{c}]}"> is at most a constant factor away from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d46bd564a6f77db38e12b156443bdbc3262cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.801ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]^{c}}"> <div class="mw-heading mw-heading3"><h3 id="Mode">Mode</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=8" title="Edit section: Mode">edit</a>]</div> Usually the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> of a binomial B(n, p) distribution is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor (n+1)p\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor (n+1)p\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcda6d9602cb8abc4e83e4a1b19ed23bc5062932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle \lfloor (n+1)p\rfloor }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \cdot \rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \cdot \rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb5178c0fbe4ac275f1083ee8f1a6e0feb8f872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.712ex; height:2.843ex;" alt="{\displaystyle \lfloor \cdot \rfloor }"> is the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a>. However, when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\(n+1)\,p\ {\text{ and }}\ (n+1)\,p-1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mode</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is 0 or a noninteger</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\(n+1)\,p\ {\text{ and }}\ (n+1)\,p-1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74519d31faaf11d1b8e90572aeaa12fae140584e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:71.82ex; height:8.843ex;" alt="{\displaystyle {\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\(n+1)\,p\ {\text{ and }}\ (n+1)\,p-1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}}"></dd></dl> Proof: Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k)={\binom {n}{k}}p^{k}q^{n-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k)={\binom {n}{k}}p^{k}q^{n-k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f3731d97f73b24bb16c939f34ad4bc0324ab1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.551ex; height:6.176ex;" alt="{\displaystyle f(k)={\binom {n}{k}}p^{k}q^{n-k}.}"></dd></dl> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6ac10fa45fb984d886065f959a6bdd467b5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}"> only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc54adf96acf7e2466de00b7739144d36840108f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle f(0)}"> has a nonzero value with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00884d9c3ef0dbc18fe185a4444f1da931c8ba0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=1}">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"> we find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18729f39cdba4b90c6c24c34cf558e826d0d1f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.744ex; height:2.843ex;" alt="{\displaystyle f(n)=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7d839a238fc7f702f16f17238b97af187713f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.56ex; height:2.843ex;" alt="{\displaystyle f(k)=0}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\neq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≠</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\neq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe6b432697569196d816b5f1c4582a685ab6b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.704ex; height:2.676ex;" alt="{\displaystyle k\neq n}">. This proves that the mode is 0 for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6ac10fa45fb984d886065f959a6bdd467b5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}">. We find <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(k+1)}{f(k)}}={\frac {(n-k)p}{(k+1)(1-p)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(k+1)}{f(k)}}={\frac {(n-k)p}{(k+1)(1-p)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cd7ba8eb9f47c813648663c1937f5f01ecb0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.078ex; height:6.509ex;" alt="{\displaystyle {\frac {f(k+1)}{f(k)}}={\frac {(n-k)p}{(k+1)(1-p)}}}">.</dd></dl> From this follows <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}k>(n+1)p-1\Rightarrow f(k+1)<f(k)\\k=(n+1)p-1\Rightarrow f(k+1)=f(k)\\k<(n+1)p-1\Rightarrow f(k+1)>f(k)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>k</mi> <mo>></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo><</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k>(n+1)p-1\Rightarrow f(k+1)<f(k)\\k=(n+1)p-1\Rightarrow f(k+1)=f(k)\\k<(n+1)p-1\Rightarrow f(k+1)>f(k)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3d2862e74e83011904c0e11b0257d678441df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:36.754ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}k>(n+1)p-1\Rightarrow f(k+1)<f(k)\\k=(n+1)p-1\Rightarrow f(k+1)=f(k)\\k<(n+1)p-1\Rightarrow f(k+1)>f(k)\end{aligned}}}"></dd></dl> So when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)p-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd23b263c6e010f3bad0f3f6fed1e55b94cd440a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.379ex; height:2.843ex;" alt="{\displaystyle (n+1)p-1}"> is an integer, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)p-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd23b263c6e010f3bad0f3f6fed1e55b94cd440a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.379ex; height:2.843ex;" alt="{\displaystyle (n+1)p-1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0dc4e70c7a15d023d2652d5b69f22203e6cb476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.376ex; height:2.843ex;" alt="{\displaystyle (n+1)p}"> is a mode. In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)p-1\notin \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)p-1\notin \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5c94d4e8ee065391e22672e3551bb6cef768ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.77ex; height:2.843ex;" alt="{\displaystyle (n+1)p-1\notin \mathbb {Z} }">, then only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor (n+1)p-1\rfloor +1=\lfloor (n+1)p\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor (n+1)p-1\rfloor +1=\lfloor (n+1)p\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2a5233bb21b7f40b1dbc83747e05e19eef2b90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.986ex; height:2.843ex;" alt="{\displaystyle \lfloor (n+1)p-1\rfloor +1=\lfloor (n+1)p\rfloor }"> is a mode.<a href="#cite_note-10">[9]</a> <div class="mw-heading mw-heading3"><h3 id="Median">Median</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=9" title="Edit section: Median">edit</a>]</div> In general, there is no single formula to find the <a href="/wiki/Median" title="Median">median</a> for a binomial distribution, and it may even be non-unique. However, several special results have been established: <ul><li>If np is an integer, then the mean, median, and mode coincide and equal np.<a href="#cite_note-11">[10]</a><a href="#cite_note-12">[11]</a></li> <li>Any median m must lie within the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor np\rfloor \leq m\leq \lceil np\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mi>p</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mi>p</mi> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor np\rfloor \leq m\leq \lceil np\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d225e6448db67792cc4a040fbee79c3d8289b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.495ex; height:2.843ex;" alt="{\displaystyle \lfloor np\rfloor \leq m\leq \lceil np\rceil }">.<a href="#cite_note-KaasBuhrman-13">[12]</a></li> <li>A median m cannot lie too far away from the mean:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |m-np|\leq \min\{{\ln 2},\max\{p,1-p\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mo>−</mo> <mi>n</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |m-np|\leq \min\{{\ln 2},\max\{p,1-p\}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c522650ed3e564296011ddb76ad36cf8ac979a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.586ex; height:2.843ex;" alt="{\displaystyle |m-np|\leq \min\{{\ln 2},\max\{p,1-p\}\}}"> .<a href="#cite_note-Hamza-14">[13]</a></li> <li>The median is unique and equal to m = <a href="/wiki/Rounding" title="Rounding">round</a>(np) when |m − np| ≤ min{p, 1 − p} (except for the case when p = 1/2 and n is odd).<a href="#cite_note-KaasBuhrman-13">[12]</a></li> <li>When p is a rational number (with the exception of p = 1/2\ and n odd) the median is unique.<a href="#cite_note-Nowakowski-15">[14]</a></li> <li>When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008c655cf27de670fe571a82643461724e00977b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.356ex; height:5.176ex;" alt="{\displaystyle p={\frac {1}{2}}}"> and n is odd, any number m in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{\bigl (}n-1{\bigr )}\leq m\leq {\frac {1}{2}}{\bigl (}n+1{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\bigl (}n-1{\bigr )}\leq m\leq {\frac {1}{2}}{\bigl (}n+1{\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196fcb881a40cabb4d7be54ba1bbaa25b3303e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.289ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}{\bigl (}n-1{\bigr )}\leq m\leq {\frac {1}{2}}{\bigl (}n+1{\bigr )}}"> is a median of the binomial distribution. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008c655cf27de670fe571a82643461724e00977b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.356ex; height:5.176ex;" alt="{\displaystyle p={\frac {1}{2}}}"> and n is even, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {n}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {n}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55789405263d50f683a0b51b06e5f8eb12c9c749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.37ex; height:4.676ex;" alt="{\displaystyle m={\frac {n}{2}}}"> is the unique median.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Tail_bounds">Tail bounds</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=10" title="Edit section: Tail bounds">edit</a>]</div> For k ≤ np, upper bounds can be derived for the lower tail of the cumulative distribution function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)=\Pr(X\leq k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)=\Pr(X\leq k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cff50b24b2c6bb76c17d87066dd440a7f62c61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.085ex; height:2.843ex;" alt="{\displaystyle F(k;n,p)=\Pr(X\leq k)}">, the probability that there are at most k successes. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298cf38295eca689d2ab8ade2c5f6c8a38278628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.323ex; height:2.843ex;" alt="{\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)}">, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for k ≥ np. <a href="/wiki/Hoeffding%27s_inequality" title="Hoeffding's inequality">Hoeffding's inequality</a> yields the simple bound <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)\leq \exp \left(-2n\left(p-{\frac {k}{n}}\right)^{2}\right),\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mi>n</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)\leq \exp \left(-2n\left(p-{\frac {k}{n}}\right)^{2}\right),\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635aca348a43ff429a04bb8deb156df33a9b162b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.229ex; width:35.296ex; height:7.509ex;" alt="{\displaystyle F(k;n,p)\leq \exp \left(-2n\left(p-{\frac {k}{n}}\right)^{2}\right),\!}"></dd></dl> which is however not very tight. In particular, for p = 1, we have that F(k; n, p) = 0 (for fixed k, n with k < n), but Hoeffding's bound evaluates to a positive constant. A sharper bound can be obtained from the <a href="/wiki/Chernoff_bound" title="Chernoff bound">Chernoff bound</a>:<a href="#cite_note-ag-16">[15]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)\leq \exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>n</mi> <mi>D</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mo>∥</mo> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)\leq \exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/676c3163837a46726f25e1aa2954d7e4d66f91ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.254ex; height:6.176ex;" alt="{\displaystyle F(k;n,p)\leq \exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right)}"></dd></dl> where D(a ∥ p) is the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">relative entropy (or Kullback-Leibler divergence)</a> between an a-coin and a p-coin (i.e. between the Bernoulli(a) and Bernoulli(p) distribution): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(a\parallel p)=(a)\ln {\frac {a}{p}}+(1-a)\ln {\frac {1-a}{1-p}}.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∥</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>p</mi> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(a\parallel p)=(a)\ln {\frac {a}{p}}+(1-a)\ln {\frac {1-a}{1-p}}.\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78138dc698bf7b307d8e415f614120992df4228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.204ex; width:38.631ex; height:5.676ex;" alt="{\displaystyle D(a\parallel p)=(a)\ln {\frac {a}{p}}+(1-a)\ln {\frac {1-a}{1-p}}.\!}"></dd></dl> Asymptotically, this bound is reasonably tight; see <a href="#cite_note-ag-16">[15]</a> for details. One can also obtain lower bounds on the tail F(k; n, p), known as anti-concentration bounds. By approximating the binomial coefficient with <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's formula</a> it can be shown that<a href="#cite_note-17">[16]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {8n{\tfrac {k}{n}}(1-{\tfrac {k}{n}})}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>8</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>n</mi> <mi>D</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mo>∥</mo> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {8n{\tfrac {k}{n}}(1-{\tfrac {k}{n}})}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22aafed1f95ec195f0eb30627ecd1dc9c7751d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:50.462ex; height:8.343ex;" alt="{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {8n{\tfrac {k}{n}}(1-{\tfrac {k}{n}})}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right),}"></dd></dl> which implies the simpler but looser bound <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {2n}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>n</mi> </msqrt> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>n</mi> <mi>D</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> <mo>∥</mo> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {2n}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219600d2b4a4c5eaf6ba269a6ec0a5af2be1dfd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.617ex; height:6.509ex;" alt="{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {2n}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right).}"></dd></dl> For p = 1/2 and k ≥ 3n/8 for even n, it is possible to make the denominator constant:<a href="#cite_note-18">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;n,{\tfrac {1}{2}})\geq {\frac {1}{15}}\exp \left(-16n\left({\frac {1}{2}}-{\frac {k}{n}}\right)^{2}\right).\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>16</mn> <mi>n</mi> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k;n,{\tfrac {1}{2}})\geq {\frac {1}{15}}\exp \left(-16n\left({\frac {1}{2}}-{\frac {k}{n}}\right)^{2}\right).\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3720d25b301005114222afac7e21694a36120de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.204ex; width:41.299ex; height:7.509ex;" alt="{\displaystyle F(k;n,{\tfrac {1}{2}})\geq {\frac {1}{15}}\exp \left(-16n\left({\frac {1}{2}}-{\frac {k}{n}}\right)^{2}\right).\!}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Statistical_inference">Statistical inference</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=11" title="Edit section: Statistical inference">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_parameters">Estimation of parameters</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=12" title="Edit section: Estimation of parameters">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Beta_distribution#Bayesian_inference" title="Beta distribution">Beta distribution § Bayesian inference</a></div> When n is known, the parameter p can be estimated using the proportion of successes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}={\frac {x}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}={\frac {x}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0c61b30ac9345956b8060d277981ad355e4fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:7.425ex; height:4.676ex;" alt="{\displaystyle {\widehat {p}}={\frac {x}{n}}.}"></dd></dl> This estimator is found using <a href="/wiki/Maximum_likelihood_estimator" class="mw-redirect" title="Maximum likelihood estimator">maximum likelihood estimator</a> and also the <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a>. This estimator is <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> and uniformly with <a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum variance</a>, proven using <a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a>, since it is based on a <a href="/wiki/Minimal_sufficient" class="mw-redirect" title="Minimal sufficient">minimal sufficient</a> and <a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">complete</a> statistic (i.e.: x). It is also <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent</a> both in probability and in <a href="/wiki/Mean_squared_error" title="Mean squared error">MSE</a>. This statistic is <a href="/wiki/Asymptotic_distribution" title="Asymptotic distribution">asymptotically</a> <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a> thanks to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, because it is the same as taking the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean</a> over Bernoulli samples. It has a variance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle var({\widehat {p}})={\frac {p(1-p)}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mi>a</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle var({\widehat {p}})={\frac {p(1-p)}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8675d99cf6368fb2399becfb9a21aa0b763fec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.66ex; height:5.676ex;" alt="{\displaystyle var({\widehat {p}})={\frac {p(1-p)}{n}}}">, a property which is used in various ways, such as in <a href="/wiki/Binomial_proportion_confidence_interval#Wald_interval" title="Binomial proportion confidence interval">Wald's confidence intervals</a>. A closed form <a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayes estimator</a> for p also exists when using the <a href="/wiki/Beta_distribution" title="Beta distribution">Beta distribution</a> as a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate</a> <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a>. When using a general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (\alpha ,\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a028b0242ef2a39b1118715997163faedb09dd87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.408ex; height:2.843ex;" alt="{\displaystyle \operatorname {Beta} (\alpha ,\beta )}"> as a prior, the <a href="/wiki/Bayes_estimator#Posterior_mean" title="Bayes estimator">posterior mean</a> estimator is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{b}={\frac {x+\alpha }{n+\alpha +\beta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{b}={\frac {x+\alpha }{n+\alpha +\beta }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0384bcce10ab80362df2b483fe0ed07eda3f717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:16.863ex; height:5.509ex;" alt="{\displaystyle {\widehat {p}}_{b}={\frac {x+\alpha }{n+\alpha +\beta }}.}"></dd></dl> The Bayes estimator is <a href="/wiki/Asymptotic_efficiency_(Bayes)" class="mw-redirect" title="Asymptotic efficiency (Bayes)">asymptotically efficient</a> and as the sample size approaches infinity (n → ∞), it approaches the <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">MLE</a> solution.<a href="#cite_note-19">[18]</a> The Bayes estimator is <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">biased</a> (how much depends on the priors), <a href="/wiki/Bayes_estimator#Admissibility" title="Bayes estimator">admissible</a> and <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent</a> in probability. Using the Bayesian estimator with the Beta distribution can be used with <a href="/wiki/Thompson_sampling" title="Thompson sampling">Thompson sampling</a>. For the special case of using the <a href="/wiki/Standard_uniform_distribution" class="mw-redirect" title="Standard uniform distribution">standard uniform distribution</a> as a <a href="/wiki/Non-informative_prior" class="mw-redirect" title="Non-informative prior">non-informative prior</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (\alpha =1,\beta =1)=U(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (\alpha =1,\beta =1)=U(0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9759f630300f34fb942fd7a706465f1b6c9a61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.979ex; height:2.843ex;" alt="{\displaystyle \operatorname {Beta} (\alpha =1,\beta =1)=U(0,1)}">, the posterior mean estimator becomes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{b}={\frac {x+1}{n+2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{b}={\frac {x+1}{n+2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bae5563c78c303f0b2f1aaba1785f561d9f5399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:12.365ex; height:5.343ex;" alt="{\displaystyle {\widehat {p}}_{b}={\frac {x+1}{n+2}}.}"></dd></dl> (A <a href="/wiki/Bayes_estimator#Posterior_mode" title="Bayes estimator">posterior mode</a> should just lead to the standard estimator.) This method is called the <a href="/wiki/Rule_of_succession" title="Rule of succession">rule of succession</a>, which was introduced in the 18th century by <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>. When relying on <a href="/wiki/Jeffreys_prior" title="Jeffreys prior">Jeffreys prior</a>, the prior is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (\alpha ={\frac {1}{2}},\beta ={\frac {1}{2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (\alpha ={\frac {1}{2}},\beta ={\frac {1}{2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1814522a54e5770330bee8cd04faa202ee111440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.602ex; height:5.176ex;" alt="{\displaystyle \operatorname {Beta} (\alpha ={\frac {1}{2}},\beta ={\frac {1}{2}})}">,<a href="#cite_note-20">[19]</a> which leads to the estimator: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{Jeffreys}={\frac {x+{\frac {1}{2}}}{n+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> <mi>e</mi> <mi>f</mi> <mi>f</mi> <mi>r</mi> <mi>e</mi> <mi>y</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{Jeffreys}={\frac {x+{\frac {1}{2}}}{n+1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d910b4ab13feed021c4ffeae6183662b31b3f35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:18.801ex; height:6.509ex;" alt="{\displaystyle {\widehat {p}}_{Jeffreys}={\frac {x+{\frac {1}{2}}}{n+1}}.}"></dd></dl> When estimating p with very rare events and a small n (e.g.: if x = 0), then using the standard estimator leads to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2f106c2630fab48ac3c4fc2b59a70b3b67a45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.357ex; height:2.509ex;" alt="{\displaystyle {\widehat {p}}=0,}"> which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators.<a href="#cite_note-21">[20]</a> One way is to use the Bayes estimator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{b}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c962fa336a394ce7cc1591b7fd38b5971e53e101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:2.386ex; height:2.676ex;" alt="{\displaystyle {\widehat {p}}_{b}}">, leading to: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{b}={\frac {1}{n+2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{b}={\frac {1}{n+2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9a3a404ea6597b660bfbb2d38461448bce8731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:12.365ex; height:5.343ex;" alt="{\displaystyle {\widehat {p}}_{b}={\frac {1}{n+2}}.}"></dd></dl> Another method is to use the upper bound of the <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> obtained using the <a href="/wiki/Rule_of_three_(statistics)" title="Rule of three (statistics)">rule of three</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p}}_{\text{rule of 3}}={\frac {3}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rule of 3</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p}}_{\text{rule of 3}}={\frac {3}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3ae5c0117971e485de1f9e78bb6640ad8fc8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:13.712ex; height:5.176ex;" alt="{\displaystyle {\widehat {p}}_{\text{rule of 3}}={\frac {3}{n}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Confidence_intervals_for_the_parameter_p">Confidence intervals for the parameter p</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=13" title="Edit section: Confidence intervals for the parameter p">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_proportion_confidence_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Z-test#Comparing_the_Proportions_of_Two_Binomials" title="Z-test">Z-test § Comparing the Proportions of Two Binomials</a></div> Even for quite large values of n, the actual distribution of the mean is significantly nonnormal.<a href="#cite_note-Brown2001-22">[21]</a> Because of this problem several methods to estimate confidence intervals have been proposed. In the equations for confidence intervals below, the variables have the following meaning: <ul><li>n1 is the number of successes out of n, the total number of trials</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p\,}}={\frac {n_{1}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p\,}}={\frac {n_{1}}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b69e98db6d3007b699801c5e8d7f363e1d46059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.029ex; height:4.676ex;" alt="{\displaystyle {\widehat {p\,}}={\frac {n_{1}}{n}}}"> is the proportion of successes</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\tfrac {1}{2}}\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\tfrac {1}{2}}\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e26f1c45ea84772da6e541b6303f6af26c33e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.149ex; height:3.509ex;" alt="{\displaystyle 1-{\tfrac {1}{2}}\alpha }"> <a href="/wiki/Quantile" title="Quantile">quantile</a> of a <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a> (i.e., <a href="/wiki/Probit" title="Probit">probit</a>) corresponding to the target error rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">. For example, for a 95% <a href="/wiki/Confidence_level" class="mw-redirect" title="Confidence level">confidence level</a> the error <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> = 0.05, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\tfrac {1}{2}}\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\tfrac {1}{2}}\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e26f1c45ea84772da6e541b6303f6af26c33e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.149ex; height:3.509ex;" alt="{\displaystyle 1-{\tfrac {1}{2}}\alpha }"> = 0.975 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> = 1.96.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Wald_method">Wald method</h4>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=14" title="Edit section: Wald method">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_proportion_confidence_interval#Wald_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval § Wald interval</a></div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p\,}}\pm z{\sqrt {\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>±</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p\,}}\pm z{\sqrt {\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135d45c9aa84fb7acdd0605a076eed8cfdfa364c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:18.306ex; height:7.676ex;" alt="{\displaystyle {\widehat {p\,}}\pm z{\sqrt {\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}}.}"></dd></dl> A <a href="/wiki/Continuity_correction" title="Continuity correction">continuity correction</a> of 0.5/n may be added.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] <div class="mw-heading mw-heading4"><h4 id="Agresti–Coull_method">Agresti–Coull method</h4>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=15" title="Edit section: Agresti–Coull method">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_proportion_confidence_interval#Agresti–Coull_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval § Agresti–Coull interval</a></div> <a href="#cite_note-Agresti1988-23">[22]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {p}}\pm z{\sqrt {\frac {{\tilde {p}}(1-{\tilde {p}})}{n+z^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>±</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {p}}\pm z{\sqrt {\frac {{\tilde {p}}(1-{\tilde {p}})}{n+z^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290a7265491fabdde9b7cf4c816499fab7884ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:17.068ex; height:7.676ex;" alt="{\displaystyle {\tilde {p}}\pm z{\sqrt {\frac {{\tilde {p}}(1-{\tilde {p}})}{n+z^{2}}}}}"></dd></dl> Here the estimate of p is modified to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {p}}={\frac {n_{1}+{\frac {1}{2}}z^{2}}{n+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {p}}={\frac {n_{1}+{\frac {1}{2}}z^{2}}{n+z^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb017e9c9b6c319d82e8772a8b8fcffa857f82ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:14.475ex; height:6.843ex;" alt="{\displaystyle {\tilde {p}}={\frac {n_{1}+{\frac {1}{2}}z^{2}}{n+z^{2}}}}"></dd></dl> This method works well for n > 10 and n1 ≠ 0, n.<a href="#cite_note-24">[23]</a> See here for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq 10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a7cb82f506cef64150cbde33514a8d87ed40ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.818ex; height:2.343ex;" alt="{\displaystyle n\leq 10}">.<a href="#cite_note-25">[24]</a> For n1 = 0, n use the Wilson (score) method below. <div class="mw-heading mw-heading4"><h4 id="Arcsine_method">Arcsine method</h4>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=16" title="Edit section: Arcsine method">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_proportion_confidence_interval#Arcsine_transformation" title="Binomial proportion confidence interval">Binomial proportion confidence interval § Arcsine transformation</a></div> <a href="#cite_note-Pires00-26">[25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\left(\arcsin \left({\sqrt {\widehat {p\,}}}\right)\pm {\frac {z}{2{\sqrt {n}}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\left(\arcsin \left({\sqrt {\widehat {p\,}}}\right)\pm {\frac {z}{2{\sqrt {n}}}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187fa36ccf594df63b4f3ffddbad4fc7d162c10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.12ex; height:6.509ex;" alt="{\displaystyle \sin ^{2}\left(\arcsin \left({\sqrt {\widehat {p\,}}}\right)\pm {\frac {z}{2{\sqrt {n}}}}\right).}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Wilson_(score)_method">Wilson (score) method</h4>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=17" title="Edit section: Wilson (score) method">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Binomial_proportion_confidence_interval#Wilson_score_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval § Wilson score interval</a></div> The notation in the formula below differs from the previous formulas in two respects:<a href="#cite_note-Wilson1927-27">[26]</a> <ul><li>Firstly, zx has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the xth quantile of the standard normal distribution', rather than being a shorthand for 'the (1 − x)th quantile'.</li> <li>Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z_{\alpha /2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z_{\alpha /2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c222ea3cd0e608193653d94bed365b0bcad8f6e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.196ex; height:2.509ex;" alt="{\displaystyle z=z_{\alpha /2}}"> to get the lower bound, or use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z_{1-\alpha /2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z_{1-\alpha /2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c307e50a7e05de6301d2c8683121c800d837cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.296ex; height:2.509ex;" alt="{\displaystyle z=z_{1-\alpha /2}}"> to get the upper bound. For example: for a 95% confidence level the error <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> = 0.05, so one gets the lower bound by using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z_{\alpha /2}=z_{0.025}=-1.96}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.025</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1.96</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z_{\alpha /2}=z_{0.025}=-1.96}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c0d5cf99cb562c9ee4821e146c5ff4d3a3541bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.394ex; height:3.009ex;" alt="{\displaystyle z=z_{\alpha /2}=z_{0.025}=-1.96}">, and one gets the upper bound by using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z_{1-\alpha /2}=z_{0.975}=1.96}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.975</mn> </mrow> </msub> <mo>=</mo> <mn>1.96</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z_{1-\alpha /2}=z_{0.975}=1.96}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c2f0af5cfa66de895ba743ca3d0fabed3bc42b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.686ex; height:3.009ex;" alt="{\displaystyle z=z_{1-\alpha /2}=z_{0.975}=1.96}">.</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {p\,}}+{\frac {z^{2}}{2n}}+z{\sqrt {{\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}+{\frac {z^{2}}{4n^{2}}}}}}{1+{\frac {z^{2}}{n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {p\,}}+{\frac {z^{2}}{2n}}+z{\sqrt {{\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}+{\frac {z^{2}}{4n^{2}}}}}}{1+{\frac {z^{2}}{n}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1964bbe6154395fd9ef387633409a484b42fb88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.09ex; height:10.676ex;" alt="{\displaystyle {\frac {{\widehat {p\,}}+{\frac {z^{2}}{2n}}+z{\sqrt {{\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}+{\frac {z^{2}}{4n^{2}}}}}}{1+{\frac {z^{2}}{n}}}}}"><a href="#cite_note-28">[27]</a></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Comparison">Comparison</h4>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=18" title="Edit section: Comparison">edit</a>]</div> The so-called "exact" (<a href="/wiki/Binomial_proportion_confidence_interval#Clopper–Pearson_interval" title="Binomial proportion confidence interval">Clopper–Pearson</a>) method is the most conservative.<a href="#cite_note-Brown2001-22">[21]</a> (Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.) The Wald method, although commonly recommended in textbooks, is the most biased.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=19" title="Edit section: Related distributions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Sums_of_binomials">Sums of binomials</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=20" title="Edit section: Sums of binomials">edit</a>]</div> If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is Z = X + Y ~ B(n + m, p):<a href="#cite_note-29">[28]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {P} (Z=k)&=\sum _{i=0}^{k}\left[{\binom {n}{i}}p^{i}(1-p)^{n-i}\right]\left[{\binom {m}{k-i}}p^{k-i}(1-p)^{m-k+i}\right]\\&={\binom {n+m}{k}}p^{k}(1-p)^{n+m-k}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>i</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mrow> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</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>m</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mi>i</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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>m</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {P} (Z=k)&=\sum _{i=0}^{k}\left[{\binom {n}{i}}p^{i}(1-p)^{n-i}\right]\left[{\binom {m}{k-i}}p^{k-i}(1-p)^{m-k+i}\right]\\&={\binom {n+m}{k}}p^{k}(1-p)^{n+m-k}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38fc38e9a5e2c49743f45b4dab5dae6230ab2ad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:64.119ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {P} (Z=k)&=\sum _{i=0}^{k}\left[{\binom {n}{i}}p^{i}(1-p)^{n-i}\right]\left[{\binom {m}{k-i}}p^{k-i}(1-p)^{m-k+i}\right]\\&={\binom {n+m}{k}}p^{k}(1-p)^{n+m-k}\end{aligned}}}"></dd></dl> A Binomial distributed random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables X ~ B(n, p) and Y ~ B(m, p) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z = X + Y ~ B(n + m, p). This can also be proven directly using the addition rule. However, if X and Y do not have the same probability p, then the variance of the sum will be <a href="/wiki/Binomial_sum_variance_inequality" title="Binomial sum variance inequality">smaller than the variance of a binomial variable</a> distributed as B(n + m, p). <div class="mw-heading mw-heading3"><h3 id="Poisson_binomial_distribution">Poisson binomial distribution</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=21" title="Edit section: Poisson binomial distribution">edit</a>]</div> The binomial distribution is a special case of the <a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial distribution</a>, which is the distribution of a sum of n independent non-identical <a href="/wiki/Bernoulli_trials" class="mw-redirect" title="Bernoulli trials">Bernoulli trials</a> B(pi).<a href="#cite_note-30">[29]</a> <div class="mw-heading mw-heading3"><h3 id="Ratio_of_two_binomial_distributions">Ratio of two binomial distributions</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=22" title="Edit section: Ratio of two binomial distributions">edit</a>]</div> This result was first derived by Katz and coauthors in 1978.<a href="#cite_note-Katz1978-31">[30]</a> Let X ~ B(n, p1) and Y ~ B(m, p2) be independent. Let T = (X/n) / (Y/m). Then log(T) is approximately normally distributed with mean log(p1/p2) and variance ((1/p1) − 1)/n + ((1/p2) − 1)/m. <div class="mw-heading mw-heading3"><h3 id="Conditional_binomials">Conditional binomials</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=23" title="Edit section: Conditional binomials">edit</a>]</div> If X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq). For example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq). <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px #aaa solid; width:60%"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">[Proof]</div><div class="hidden-content mw-collapsible-content" style=""> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim B(n,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim B(n,p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d561b3cbc6297b3cd2313ac82686b3e553613d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.25ex; height:2.843ex;" alt="{\displaystyle X\sim B(n,p)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim B(X,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim B(X,q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80067d26d13b2cfe5aaf9d681d31d9dff37ab15c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.529ex; height:2.843ex;" alt="{\displaystyle Y\sim B(X,q)}">, by the <a href="/wiki/Law_of_total_probability" title="Law of total probability">law of total probability</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr[Y=m]&=\sum _{k=m}^{n}\Pr[Y=m\mid X=k]\Pr[X=k]\\[2pt]&=\sum _{k=m}^{n}{\binom {n}{k}}{\binom {k}{m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</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>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr[Y=m]&=\sum _{k=m}^{n}\Pr[Y=m\mid X=k]\Pr[X=k]\\[2pt]&=\sum _{k=m}^{n}{\binom {n}{k}}{\binom {k}{m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8f896e04f3bc2c13d2eed61e48bd43e63f6406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:54.832ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\Pr[Y=m]&=\sum _{k=m}^{n}\Pr[Y=m\mid X=k]\Pr[X=k]\\[2pt]&=\sum _{k=m}^{n}{\binom {n}{k}}{\binom {k}{m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}\end{aligned}}}"></dd></dl> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}{\tbinom {k}{m}}={\tbinom {n}{m}}{\tbinom {n-m}{k-m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}{\tbinom {k}{m}}={\tbinom {n}{m}}{\tbinom {n-m}{k-m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ba6b905e89e0126dcfd8efb68da4343816a6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.844ex; height:3.676ex;" alt="{\displaystyle {\tbinom {n}{k}}{\tbinom {k}{m}}={\tbinom {n}{m}}{\tbinom {n-m}{k-m}},}"> the equation above can be expressed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[Y=m]=\sum _{k=m}^{n}{\binom {n}{m}}{\binom {n-m}{k-m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</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>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[Y=m]=\sum _{k=m}^{n}{\binom {n}{m}}{\binom {n-m}{k-m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8369ef846ffda72900efc67b334923f70ce48ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.961ex; height:6.843ex;" alt="{\displaystyle \Pr[Y=m]=\sum _{k=m}^{n}{\binom {n}{m}}{\binom {n-m}{k-m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}}"></dd></dl> Factoring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}=p^{m}p^{k-m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{k}=p^{m}p^{k-m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ea54fd82c24fc3d2ff34aa753e4a8d0f3e1928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.27ex; height:3.009ex;" alt="{\displaystyle p^{k}=p^{m}p^{k-m}}"> and pulling all the terms that don't depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> out of the sum now yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}p^{m}q^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}p^{k-m}(1-p)^{n-k}(1-q)^{k-m}\right)\\[2pt]&={\binom {n}{m}}(pq)^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}\left(p(1-q)\right)^{k-m}(1-p)^{n-k}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</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>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>m</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>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}p^{m}q^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}p^{k-m}(1-p)^{n-k}(1-q)^{k-m}\right)\\[2pt]&={\binom {n}{m}}(pq)^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}\left(p(1-q)\right)^{k-m}(1-p)^{n-k}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caaa36dbcb3b4c43d7d5310533ef9d4809ba9db8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:69.347ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}p^{m}q^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}p^{k-m}(1-p)^{n-k}(1-q)^{k-m}\right)\\[2pt]&={\binom {n}{m}}(pq)^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}\left(p(1-q)\right)^{k-m}(1-p)^{n-k}\right)\end{aligned}}}"></dd></dl> After substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=k-m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>−</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=k-m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf7683643649a372695b1d64a30c53b9b97ed61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.993ex; height:2.343ex;" alt="{\displaystyle i=k-m}"> in the expression above, we get <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[Y=m]={\binom {n}{m}}(pq)^{m}\left(\sum _{i=0}^{n-m}{\binom {n-m}{i}}(p-pq)^{i}(1-p)^{n-m-i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>m</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[Y=m]={\binom {n}{m}}(pq)^{m}\left(\sum _{i=0}^{n-m}{\binom {n-m}{i}}(p-pq)^{i}(1-p)^{n-m-i}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54401b2dd0936ca0904832912df2fe9b2c3d5153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.468ex; height:7.509ex;" alt="{\displaystyle \Pr[Y=m]={\binom {n}{m}}(pq)^{m}\left(\sum _{i=0}^{n-m}{\binom {n-m}{i}}(p-pq)^{i}(1-p)^{n-m-i}\right)}"></dd></dl> Notice that the sum (in the parentheses) above equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p-pq+1-p)^{n-m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mo>+</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>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p-pq+1-p)^{n-m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23cb4a5499f3511ab15ddbea5052ee0c5a9faa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.011ex; height:3.009ex;" alt="{\displaystyle (p-pq+1-p)^{n-m}}"> by the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>. Substituting this in finally yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}(pq)^{m}(p-pq+1-p)^{n-m}\\[4pt]&={\binom {n}{m}}(pq)^{m}(1-pq)^{n-m}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mo>+</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>m</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}(pq)^{m}(p-pq+1-p)^{n-m}\\[4pt]&={\binom {n}{m}}(pq)^{m}(1-pq)^{n-m}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324b106ed5f362da979eebfc0feaa94b44b713b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:45.746ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\Pr[Y=m]&={\binom {n}{m}}(pq)^{m}(p-pq+1-p)^{n-m}\\[4pt]&={\binom {n}{m}}(pq)^{m}(1-pq)^{n-m}\end{aligned}}}"></dd></dl> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim B(n,pq)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim B(n,pq)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96528879e268db412c4037915b32d7b30097251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.113ex; height:2.843ex;" alt="{\displaystyle Y\sim B(n,pq)}"> as desired. </div></div> <div class="mw-heading mw-heading3"><h3 id="Bernoulli_distribution">Bernoulli distribution</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=24" title="Edit section: Bernoulli distribution">edit</a>]</div> The <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a> is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bernoulli(p). Conversely, any binomial distribution, B(n, p), is the distribution of the sum of n independent <a href="/wiki/Bernoulli_trials" class="mw-redirect" title="Bernoulli trials">Bernoulli trials</a>, Bernoulli(p), each with the same probability p.<a href="#cite_note-32">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Normal_approximation">Normal approximation</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=25" title="Edit section: Normal approximation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval" title="Binomial proportion confidence interval">Binomial proportion confidence interval § Normal approximation interval</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binomial_Distribution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Binomial_Distribution.svg/250px-Binomial_Distribution.svg.png" decoding="async" width="250" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Binomial_Distribution.svg/375px-Binomial_Distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Binomial_Distribution.svg/500px-Binomial_Distribution.svg.png 2x" data-file-width="1008" data-file-height="962" /></a><figcaption>Binomial <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> and normal <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> approximation for n = 6 and p = 0.5</figcaption></figure> If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(n, p) is given by the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(np,\,np(1-p)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>p</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(np,\,np(1-p)),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395141e8dc8a7820bd37c569a38cd91d1ff3b22c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:18.324ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(np,\,np(1-p)),}"></dd></dl> and this basic approximation can be improved in a simple way by using a suitable <a href="/wiki/Continuity_correction" title="Continuity correction">continuity correction</a>. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1.<a href="#cite_note-bhh-33">[32]</a> Various <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rules of thumb</a> may be used to decide whether n is large enough, and p is far enough from the extremes of zero or one: <ul><li>One rule<a href="#cite_note-bhh-33">[32]</a> is that for n > 5 the normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|1-2p|}{\sqrt {np(1-p)}}}={\frac {1}{\sqrt {n}}}\left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<0.3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mn>0.3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|1-2p|}{\sqrt {np(1-p)}}}={\frac {1}{\sqrt {n}}}\left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<0.3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67b44fc8960728dec49878de2b4c9428233cee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.26ex; height:7.509ex;" alt="{\displaystyle {\frac {|1-2p|}{\sqrt {np(1-p)}}}={\frac {1}{\sqrt {n}}}\left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<0.3.}"></dd></dl></li></ul> This can be made precise using the <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen theorem</a>. <ul><li>A stronger rule states that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values; that is, only if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \pm 3\sigma =np\pm 3{\sqrt {np(1-p)}}\in (0,n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>±</mo> <mn>3</mn> <mi>σ</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo>±</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \pm 3\sigma =np\pm 3{\sqrt {np(1-p)}}\in (0,n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0827e237f6a46cbfd1dd8ec2419d5af49cde0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.157ex; height:4.843ex;" alt="{\displaystyle \mu \pm 3\sigma =np\pm 3{\sqrt {np(1-p)}}\in (0,n).}"></dd></dl></li></ul> <dl><dd>This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c6e9fbd88e78967f28dc5d9e1081181c441a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.758ex; height:6.176ex;" alt="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"></dd></dl></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px #aaa solid; width:66%"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">[Proof]</div><div class="hidden-content mw-collapsible-content" style=""> The rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np\pm 3{\sqrt {np(1-p)}}\in (0,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> <mo>±</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np\pm 3{\sqrt {np(1-p)}}\in (0,n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd5334c1ccc22971961aa61efa72e5a3c936191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.678ex; height:4.843ex;" alt="{\displaystyle np\pm 3{\sqrt {np(1-p)}}\in (0,n)}"> is totally equivalent to request that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np-3{\sqrt {np(1-p)}}>0\quad {\text{and}}\quad np+3{\sqrt {np(1-p)}}<n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mi>p</mi> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo><</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np-3{\sqrt {np(1-p)}}>0\quad {\text{and}}\quad np+3{\sqrt {np(1-p)}}<n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edeb2608a88760fc7bc8c49cf6a1544b4535aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.667ex; height:4.843ex;" alt="{\displaystyle np-3{\sqrt {np(1-p)}}>0\quad {\text{and}}\quad np+3{\sqrt {np(1-p)}}<n.}"></dd></dl> Moving terms around yields: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np>3{\sqrt {np(1-p)}}\quad {\text{and}}\quad n(1-p)>3{\sqrt {np(1-p)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> <mo>></mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np>3{\sqrt {np(1-p)}}\quad {\text{and}}\quad n(1-p)>3{\sqrt {np(1-p)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece1ff0b306f376e61399a9b497254d78d458e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.241ex; height:4.843ex;" alt="{\displaystyle np>3{\sqrt {np(1-p)}}\quad {\text{and}}\quad n(1-p)>3{\sqrt {np(1-p)}}.}"></dd></dl> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}">, we can apply the square power and divide by the respective factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e608560d4c63c8192e3cec7075696cd7e566d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.618ex; height:3.009ex;" alt="{\displaystyle np^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(1-p)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(1-p)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d909137fd4ebb23868930aed92a67d3d02e1b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.431ex; height:3.176ex;" alt="{\displaystyle n(1-p)^{2}}">, to obtain the desired conditions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c6e9fbd88e78967f28dc5d9e1081181c441a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.758ex; height:6.176ex;" alt="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"></dd></dl> Notice that these conditions automatically imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2966b368b6eeb84ae2104cffbcf74e35a6d17e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>9}">. On the other hand, apply again the square root and divide by 3, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}>0\quad {\text{and}}\quad {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {p}{1-p}}}>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mn>3</mn> </mfrac> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mn>3</mn> </mfrac> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}>0\quad {\text{and}}\quad {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {p}{1-p}}}>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc495e4a45bc89d43ac89d9f899e65dfec47bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.756ex; height:7.509ex;" alt="{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}>0\quad {\text{and}}\quad {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {p}{1-p}}}>0.}"></dd></dl> Subtracting the second set of inequalities from the first one yields: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}>-{\frac {\sqrt {n}}{3}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mn>3</mn> </mfrac> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>></mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mn>3</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}>-{\frac {\sqrt {n}}{3}};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b21050d0dff35d3a9baaa3b2b153cd7eff39d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.49ex; height:7.509ex;" alt="{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}>-{\frac {\sqrt {n}}{3}};}"></dd></dl> and so, the desired first rule is satisfied, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<{\frac {\sqrt {n}}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<{\frac {\sqrt {n}}{3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c777de17db149740cfbcb5d0ebfba3518d33e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.098ex; height:7.509ex;" alt="{\displaystyle \left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<{\frac {\sqrt {n}}{3}}.}"></dd></dl> </div></div> <ul><li>Another commonly used rule is that both values np and n(1 − p) must be greater than<a href="#cite_note-34">[33]</a><a href="#cite_note-35">[34]</a> or equal to 5. However, the specific number varies from source to source, and depends on how good an approximation one wants. In particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px #aaa solid; width:66%"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">[Proof]</div><div class="hidden-content mw-collapsible-content" style=""> Assume that both values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.009ex;" alt="{\displaystyle np}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(1-p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66f82bbfdaa65adface0109e2254643c8303de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.376ex; height:2.843ex;" alt="{\displaystyle n(1-p)}"> are greater than 9. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}">, we easily have that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np\geq 9>9(1-p)\quad {\text{and}}\quad n(1-p)\geq 9>9p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>p</mi> <mo>≥</mo> <mn>9</mn> <mo>></mo> <mn>9</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>9</mn> <mo>></mo> <mn>9</mn> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np\geq 9>9(1-p)\quad {\text{and}}\quad n(1-p)\geq 9>9p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09bcf26e04790231ae63b77e394ab9549773ca28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.175ex; height:2.843ex;" alt="{\displaystyle np\geq 9>9(1-p)\quad {\text{and}}\quad n(1-p)\geq 9>9p.}"></dd></dl> We only have to divide now by the respective factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}">, to deduce the alternative form of the 3-standard-deviation rule: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>></mo> <mn>9</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c6e9fbd88e78967f28dc5d9e1081181c441a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.758ex; height:6.176ex;" alt="{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}"></dd></dl> </div></div> The following is an example of applying a <a href="/wiki/Continuity_correction" title="Continuity correction">continuity correction</a>. Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results. This approximation, known as <a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">de Moivre–Laplace theorem</a>, is a huge time-saver when undertaking calculations by hand (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>'s book <a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances">The Doctrine of Chances</a> in 1738. Nowadays, it can be seen as a consequence of the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> since B(n, p) is a sum of n independent, identically distributed <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli variables</a> with parameter p. This fact is the basis of a <a href="/wiki/Hypothesis_test" class="mw-redirect" title="Hypothesis test">hypothesis test</a>, a "proportion z-test", for the value of p using x/n, the sample proportion and estimator of p, in a <a href="/wiki/Common_test_statistics" class="mw-redirect" title="Common test statistics">common test statistic</a>.<a href="#cite_note-36">[35]</a> For example, suppose one randomly samples n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If groups of n people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\sqrt {\frac {p(1-p)}{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\sqrt {\frac {p(1-p)}{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efa9446419873f73391bd63522026579865e1b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.739ex; height:7.676ex;" alt="{\displaystyle \sigma ={\sqrt {\frac {p(1-p)}{n}}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Poisson_approximation">Poisson approximation</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=26" title="Edit section: Poisson approximation">edit</a>]</div> The binomial distribution converges towards the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> as the number of trials goes to infinity while the product np converges to a finite limit. Therefore, the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05<a href="#cite_note-37">[36]</a> such that np ≤ 1, or if n > 50 and p < 0.1 such that np < 5,<a href="#cite_note-38">[37]</a> or if n ≥ 100 and np ≤ 10.<a href="#cite_note-nist-39">[38]</a><a href="#cite_note-40">[39]</a> Concerning the accuracy of Poisson approximation, see Novak,<a href="#cite_note-41">[40]</a> ch. 4, and references therein. <div class="mw-heading mw-heading3"><h3 id="Limiting_distributions">Limiting distributions</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=27" title="Edit section: Limiting distributions">edit</a>]</div> <ul><li><a href="/wiki/Poisson_limit_theorem" title="Poisson limit theorem">Poisson limit theorem</a>: As n approaches ∞ and p approaches 0 with the product np held fixed, the Binomial(n, p) distribution approaches the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> with <a href="/wiki/Expected_value" title="Expected value">expected value</a> λ = np.<a href="#cite_note-nist-39">[38]</a></li> <li><a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">de Moivre–Laplace theorem</a>: As n approaches ∞ while p remains fixed, the distribution of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X-np}{\sqrt {np(1-p)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>X</mi> <mo>−</mo> <mi>n</mi> <mi>p</mi> </mrow> <msqrt> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X-np}{\sqrt {np(1-p)}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cac4977ac98028e7199c2665e93b80e847c0ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.706ex; height:6.676ex;" alt="{\displaystyle {\frac {X-np}{\sqrt {np(1-p)}}}}"></dd></dl></li></ul> <dl><dd>approaches the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with expected value 0 and <a href="/wiki/Variance" title="Variance">variance</a> 1. This result is sometimes loosely stated by saying that the distribution of X is <a href="/wiki/Asymptotic_normality" class="mw-redirect" title="Asymptotic normality">asymptotically normal</a> with expected value 0 and <a href="/wiki/Variance" title="Variance">variance</a> 1. This result is a specific case of the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Beta_distribution">Beta distribution</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=28" title="Edit section: Beta distribution">edit</a>]</div> The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution is the <a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a> of k successes given n independent events each with a probability p of success. Mathematically, when α = k + 1 and β = n − k + 1, the beta distribution and the binomial distribution are related by[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Is the left hand side referring to a probability density, and the right hand side to a probability mass function? Clearly a beta distributed random variable can not be a scalar multiple of a binomial random variable given that the former is continuous and the latter discrete. In any case, it would seem to be more correct to say that this relationship means that the PDF of one is related to the PMF of the other, rather than appearing to say that the _distributions_ (often interchangeable with their CDFs) are directly related to one another. (March 2023)">clarification needed</a>] a factor of n + 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (p;\alpha ;\beta )=(n+1)B(k;n;p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>α</mi> <mo>;</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>;</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (p;\alpha ;\beta )=(n+1)B(k;n;p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961beb7816eba573d7bccf5de9b90fdd1b1f3997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.333ex; height:2.843ex;" alt="{\displaystyle \operatorname {Beta} (p;\alpha ;\beta )=(n+1)B(k;n;p)}"></dd></dl> <a href="/wiki/Beta_distribution" title="Beta distribution">Beta distributions</a> also provide a family of <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior probability distributions</a> for binomial distributions in <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>:<a href="#cite_note-MacKay-42">[41]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(p;\alpha ,\beta )={\frac {p^{\alpha -1}(1-p)^{\beta -1}}{\operatorname {Beta} (\alpha ,\beta )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </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>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(p;\alpha ,\beta )={\frac {p^{\alpha -1}(1-p)^{\beta -1}}{\operatorname {Beta} (\alpha ,\beta )}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470eca5417087ae4bc2425a8188b32f50b3aa437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.003ex; height:6.676ex;" alt="{\displaystyle P(p;\alpha ,\beta )={\frac {p^{\alpha -1}(1-p)^{\beta -1}}{\operatorname {Beta} (\alpha ,\beta )}}.}"></dd></dl> Given a uniform prior, the posterior distribution for the probability of success p given n independent events with k observed successes is a beta distribution.<a href="#cite_note-43">[42]</a> <div class="mw-heading mw-heading2"><h2 id="Computational_methods">Computational methods</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=29" title="Edit section: Computational methods">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Random_number_generation">Random number generation</h3>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=30" title="Edit section: Random number generation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Pseudo-random_number_sampling" class="mw-redirect" title="Pseudo-random number sampling">Pseudo-random number sampling</a></div> Methods for <a href="/wiki/Random_number_generation" title="Random number generation">random number generation</a> where the <a href="/wiki/Marginal_distribution" title="Marginal distribution">marginal distribution</a> is a binomial distribution are well-established.<a href="#cite_note-44">[43]</a><a href="#cite_note-45">[44]</a> One way to generate <a href="/wiki/Random_variate" title="Random variate">random variates</a> samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that Pr(X = k) for all values k from 0 through n. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generator</a> to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=31" title="Edit section: History">edit</a>]</div> This distribution was derived by <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>. He considered the case where p = r/(r + s) where p is the probability of success and r and s are positive integers. <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> had earlier considered the case where p = 1/2, tabulating the corresponding binomial coefficients in what is now recognized as <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>.<a href="#cite_note-:1-46">[45]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=32" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <ul><li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial distribution</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial distribution</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial distribution</a></li> <li>Binomial measure, an example of a <a href="/wiki/Multifractal_system" title="Multifractal system">multifractal</a> <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>.<a href="#cite_note-47">[46]</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li> <li><a href="/wiki/Piling-up_lemma" title="Piling-up lemma">Piling-up lemma</a>, the resulting probability when <a href="/wiki/XOR" class="mw-redirect" title="XOR">XOR</a>-ing independent Boolean variables</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=33" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output 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.id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWestland2020" 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New York: McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontopr0000wads/page/52">52</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Probability+and+Random+Variables&rft.place=New+York&rft.pages=52&rft.pub=McGraw-Hill&rft.date=1960&rft.aulast=Wadsworth&rft.aufirst=G.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontopr0000wads&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJowett1963" class="citation journal cs1">Jowett, G. H. (1963). "The Relationship Between the Binomial and F Distributions". Journal of the Royal Statistical Society, Series D. 13 (1): 55–57. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2986663">10.2307/2986663</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2986663">2986663</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+D&rft.atitle=The+Relationship+Between+the+Binomial+and+F+Distributions&rft.volume=13&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E55-%3C%2Fspan%3E57&rft.date=1963&rft_id=info%3Adoi%2F10.2307%2F2986663&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2986663%23id-name%3DJSTOR&rft.aulast=Jowett&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> See <a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Expectation_of_Binomial_Distribution">Proof Wiki</a> </li> <li id="cite_note-Andreas2008-7"><a href="#cite_ref-Andreas2008_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnoblauch2008" class="citation cs2">Knoblauch, Andreas (2008), <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40233780">"Closed-Form Expressions for the Moments of the Binomial Probability Distribution"</a>, SIAM Journal on Applied Mathematics, 69 (1): 197–204, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F070700024">10.1137/070700024</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40233780">40233780</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Applied+Mathematics&rft.atitle=Closed-Form+Expressions+for+the+Moments+of+the+Binomial+Probability+Distribution&rft.volume=69&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E204&rft.date=2008&rft_id=info%3Adoi%2F10.1137%2F070700024&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F40233780%23id-name%3DJSTOR&rft.aulast=Knoblauch&rft.aufirst=Andreas&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F40233780&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-Nguyen2021-8"><a href="#cite_ref-Nguyen2021_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNguyen2021" class="citation cs2">Nguyen, Duy (2021), <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00031305.2019.1679257?journalCode=utas20">"A probabilistic approach to the moments of binomial random variables and application"</a>, The American Statistician, 75 (1): 101–103, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.2019.1679257">10.1080/00031305.2019.1679257</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:209923008">209923008</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=A+probabilistic+approach+to+the+moments+of+binomial+random+variables+and+application&rft.volume=75&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E101-%3C%2Fspan%3E103&rft.date=2021&rft_id=info%3Adoi%2F10.1080%2F00031305.2019.1679257&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A209923008%23id-name%3DS2CID&rft.aulast=Nguyen&rft.aufirst=Duy&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00031305.2019.1679257%3FjournalCode%3Dutas20&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._Ahle2022" class="citation cs2">D. Ahle, Thomas (2022), "Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions", Statistics & Probability Letters, 182: 109306, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2103.17027">2103.17027</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spl.2021.109306">10.1016/j.spl.2021.109306</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+%26+Probability+Letters&rft.atitle=Sharp+and+Simple+Bounds+for+the+raw+Moments+of+the+Binomial+and+Poisson+Distributions&rft.volume=182&rft.pages=109306&rft.date=2022&rft_id=info%3Aarxiv%2F2103.17027&rft_id=info%3Adoi%2F10.1016%2Fj.spl.2021.109306&rft.aulast=D.+Ahle&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNicolas2019" class="citation web cs1">Nicolas, André (January 7, 2019). <a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/117940">"Finding mode in Binomial distribution"</a>. <a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Exchange&rft.atitle=Finding+mode+in+Binomial+distribution&rft.date=2019-01-07&rft.aulast=Nicolas&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F117940&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeumann1966" class="citation journal cs1 cs1-prop-foreign-lang-source">Neumann, P. (1966). "Über den Median der Binomial- and Poissonverteilung". Wissenschaftliche Zeitschrift der Technischen Universität Dresden (in German). 19: 29–33.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wissenschaftliche+Zeitschrift+der+Technischen+Universit%C3%A4t+Dresden&rft.atitle=%C3%9Cber+den+Median+der+Binomial-+and+Poissonverteilung&rft.volume=19&rft.pages=%3Cspan+class%3D%22nowrap%22%3E29-%3C%2Fspan%3E33&rft.date=1966&rft.aulast=Neumann&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", <a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a> 94, 331-332. </li> <li id="cite_note-KaasBuhrman-13">^ <a href="#cite_ref-KaasBuhrman_13-0">a</a> <a href="#cite_ref-KaasBuhrman_13-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaasBuhrman1980" class="citation journal cs1">Kaas, R.; Buhrman, J.M. (1980). "Mean, Median and Mode in Binomial Distributions". Statistica Neerlandica. 34 (1): 13–18. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1467-9574.1980.tb00681.x">10.1111/j.1467-9574.1980.tb00681.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistica+Neerlandica&rft.atitle=Mean%2C+Median+and+Mode+in+Binomial+Distributions&rft.volume=34&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E13-%3C%2Fspan%3E18&rft.date=1980&rft_id=info%3Adoi%2F10.1111%2Fj.1467-9574.1980.tb00681.x&rft.aulast=Kaas&rft.aufirst=R.&rft.au=Buhrman%2C+J.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-Hamza-14"><a href="#cite_ref-Hamza_14-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamza1995" class="citation journal cs1">Hamza, K. (1995). 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Statistics & Probability Letters. 23: 21–25. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-7152%2894%2900090-U">10.1016/0167-7152(94)00090-U</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+%26+Probability+Letters&rft.atitle=The+smallest+uniform+upper+bound+on+the+distance+between+the+mean+and+the+median+of+the+binomial+and+Poisson+distributions&rft.volume=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E21-%3C%2Fspan%3E25&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0167-7152%2894%2900090-U&rft.aulast=Hamza&rft.aufirst=K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-Nowakowski-15"><a href="#cite_ref-Nowakowski_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNowakowski2021" class="citation journal cs1">Nowakowski, Sz. 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Bulletin of Mathematical Biology. 51 (1): 125–131. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02458840">10.1007/BF02458840</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/2706397">2706397</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:189884382">189884382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Mathematical+Biology&rft.atitle=Tutorial+on+large+deviations+for+the+binomial+distribution&rft.volume=51&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E125-%3C%2Fspan%3E131&rft.date=1989&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189884382%23id-name%3DS2CID&rft_id=info%3Apmid%2F2706397&rft_id=info%3Adoi%2F10.1007%2FBF02458840&rft.aulast=Arratia&rft.aufirst=R.&rft.au=Gordon%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_B._Ash1990" class="citation book cs1">Robert B. 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Retrieved 2023-10-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=6.4%3A+Normal+Approximation+to+the+Binomial+Distribution+-+Statistics+LibreTexts&rft.date=2023-05-29&rft_id=https%3A%2F%2Fstats.libretexts.org%2FCourses%2FLas_Positas_College%2FMath_40%3A_Statistics_and_Probability%2F06%3A_Continuous_Random_Variables_and_the_Normal_Distribution%2F6.04%3A_Normal_Approximation_to_the_Binomial_Distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>) </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>/<a href="/wiki/SEMATECH" title="SEMATECH">SEMATECH</a>, <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc24.htm">"7.2.4. Does the proportion of defectives meet requirements?"</a> e-Handbook of Statistical Methods. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20230328081322/https://online.stat.psu.edu/stat414/lesson/12/12.4">"12.4 – Approximating the Binomial Distribution | STAT 414"</a>. Pennstate: Statistics Online Courses. 2023-03-28. Archived from the original on 2023-03-28. Retrieved 2023-10-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pennstate%3A+Statistics+Online+Courses&rft.atitle=12.4+%E2%80%93+Approximating+the+Binomial+Distribution+%7C+STAT+414&rft.date=2023-03-28&rft_id=https%3A%2F%2Fonline.stat.psu.edu%2Fstat414%2Flesson%2F12%2F12.4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_news" title="Template:Cite news">cite news</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>) </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen2011" class="citation book cs1">Chen, Zac (2011). H2 mathematics handbook (1 ed.). Singapore: Educational publishing house. p. 348. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789814288484" title="Special:BookSources/9789814288484"><bdi>9789814288484</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=H2+mathematics+handbook&rft.place=Singapore&rft.pages=348&rft.edition=1&rft.pub=Educational+publishing+house&rft.date=2011&rft.isbn=9789814288484&rft.aulast=Chen&rft.aufirst=Zac&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-nist-39">^ <a href="#cite_ref-nist_39-0">a</a> <a href="#cite_ref-nist_39-1">b</a> <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>/<a href="/wiki/SEMATECH" title="SEMATECH">SEMATECH</a>, <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm">"6.3.3.1. Counts Control Charts"</a>, e-Handbook of Statistical Methods. </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20230313085931/https://mathcenter.oxford.emory.edu/site/math117/connectingPoissonAndBinomial/">"The Connection Between the Poisson and Binomial Distributions"</a>. 2023-03-13. Archived from the original on 2023-03-13. Retrieved 2023-10-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Connection+Between+the+Poisson+and+Binomial+Distributions&rft.date=2023-03-13&rft_id=https%3A%2F%2Fmathcenter.oxford.emory.edu%2Fsite%2Fmath117%2FconnectingPoissonAndBinomial%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>) </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Novak S.Y. (2011) Extreme value methods with applications to finance. London: CRC/ Chapman & Hall/Taylor & Francis. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781-43983-5746" title="Special:BookSources/9781-43983-5746">9781-43983-5746</a>. </li> <li id="cite_note-MacKay-42"><a href="#cite_ref-MacKay_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacKay2003" class="citation book cs1">MacKay, David (2003). Information Theory, Inference and Learning Algorithms. Cambridge University Press; First Edition. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521642989" title="Special:BookSources/978-0521642989"><bdi>978-0521642989</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory%2C+Inference+and+Learning+Algorithms&rft.pub=Cambridge+University+Press%3B+First+Edition&rft.date=2003&rft.isbn=978-0521642989&rft.aulast=MacKay&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.statlect.com/probability-distributions/beta-distribution">"Beta distribution"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Beta+distribution&rft_id=https%3A%2F%2Fwww.statlect.com%2Fprobability-distributions%2Fbeta-distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Devroye, Luc (1986) Non-Uniform Random Variate Generation, New York: Springer-Verlag. (See especially <a rel="nofollow" class="external text" href="http://luc.devroye.org/chapter_ten.pdf">Chapter X, Discrete Univariate Distributions</a>) </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKachitvichyanukulSchmeiser1988" class="citation journal cs1">Kachitvichyanukul, V.; Schmeiser, B. W. (1988). "Binomial random variate generation". Communications of the ACM. 31 (2): 216–222. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F42372.42381">10.1145/42372.42381</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18698828">18698828</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Binomial+random+variate+generation&rft.volume=31&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E216-%3C%2Fspan%3E222&rft.date=1988&rft_id=info%3Adoi%2F10.1145%2F42372.42381&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18698828%23id-name%3DS2CID&rft.aulast=Kachitvichyanukul&rft.aufirst=V.&rft.au=Schmeiser%2C+B.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-:1-46"><a href="#cite_ref-:1_46-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2009" class="citation book cs1">Katz, Victor (2009). "14.3: Elementary Probability". A History of Mathematics: An Introduction. Addison-Wesley. p. 491. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-38700-4" title="Special:BookSources/978-0-321-38700-4"><bdi>978-0-321-38700-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=14.3%3A+Elementary+Probability&rft.btitle=A+History+of+Mathematics%3A+An+Introduction&rft.pages=491&rft.pub=Addison-Wesley&rft.date=2009&rft.isbn=978-0-321-38700-4&rft.aulast=Katz&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns. 3.2 The Binomial Measure is the Simplest Example of a Multifractal </li> </ol></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Except the trivial case p = 0, which must be checked separately. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=34" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHirsch1957" class="citation book cs1">Hirsch, Werner Z. (1957). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KostAAAAIAAJ&pg=PA140">"Binomial Distribution—Success or Failure, How Likely Are They?"</a>. Introduction to Modern Statistics. New York: MacMillan. pp. 140–153.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Binomial+Distribution%E2%80%94Success+or+Failure%2C+How+Likely+Are+They%3F&rft.btitle=Introduction+to+Modern+Statistics&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E140-%3C%2Fspan%3E153&rft.pub=MacMillan&rft.date=1957&rft.aulast=Hirsch&rft.aufirst=Werner+Z.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKostAAAAIAAJ%26pg%3DPA140&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeterWassermanWhitmore1988" class="citation book cs1">Neter, John; Wasserman, William; Whitmore, G. A. (1988). Applied Statistics (Third ed.). Boston: Allyn & Bacon. pp. 185–192. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-205-10328-6" title="Special:BookSources/0-205-10328-6"><bdi>0-205-10328-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Statistics&rft.place=Boston&rft.pages=%3Cspan+class%3D%22nowrap%22%3E185-%3C%2Fspan%3E192&rft.edition=Third&rft.pub=Allyn+%26+Bacon&rft.date=1988&rft.isbn=0-205-10328-6&rft.aulast=Neter&rft.aufirst=John&rft.au=Wasserman%2C+William&rft.au=Whitmore%2C+G.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinomial+distribution" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Binomial_distribution&action=edit&section=35" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output 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text" href="https://math.stackexchange.com/q/1065487">X-Y</a> or <a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/562119">|X-Y|</a></li> <li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=Prob+x+%3E+19+if+x+is+binomial+with+n+%3D+36++and+p+%3D+.6">Querying the binomial probability distribution in WolframAlpha</a></li> <li>Confidence (credible) intervals for binomial probability, p: <a rel="nofollow" class="external text" href="https://causascientia.org/math_stat/ProportionCI.html">online calculator</a> available at <a rel="nofollow" class="external text" href="https://causascientia.org">causaScientia.org</a></li></ul> <div style="clear:both;" class=""></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)587" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a class="mw-selflink selflink">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a semi-infinite interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution">F</a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's T-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on the whole real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's z</a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q</a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU</a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral t</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t</a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis κ-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis κ-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis κ-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis κ-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis κ-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution">q-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution">q-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous- discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate (joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Discrete: </li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li>Continuous: </li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate t</a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_variate_beta_distribution" title="Matrix variate beta distribution">Matrix beta</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix t</a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt>Bivariate (spherical)</dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt>Bivariate (toroidal)</dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt>Multivariate</dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Degenerate</dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt>Singular</dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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Binomial distribution - Wikipedia