Poisson distribution

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id="toc-Probability_mass_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_mass_function"> <div class="vector-toc-text"> 2.1 Probability mass function </div> </a> <ul id="toc-Probability_mass_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 2.2 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assumptions_and_validity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Assumptions_and_validity"> <div class="vector-toc-text"> 2.3 Assumptions and validity </div> </a> <ul id="toc-Assumptions_and_validity-sublist" class="vector-toc-list"> <li id="toc-Examples_of_probability_for_Poisson_distributions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_of_probability_for_Poisson_distributions"> <div class="vector-toc-text"> 2.3.1 Examples of probability for Poisson distributions </div> </a> <ul id="toc-Examples_of_probability_for_Poisson_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Once_in_an_interval_events:_The_special_case_of_λ_=_1_and_k_=_0" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Once_in_an_interval_events:_The_special_case_of_λ_=_1_and_k_=_0"> <div class="vector-toc-text"> 2.3.2 Once in an interval events: The special case of λ = 1 and k = 0 </div> </a> <ul id="toc-Once_in_an_interval_events:_The_special_case_of_λ_=_1_and_k_=_0-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples_that_violate_the_Poisson_assumptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_that_violate_the_Poisson_assumptions"> <div class="vector-toc-text"> 2.4 Examples that violate the Poisson assumptions </div> </a> <ul id="toc-Examples_that_violate_the_Poisson_assumptions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 3 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Descriptive_statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Descriptive_statistics"> <div class="vector-toc-text"> 3.1 Descriptive statistics </div> </a> <ul id="toc-Descriptive_statistics-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"> 3.2 Median </div> </a> <ul id="toc-Median-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"> 3.3 Higher moments </div> </a> <ul id="toc-Higher_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sums_of_Poisson-distributed_random_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sums_of_Poisson-distributed_random_variables"> <div class="vector-toc-text"> 3.4 Sums of Poisson-distributed random variables </div> </a> <ul id="toc-Sums_of_Poisson-distributed_random_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximum_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximum_entropy"> <div class="vector-toc-text"> 3.5 Maximum entropy </div> </a> <ul id="toc-Maximum_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_properties"> <div class="vector-toc-text"> 3.6 Other properties </div> </a> <ul id="toc-Other_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_races" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_races"> <div class="vector-toc-text"> 3.7 Poisson races </div> </a> <ul id="toc-Poisson_races-sublist" class="vector-toc-list"> </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-As_a_Binomial_distribution_with_infinitesimal_time-steps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_Binomial_distribution_with_infinitesimal_time-steps"> <div class="vector-toc-text"> 4.1 As a Binomial distribution with infinitesimal time-steps </div> </a> <ul id="toc-As_a_Binomial_distribution_with_infinitesimal_time-steps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> 4.2 General </div> </a> <ul id="toc-General-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"> 4.3 Poisson approximation </div> </a> <ul id="toc-Poisson_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bivariate_Poisson_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bivariate_Poisson_distribution"> <div class="vector-toc-text"> 4.4 Bivariate Poisson distribution </div> </a> <ul id="toc-Bivariate_Poisson_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_Poisson_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_Poisson_distribution"> <div class="vector-toc-text"> 4.5 Free Poisson distribution </div> </a> <ul id="toc-Free_Poisson_distribution-sublist" class="vector-toc-list"> <li id="toc-Some_transforms_of_this_law" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Some_transforms_of_this_law"> <div class="vector-toc-text"> 4.5.1 Some transforms of this law </div> </a> <ul id="toc-Some_transforms_of_this_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Weibull_and_stable_count" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weibull_and_stable_count"> <div class="vector-toc-text"> 4.6 Weibull and stable count </div> </a> <ul id="toc-Weibull_and_stable_count-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"> 5 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-Parameter_estimation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parameter_estimation"> <div class="vector-toc-text"> 5.1 Parameter estimation </div> </a> <ul id="toc-Parameter_estimation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Confidence_interval" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Confidence_interval"> <div class="vector-toc-text"> 5.2 Confidence interval </div> </a> <ul id="toc-Confidence_interval-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bayesian_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bayesian_inference"> <div class="vector-toc-text"> 5.3 Bayesian inference </div> </a> <ul id="toc-Bayesian_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simultaneous_estimation_of_multiple_Poisson_means" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simultaneous_estimation_of_multiple_Poisson_means"> <div class="vector-toc-text"> 5.4 Simultaneous estimation of multiple Poisson means </div> </a> <ul id="toc-Simultaneous_estimation_of_multiple_Poisson_means-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Occurrence_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Occurrence_and_applications"> <div class="vector-toc-text"> 6 Occurrence and applications </div> </a> <button aria-controls="toc-Occurrence_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Occurrence and applications subsection </button> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> <li id="toc-Law_of_rare_events" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Law_of_rare_events"> <div class="vector-toc-text"> 6.1 Law of rare events </div> </a> <ul id="toc-Law_of_rare_events-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_point_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_point_process"> <div class="vector-toc-text"> 6.2 Poisson point process </div> </a> <ul id="toc-Poisson_point_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_regression_and_negative_binomial_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_regression_and_negative_binomial_regression"> <div class="vector-toc-text"> 6.3 Poisson regression and negative binomial regression </div> </a> <ul id="toc-Poisson_regression_and_negative_binomial_regression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> 6.4 Biology </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_applications_in_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_applications_in_science"> <div class="vector-toc-text"> 6.5 Other applications in science </div> </a> <ul id="toc-Other_applications_in_science-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"> 7 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-Evaluating_the_Poisson_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evaluating_the_Poisson_distribution"> <div class="vector-toc-text"> 7.1 Evaluating the Poisson distribution </div> </a> <ul id="toc-Evaluating_the_Poisson_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_variate_generation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_variate_generation"> <div class="vector-toc-text"> 7.2 Random variate generation </div> </a> <ul id="toc-Random_variate_generation-sublist" class="vector-toc-list"> </ul> </li> </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"> 8 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"> 9 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 9.1 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 9.2 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </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">Poisson 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 51 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-51" aria-hidden="true" > 51 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%A8%D9%88%D8%A7%D8%B3%D9%88%D9%86" title="توزيع بواسون – Arabic" lang="ar" hreflang="ar" data-title="توزيع بواسون" 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_de_Poisson" title="Distribución de Poisson – Asturian" lang="ast" hreflang="ast" data-title="Distribución de Poisson" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%BE%D9%88%D8%A7%D8%B3%D9%88%D9%86_%D8%AF%D8%A7%D8%BA%DB%8C%D9%84%DB%8C%D9%85%DB%8C" title="پواسون داغیلیمی – South Azerbaijani" lang="azb" hreflang="azb" data-title="پواسون داغیلیمی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%AF%E0%A6%BC%E0%A6%B8%E0%A7%8B%E0%A6%81_%E0%A6%AC%E0%A6%BF%E0%A6%A8%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8" title="পয়সোঁ বিন্যাস – Bangla" lang="bn" hreflang="bn" data-title="পয়সোঁ বিন্যাস" 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%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5_%D0%9F%D1%83%D0%B0%D1%81%D0%BE%D0%BD%D0%B0" title="Размеркаванне Пуасона – Belarusian" lang="be" hreflang="be" data-title="Размеркаванне Пуасона" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%9F%D0%BE%D0%B0%D1%81%D0%BE%D0%BD" title="Разпределение на Поасон – Bulgarian" lang="bg" hreflang="bg" data-title="Разпределение на Поасон" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_de_Poisson" title="Distribució de Poisson – Catalan" lang="ca" hreflang="ca" data-title="Distribució de Poisson" 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/Poissonovo_rozd%C4%9Blen%C3%AD" title="Poissonovo rozdělení – Czech" lang="cs" hreflang="cs" data-title="Poissonovo 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/Poissonfordeling" title="Poissonfordeling – Danish" lang="da" hreflang="da" data-title="Poissonfordeling" 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/Poisson-Verteilung" title="Poisson-Verteilung – German" lang="de" hreflang="de" data-title="Poisson-Verteilung" 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/Poissoni_jaotus" title="Poissoni jaotus – Estonian" lang="et" hreflang="et" data-title="Poissoni jaotus" 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%9A%CE%B1%CF%84%CE%B1%CE%BD%CE%BF%CE%BC%CE%AE_%CE%A0%CE%BF%CF%85%CE%B1%CF%83%CF%83%CF%8C%CE%BD" title="Κατανομή Πουασσόν – Greek" lang="el" hreflang="el" data-title="Κατανομή Πουασσόν" 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_de_Poisson" title="Distribución de Poisson – Spanish" lang="es" hreflang="es" data-title="Distribución de Poisson" 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/Poissonen_banaketa" title="Poissonen banaketa – Basque" lang="eu" hreflang="eu" data-title="Poissonen banaketa" 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_%D9%BE%D9%88%D8%A7%D8%B3%D9%88%D9%86" title="توزیع پواسون – Persian" lang="fa" hreflang="fa" data-title="توزیع پواسون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_de_Poisson" title="Loi de Poisson – French" lang="fr" hreflang="fr" data-title="Loi de Poisson" 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_de_Poisson" title="Distribución de Poisson – Galician" lang="gl" hreflang="gl" data-title="Distribución de Poisson" 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/%ED%91%B8%EC%95%84%EC%86%A1_%EB%B6%84%ED%8F%AC" title="푸아송 분포 – Korean" lang="ko" hreflang="ko" data-title="푸아송 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%B8%D6%82%D5%A1%D5%BD%D5%B8%D5%B6%D5%AB_%D5%A2%D5%A1%D5%B7%D5%AD%D5%B8%D6%82%D5%B4" title="Պուասոնի բաշխում – Armenian" lang="hy" hreflang="hy" data-title="Պուասոնի բաշխում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Distribusi_Poisson" title="Distribusi Poisson – Indonesian" lang="id" hreflang="id" data-title="Distribusi Poisson" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Poisson-dreifing" title="Poisson-dreifing – Icelandic" lang="is" hreflang="is" data-title="Poisson-dreifing" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_di_Poisson" title="Distribuzione di Poisson – Italian" lang="it" hreflang="it" data-title="Distribuzione di Poisson" 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%A4%D7%95%D7%90%D7%A1%D7%95%D7%9F" title="התפלגות פואסון – Hebrew" lang="he" hreflang="he" data-title="התפלגות פואסון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%A3%E1%83%90%E1%83%A1%E1%83%9D%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%92%E1%83%90%E1%83%9C%E1%83%90%E1%83%AC%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="პუასონის განაწილება – Georgian" lang="ka" hreflang="ka" data-title="პუასონის განაწილება" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Puasona_sadal%C4%ABjums" title="Puasona sadalījums – Latvian" lang="lv" hreflang="lv" data-title="Puasona 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/Puasono_skirstinys" title="Puasono skirstinys – Lithuanian" lang="lt" hreflang="lt" data-title="Puasono 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/Poisson-eloszl%C3%A1s" title="Poisson-eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Poisson-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%9F%D0%BE%D0%B0%D1%81%D0%BE%D0%BD%D0%BE%D0%B2%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="Поасонова распределба – Macedonian" lang="mk" hreflang="mk" data-title="Поасонова 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_Poisson" title="Taburan Poisson – Malay" lang="ms" hreflang="ms" data-title="Taburan Poisson" 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/Poissonverdeling" title="Poissonverdeling – Dutch" lang="nl" hreflang="nl" data-title="Poissonverdeling" 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/%E3%83%9D%E3%82%A2%E3%82%BD%E3%83%B3%E5%88%86%E5%B8%83" title="ポアソン分布 – Japanese" lang="ja" hreflang="ja" data-title="ポアソン分布" 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/Poisson-fordeling" title="Poisson-fordeling – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Poisson-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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_Poissona" title="Rozkład Poissona – Polish" lang="pl" hreflang="pl" data-title="Rozkład Poissona" 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_de_Poisson" title="Distribuição de Poisson – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição de Poisson" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD%D0%B0" title="Распределение Пуассона – Russian" lang="ru" hreflang="ru" data-title="Распределение Пуассона" 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_Poisson" title="Shpërndarja Poisson – Albanian" lang="sq" hreflang="sq" data-title="Shpërndarja Poisson" 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/Poisson_distribution" title="Poisson distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Poisson 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/Poissonovo_rozdelenie" title="Poissonovo rozdelenie – Slovak" lang="sk" hreflang="sk" data-title="Poissonovo 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/Poissonova_porazdelitev" title="Poissonova porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Poissonova 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/Poasonova_raspodela" title="Poasonova raspodela – Serbian" lang="sr" hreflang="sr" data-title="Poasonova 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_Poisson" title="Sebaran Poisson – Sundanese" lang="su" hreflang="su" data-title="Sebaran Poisson" 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/Poissonin_jakauma" title="Poissonin jakauma – Finnish" lang="fi" hreflang="fi" data-title="Poissonin jakauma" 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/Poissonf%C3%B6rdelning" title="Poissonfördelning – Swedish" lang="sv" hreflang="sv" data-title="Poissonfördelning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD_%D0%B1%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D1%88%D0%B5" title="Пуассон бүленеше – Tatar" lang="tt" hreflang="tt" data-title="Пуассон бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Poisson_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Poisson dağılımı – Turkish" lang="tr" hreflang="tr" data-title="Poisson 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%A0%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB_%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD%D0%B0" title="Розподіл Пуассона – Ukrainian" lang="uk" hreflang="uk" data-title="Розподіл Пуассона" 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 href="https://ur.wikipedia.org/wiki/%D9%BE%D9%88%D8%A6%DB%8C%D8%B3%D9%86_%D8%AA%D9%88%D8%B2%DB%8C%D8%B9" title="پوئیسن توزیع – Urdu" lang="ur" hreflang="ur" data-title="پوئیسن توزیع" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ph%E1%BB%91i_Poisson" title="Phân phối Poisson – Vietnamese" lang="vi" hreflang="vi" data-title="Phân phối Poisson" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%B3%8A%E6%9D%BE%E5%88%86%E5%B8%83" title="泊松分布 – Wu" lang="wuu" hreflang="wuu" data-title="泊松分布" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B3%8A%E6%B7%9E%E5%88%86%E4%BD%88" title="泊淞分佈 – Cantonese" lang="yue" hreflang="yue" data-title="泊淞分佈" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%9C%E7%93%A6%E6%9D%BE%E5%88%86%E5%B8%83" title="卜瓦松分布 – Chinese" lang="zh" hreflang="zh" data-title="卜瓦松分布" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> 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.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">Poisson distribution</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability mass function</div><a href="/wiki/File:Poisson_pmf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Poisson_pmf.svg/325px-Poisson_pmf.svg.png" decoding="async" width="325" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Poisson_pmf.svg/488px-Poisson_pmf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Poisson_pmf.svg/650px-Poisson_pmf.svg.png 2x" data-file-width="330" data-file-height="255" /></a><div class="infobox-caption">The horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye.</div></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><a href="/wiki/File:Poisson_cdf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Poisson_cdf.svg/325px-Poisson_cdf.svg.png" decoding="async" width="325" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Poisson_cdf.svg/488px-Poisson_cdf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Poisson_cdf.svg/650px-Poisson_cdf.svg.png 2x" data-file-width="327" data-file-height="260" /></a><div class="infobox-caption">The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values.</div></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 \operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2171745e86cc9d5c28f0e2595db6b5532fe8bb47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.473ex; height:2.843ex;" alt="{\displaystyle \operatorname {Pois} (\lambda )}"></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 \lambda \in (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in (0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66704521911e7bce5122e668683005f6d22207ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.525ex; height:2.843ex;" alt="{\displaystyle \lambda \in (0,\infty )}"> (rate)</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 \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bceb13f72e37bcd50b60e5fb2fa05bcf15c265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.784ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {N} _{0}}"> (<a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">Natural numbers</a> starting from 0)</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 {\frac {\lambda ^{k}e^{-\lambda }}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b15a85051528f8722d2b676a557f6c22bf85c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.833ex; height:5.843ex;" alt="{\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{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 {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44c577126e6cab854deb2586eea897e7e1380d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.413ex; height:6.509ex;" alt="{\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}},}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }{\frac {\lambda ^{j}}{j!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</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"> <mfrac> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }{\frac {\lambda ^{j}}{j!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42278d90b362f7bd56a5bdd4b726b03a2300ffc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.43ex; height:8.009ex;" alt="{\displaystyle e^{-\lambda }\sum _{j=0}^{\lfloor k\rfloor }{\frac {\lambda ^{j}}{j!}},}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7bb45bf477cfc1deaf939b475b03c0f6b02342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.316ex; height:2.843ex;" alt="{\displaystyle Q(\lfloor k+1\rfloor ,\lambda )}"> (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c10f12b66e1ef733d7d4a58e6ac0dc86550bf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k\geq 0,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a26a2719b58c1e3dc41c04ae1f927fc00e2891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.781ex; height:2.843ex;" alt="{\displaystyle \Gamma (x,y)}"> is the <a href="/wiki/Upper_incomplete_gamma_function" class="mw-redirect" title="Upper incomplete gamma function">upper incomplete gamma function</a>, <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 <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> is the <a href="/wiki/Regularized_gamma_function" class="mw-redirect" title="Regularized gamma function">regularized gamma 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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></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 \approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <mrow> <mo>⌊</mo> <mrow> <mi>λ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>50</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7096e049b66dc970bb7ad4f95bd95fe410d56608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.715ex; height:6.176ex;" alt="{\displaystyle \approx \left\lfloor \lambda +{\frac {1}{3}}-{\frac {1}{50\lambda }}\right\rfloor }"></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 \left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌈</mo> <mi>λ</mi> <mo>⌉</mo> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mo>⌊</mo> <mi>λ</mi> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39855377f4991452ccf691ed5c80b9615ca28b8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.877ex; height:2.843ex;" alt="{\displaystyle \left\lceil \lambda \right\rceil -1,\left\lfloor \lambda \right\rfloor }"></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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></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 {1}{\sqrt {\lambda }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>λ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\lambda }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89acf6db1533e1d7afef03d16012dbe8eddb0b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:4.127ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {\lambda }}}}"></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}{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d496a9397007733fdbbb2f98433e7aab4e51ff4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.191ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\lambda }}}"></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 \lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c5bb5a142a1b60e4a04c725d686557fdb238bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.518ex; height:7.176ex;" alt="{\displaystyle \lambda {\Bigl [}1-\log(\lambda ){\Bigr ]}+e^{-\lambda }\sum _{k=0}^{\infty }{\frac {\lambda ^{k}\log(k!)}{k!}}}">   or for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\approx {\frac {1}{2}}\log \left(2\pi e\lambda \right)-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}\\-{\frac {19}{360\lambda ^{3}}}+{\mathcal {O}}\left({\frac {1}{\lambda ^{4}}}\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>e</mi> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>24</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mrow> <mn>360</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\approx {\frac {1}{2}}\log \left(2\pi e\lambda \right)-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}\\-{\frac {19}{360\lambda ^{3}}}+{\mathcal {O}}\left({\frac {1}{\lambda ^{4}}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec5ecd1664d2bbc5ba8db81ed452e1f72773a67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.037ex; margin-bottom: -0.301ex; width:31.072ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\approx {\frac {1}{2}}\log \left(2\pi e\lambda \right)-{\frac {1}{12\lambda }}-{\frac {1}{24\lambda ^{2}}}\\-{\frac {19}{360\lambda ^{3}}}+{\mathcal {O}}\left({\frac {1}{\lambda ^{4}}}\right)\end{aligned}}}"></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 \exp \left[\lambda \left(e^{t}-1\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left[\lambda \left(e^{t}-1\right)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c1560d02e57d6d1f2a6223fa16061ad399bb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.276ex; height:3.176ex;" alt="{\displaystyle \exp \left[\lambda \left(e^{t}-1\right)\right]}"></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 \exp \left[\lambda \left(e^{it}-1\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left[\lambda \left(e^{it}-1\right)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d912f7d931127ba7a6d044105a647ea4590b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.844ex; height:3.343ex;" alt="{\displaystyle \exp \left[\lambda \left(e^{it}-1\right)\right]}"></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 \exp \left[\lambda \left(z-1\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left[\lambda \left(z-1\right)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b112607a91ffced05b02706979ee6651a6db3c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.489ex; height:2.843ex;" alt="{\displaystyle \exp \left[\lambda \left(z-1\right)\right]}"></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 {\frac {1}{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d496a9397007733fdbbb2f98433e7aab4e51ff4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.191ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{\lambda }}}"></td></tr></tbody></table> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the Poisson distribution (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ˈpwɑːsɒn/</a>) is a <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a> that expresses the probability of a given number of <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">events</a> occurring in a fixed interval of time if these events occur with a known constant mean rate and <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independently</a> of the time since the last event.<a href="#cite_note-Haight1967-1">[1]</a> It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after <a href="/wiki/France" title="France">French</a> mathematician <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>. It plays an important role for <a href="/wiki/Discrete-stable_distribution" title="Discrete-stable distribution">discrete-stable distributions</a>. Under a Poisson distribution with the <a href="/wiki/Expected_value" title="Expected value">expectation</a> of λ events in a given interval, the probability of k events in the same interval is:<a href="#cite_note-Yates2014-2">[2]</a>: 60  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc22b6080eef83a193028baca6189ccd675b61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.48ex; height:5.843ex;" alt="{\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.}"></dd></dl> For instance, consider a call center which receives an average of λ = 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next one will arrive. Under these assumptions, the number k of calls received during any minute has a Poisson probability distribution. Receiving k = 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23. A classic example used to motivate the Poisson distribution is the number of <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a> events during a fixed observation period.<a href="#cite_note-3">[3]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> The distribution was first introduced by <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a> (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837).<a href="#cite_note-Poisson1837-4">[4]</a>: 205-207  The work theorized about the number of wrongful convictions in a given country by focusing on certain <a href="/wiki/Random_variable" title="Random variable">random variables</a> N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a <a href="/wiki/Time" title="Time">time</a>-interval of given length. The result had already been given in 1711 by <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus .<a href="#cite_note-deMoivre1711-5">[5]</a>: 219 <a href="#cite_note-deMoivre1718-6">[6]</a>: 14-15 <a href="#cite_note-deMoivre1721-7">[7]</a>: 193 <a href="#cite_note-Johnson2005-8">[8]</a>: 157  This makes it an example of <a href="/wiki/Stigler%27s_law" class="mw-redirect" title="Stigler's law">Stigler's law</a> and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.<a href="#cite_note-Stigler1982-9">[9]</a><a href="#cite_note-Hald1984-10">[10]</a> In 1860, <a href="/wiki/Simon_Newcomb" title="Simon Newcomb">Simon Newcomb</a> fitted the Poisson distribution to the number of stars found in a unit of space.<a href="#cite_note-Newcomb1860-11">[11]</a> A further practical application was made by <a href="/wiki/Ladislaus_Bortkiewicz" title="Ladislaus Bortkiewicz">Ladislaus Bortkiewicz</a> in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution.<a href="#cite_note-vonBortkiewitsch1898-12">[12]</a>: 23-25 . <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div> <div class="mw-heading mw-heading3"><h3 id="Probability_mass_function">Probability mass function</h3></div> A discrete <a href="/wiki/Random_variable" title="Random variable">random variable</a> X is said to have a Poisson distribution with parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"> if it has a <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> given by:<a href="#cite_note-Yates2014-2">[2]</a>: 60  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;\lambda )=\Pr(X{=}k)={\frac {\lambda ^{k}e^{-\lambda }}{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>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;\lambda )=\Pr(X{=}k)={\frac {\lambda ^{k}e^{-\lambda }}{k!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0693ecaa606e3878dfa9a85552d357c690ffb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.668ex; height:5.843ex;" alt="{\displaystyle f(k;\lambda )=\Pr(X{=}k)={\frac {\lambda ^{k}e^{-\lambda }}{k!}},}"></dd></dl> where <ul><li>k is the number of occurrences (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,2,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d4b81efe1ce928c56c7b143cb2591f2100246c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.622ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,\ldots }">)</li> <li>e is <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=2.71828\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>2.71828</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=2.71828\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66c92b4dba5b8d590a9e2b01c854ca16f06927a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.914ex; height:2.176ex;" alt="{\displaystyle e=2.71828\ldots }">)</li> <li>k! = k(k–1) ··· (3)(2)(1) is the <a href="/wiki/Factorial" title="Factorial">factorial</a>.</li></ul> The positive <a href="/wiki/Real_number" title="Real number">real number</a> λ is equal to the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of X and also to its <a href="/wiki/Variance" title="Variance">variance</a>.<a href="#cite_note-13">[13]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\operatorname {E} (X)=\operatorname {Var} (X).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\operatorname {E} (X)=\operatorname {Var} (X).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2debd3f9adf97c8af4919aa69ed4a7121b47a737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.178ex; height:2.843ex;" alt="{\displaystyle \lambda =\operatorname {E} (X)=\operatorname {Var} (X).}"></dd></dl> The Poisson distribution can be applied to systems with a <a href="/wiki/Large_number_of_rare_events" title="Large number of rare events">large number of possible events, each of which is rare</a>. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution. The equation can be adapted if, instead of the average number of events <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00aebb041f4a569408e310294efcc29e0eded7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.002ex; height:2.509ex;" alt="{\displaystyle \lambda ,}"> we are given the average rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> at which events occur. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =rt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi>r</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =rt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b61e1785da67e0020862cf83133fbb3f6a7d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.989ex; height:2.509ex;" alt="{\displaystyle \lambda =rt,}"> and:<a href="#cite_note-14">[14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k{\text{ events in interval }}t)={\frac {(rt)^{k}e^{-rt}}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> events in interval </mtext> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k{\text{ events in interval }}t)={\frac {(rt)^{k}e^{-rt}}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52913ad3c9d45138ab03c8df7fe9f82728b1f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.395ex; height:6.009ex;" alt="{\displaystyle P(k{\text{ events in interval }}t)={\frac {(rt)^{k}e^{-rt}}{k!}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG" class="mw-file-description"><img alt="Chewing gum on a sidewalk in Reykjavík." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG/220px-Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG/330px-Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG/440px-Chewing_gum_on_a_sidewalk_in_Reykjav%C3%ADk.JPG 2x" data-file-width="2472" data-file-height="3296" /></a><figcaption>Chewing gum on a sidewalk. The number of pieces on a single tile is approximately Poisson distributed.</figcaption></figure> The Poisson distribution may be useful to model events such as: <ul><li>the number of meteorites greater than 1-meter diameter that strike Earth in a year;</li> <li>the number of laser photons hitting a detector in a particular time interval;</li> <li>the number of students achieving a low and high mark in an exam; and</li> <li>locations of defects and dislocations in materials.</li></ul> Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional).<a href="#cite_note-15">[15]</a> <div class="mw-heading mw-heading3"><h3 id="Assumptions_and_validity">Assumptions and validity</h3></div> The Poisson distribution is an appropriate model if the following assumptions are true: <ul><li>k, a nonnegative integer, is the number of times an event occurs in an interval.</li> <li>The occurrence of one event <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">does not affect the probability</a> of a second event.</li> <li>The average rate at which events occur is independent of any occurrences.</li> <li>Two events cannot occur at exactly the same instant.</li></ul> If these conditions are true, then k is a Poisson random variable; the distribution of k is a Poisson distribution. The Poisson distribution is also the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity (see <a href="#Related_distributions">Related distributions</a>). <div class="mw-heading mw-heading4"><h4 id="Examples_of_probability_for_Poisson_distributions">Examples of probability for Poisson distributions</h4></div> <style data-mw-deduplicate="TemplateStyles:r1216972533">.mw-parser-output .col-begin{border-collapse:collapse;padding:0;color:inherit;width:100%;border:0;margin:0}.mw-parser-output .col-begin-small{font-size:90%}.mw-parser-output .col-break{vertical-align:top;text-align:left}.mw-parser-output .col-break-2{width:50%}.mw-parser-output .col-break-3{width:33.3%}.mw-parser-output .col-break-4{width:25%}.mw-parser-output .col-break-5{width:20%}@media(max-width:720px){.mw-parser-output .col-begin,.mw-parser-output .col-begin>tbody,.mw-parser-output .col-begin>tbody>tr,.mw-parser-output .col-begin>tbody>tr>td{display:block!important;width:100%!important}.mw-parser-output .col-break{padding-left:0!important}}</style><div> <table class="col-begin" role="presentation"> <tbody><tr> <td class="col-break"> On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate. Because the average event rate is one overflow flood per 100 years, λ = 1 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k{\text{ overflow floods in 100 years}})={\frac {\lambda ^{k}e^{-\lambda }}{k!}}={\frac {1^{k}e^{-1}}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> overflow floods in 100 years</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k{\text{ overflow floods in 100 years}})={\frac {\lambda ^{k}e^{-\lambda }}{k!}}={\frac {1^{k}e^{-1}}{k!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3aa61d2a6d2e549fba969b4cfb923e5f612356a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.305ex; height:5.843ex;" alt="{\displaystyle P(k{\text{ overflow floods in 100 years}})={\frac {\lambda ^{k}e^{-\lambda }}{k!}}={\frac {1^{k}e^{-1}}{k!}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=0{\text{ overflow floods in 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {e^{-1}}{1}}\approx 0.368}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> overflow floods in 100 years</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>1</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.368</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=0{\text{ overflow floods in 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {e^{-1}}{1}}\approx 0.368}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539604237097b4b7eaf1304d9eb28c2e8b241051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:62.346ex; height:5.843ex;" alt="{\displaystyle P(k=0{\text{ overflow floods in 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {e^{-1}}{1}}\approx 0.368}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=1{\text{ overflow flood in 100 years}})={\frac {1^{1}e^{-1}}{1!}}={\frac {e^{-1}}{1}}\approx 0.368}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> overflow flood in 100 years</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>1</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.368</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=1{\text{ overflow flood in 100 years}})={\frac {1^{1}e^{-1}}{1!}}={\frac {e^{-1}}{1}}\approx 0.368}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ecb437f5dc2d6fdd04104dd682909feab34802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.43ex; height:5.843ex;" alt="{\displaystyle P(k=1{\text{ overflow flood in 100 years}})={\frac {1^{1}e^{-1}}{1!}}={\frac {e^{-1}}{1}}\approx 0.368}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=2{\text{ overflow floods in 100 years}})={\frac {1^{2}e^{-1}}{2!}}={\frac {e^{-1}}{2}}\approx 0.184}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> overflow floods in 100 years</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.184</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=2{\text{ overflow floods in 100 years}})={\frac {1^{2}e^{-1}}{2!}}={\frac {e^{-1}}{2}}\approx 0.184}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d018cf11ae3ada2c025c25c784b893ad4b057bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:62.346ex; height:5.843ex;" alt="{\displaystyle P(k=2{\text{ overflow floods in 100 years}})={\frac {1^{2}e^{-1}}{2!}}={\frac {e^{-1}}{2}}\approx 0.184}"></dd></dl> </td> <td class="col-break" style="padding-left: 5em;"> <dl><dd><table class="wikitable"> <tbody><tr> <th>k</th> <th>P(k overflow floods in 100 years) </th></tr> <tr> <td>0</td> <td>0.368 </td></tr> <tr> <td>1</td> <td>0.368 </td></tr> <tr> <td>2</td> <td>0.184 </td></tr> <tr> <td>3</td> <td>0.061 </td></tr> <tr> <td>4</td> <td>0.015 </td></tr> <tr> <td>5</td> <td>0.003 </td></tr> <tr> <td>6</td> <td>0.0005 </td></tr></tbody></table></dd></dl> The probability for 0 to 6 overflow floods in a 100-year period. </td></tr></tbody></table></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1216972533"><div> <table class="col-begin" role="presentation"> <tbody><tr> <td class="col-break"> It has been reported that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.<a href="#cite_note-Ugarte2016-16">[16]</a> Because the average event rate is 2.5 goals per match, λ = 2.5 . <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k{\text{ goals in a match}})={\frac {2.5^{k}e^{-2.5}}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> goals in a match</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k{\text{ goals in a match}})={\frac {2.5^{k}e^{-2.5}}{k!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284f2ea7482e5f7296cbe1d57a0fdfe67666dca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.259ex; height:5.843ex;" alt="{\displaystyle P(k{\text{ goals in a match}})={\frac {2.5^{k}e^{-2.5}}{k!}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=0{\text{ goals in a match}})={\frac {2.5^{0}e^{-2.5}}{0!}}={\frac {e^{-2.5}}{1}}\approx 0.082}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> goals in a match</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> <mn>1</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.082</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=0{\text{ goals in a match}})={\frac {2.5^{0}e^{-2.5}}{0!}}={\frac {e^{-2.5}}{1}}\approx 0.082}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b9f5b04a762af5cf6db5639f7c6203526379d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:55.511ex; height:5.843ex;" alt="{\displaystyle P(k=0{\text{ goals in a match}})={\frac {2.5^{0}e^{-2.5}}{0!}}={\frac {e^{-2.5}}{1}}\approx 0.082}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=1{\text{ goal in a match}})={\frac {2.5^{1}e^{-2.5}}{1!}}={\frac {2.5e^{-2.5}}{1}}\approx 0.205}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> goal in a match</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2.5</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.205</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=1{\text{ goal in a match}})={\frac {2.5^{1}e^{-2.5}}{1!}}={\frac {2.5e^{-2.5}}{1}}\approx 0.205}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8d3b627a69fcb05a0e3202cd4cd03e22cb4dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:57.567ex; height:5.843ex;" alt="{\displaystyle P(k=1{\text{ goal in a match}})={\frac {2.5^{1}e^{-2.5}}{1!}}={\frac {2.5e^{-2.5}}{1}}\approx 0.205}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k=2{\text{ goals in a match}})={\frac {2.5^{2}e^{-2.5}}{2!}}={\frac {6.25e^{-2.5}}{2}}\approx 0.257}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> goals in a match</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6.25</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.257</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k=2{\text{ goals in a match}})={\frac {2.5^{2}e^{-2.5}}{2!}}={\frac {6.25e^{-2.5}}{2}}\approx 0.257}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ffb7f0c13f51b960d1580e5cac1d4365ef4166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.645ex; height:5.843ex;" alt="{\displaystyle P(k=2{\text{ goals in a match}})={\frac {2.5^{2}e^{-2.5}}{2!}}={\frac {6.25e^{-2.5}}{2}}\approx 0.257}"></dd></dl> </td> <td class="col-break"> <dl><dd><table class="wikitable"> <tbody><tr> <th>k</th> <th>P(k goals in a World Cup soccer match) </th></tr> <tr> <td>0</td> <td>0.082 </td></tr> <tr> <td>1</td> <td>0.205 </td></tr> <tr> <td>2</td> <td>0.257 </td></tr> <tr> <td>3</td> <td>0.213 </td></tr> <tr> <td>4</td> <td>0.133 </td></tr> <tr> <td>5</td> <td>0.067 </td></tr> <tr> <td>6</td> <td>0.028 </td></tr> <tr> <td>7</td> <td>0.010 </td></tr></tbody></table></dd></dl> The probability for 0 to 7 goals in a match. </td></tr></tbody></table></div> <div class="mw-heading mw-heading4"><h4 id="Once_in_an_interval_events:_The_special_case_of_λ_=_1_and_k_=_0">Once in an interval events: The special case of λ = 1 and k = 0</h4></div> Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years ( λ = 1 event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of k = 0 meteorite hits in the next 100 years? <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k={\text{0 meteorites hit in next 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {1}{e}}\approx 0.37.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 meteorites hit in next 100 years</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> <mo>≈</mo> <mn>0.37.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k={\text{0 meteorites hit in next 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {1}{e}}\approx 0.37.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab82295afc6daa62eb2556d4931889793826c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:63.512ex; height:5.843ex;" alt="{\displaystyle P(k={\text{0 meteorites hit in next 100 years}})={\frac {1^{0}e^{-1}}{0!}}={\frac {1}{e}}\approx 0.37.}"></dd></dl> Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1 − 0.37 = 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years. In an example above, an overflow flood occurred once every 100 years (λ = 1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation. In general, if an event occurs on average once per interval (λ = 1), and the events follow a Poisson distribution, then P(0 events in next interval) = 0.37. In addition, P(exactly one event in next interval) = 0.37, as shown in the table for overflow floods. <div class="mw-heading mw-heading3"><h3 id="Examples_that_violate_the_Poisson_assumptions">Examples that violate the Poisson assumptions</h3></div> The number of students who arrive at the <a href="/wiki/Student_center" title="Student center">student union</a> per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups). The non-constant arrival rate may be modeled as a <a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">mixed Poisson distribution</a>, and the arrival of groups rather than individual students as a <a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">compound Poisson process</a>. The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution, if one large earthquake increases the probability of aftershocks of similar magnitude. Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a <a href="/wiki/Zero-truncated_Poisson_distribution" title="Zero-truncated Poisson distribution">zero-truncated Poisson distribution</a>. Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a <a href="/wiki/Zero-inflated_model" title="Zero-inflated model">zero-inflated model</a>. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div> <div class="mw-heading mw-heading3"><h3 id="Descriptive_statistics">Descriptive statistics</h3></div> <ul><li>The <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a Poisson random variable is λ.</li> <li>The <a href="/wiki/Variance" title="Variance">variance</a> of a Poisson random variable is also λ.</li> <li>The <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda ^{-1/2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda ^{-1/2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d86abd02434f5828f44e3363096013ead691cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.979ex; height:3.009ex;" alt="{\textstyle \lambda ^{-1/2},}"> while the <a href="/wiki/Index_of_dispersion" title="Index of dispersion">index of dispersion</a> is 1.<a href="#cite_note-Johnson2005-8">[8]</a>: 163 </li> <li>The <a href="/wiki/Mean_absolute_deviation" class="mw-redirect" title="Mean absolute deviation">mean absolute deviation</a> about the mean is<a href="#cite_note-Johnson2005-8">[8]</a>: 163  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\ |X-\lambda |\ ]={\frac {2\lambda ^{\lfloor \lambda \rfloor +1}e^{-\lambda }}{\lfloor \lambda \rfloor !}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>λ</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mo fence="false" stretchy="false">⌊</mo> <mi>λ</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\ |X-\lambda |\ ]={\frac {2\lambda ^{\lfloor \lambda \rfloor +1}e^{-\lambda }}{\lfloor \lambda \rfloor !}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38314ac3cdef3237f4321117fbf030a1115f6b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.91ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} [\ |X-\lambda |\ ]={\frac {2\lambda ^{\lfloor \lambda \rfloor +1}e^{-\lambda }}{\lfloor \lambda \rfloor !}}.}"></li> <li>The <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> of a Poisson-distributed random variable with non-integer λ is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \lambda \rfloor ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>λ</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \lambda \rfloor ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c849ab92d3ad00c65efed52698719af80cb4d8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.067ex; height:2.843ex;" alt="{\displaystyle \lfloor \lambda \rfloor ,}"> which is the largest integer less than or equal to λ. This is also written as <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor</a>(λ). When λ is a positive integer, the modes are λ and λ − 1.</li> <li>All of the <a href="/wiki/Cumulant" title="Cumulant">cumulants</a> of the Poisson distribution are equal to the expected value λ. The n th <a href="/wiki/Factorial_moment" title="Factorial moment">factorial moment</a> of the Poisson distribution is λ n  .</li> <li>The <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as "exposure").<a href="#cite_note-Helske2017-17">[17]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Median">Median</h3></div> Bounds for the median (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }">) of the distribution are known and are <a href="/wiki/Mathematical_jargon#sharp" class="mw-redirect" title="Mathematical jargon">sharp</a>:<a href="#cite_note-Choi1994-18">[18]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda -\ln 2\leq \nu <\lambda +{\frac {1}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo>≤</mo> <mi>ν</mi> <mo><</mo> <mi>λ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda -\ln 2\leq \nu <\lambda +{\frac {1}{3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ea94f8222d30fd83d8811640620cda70d67532" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.955ex; height:5.176ex;" alt="{\displaystyle \lambda -\ln 2\leq \nu <\lambda +{\frac {1}{3}}.}"> <div class="mw-heading mw-heading3"><h3 id="Higher_moments">Higher moments</h3></div> The higher non-centered <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> mk of the Poisson distribution are <a href="/wiki/Touchard_polynomials" title="Touchard polynomials">Touchard polynomials</a> in λ: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}=\sum _{i=0}^{k}\lambda ^{i}{\begin{Bmatrix}k\\i\end{Bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <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> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k}=\sum _{i=0}^{k}\lambda ^{i}{\begin{Bmatrix}k\\i\end{Bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407a05be1700329aa6b29dc7add0bdbd6e0cdeae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.22ex; height:7.343ex;" alt="{\displaystyle m_{k}=\sum _{i=0}^{k}\lambda ^{i}{\begin{Bmatrix}k\\i\end{Bmatrix}},}"> where the braces { } denote <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>.<a href="#cite_note-Riordan1937-19">[19]</a><a href="#cite_note-Haight1967-1">[1]</a>: 6  In other words,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X]=\lambda ,\quad E[X(X-1)]=\lambda ^{2},\quad E[X(X-1)(X-2)]=\lambda ^{3},\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>λ</mi> <mo>,</mo> <mspace width="1em" /> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X]=\lambda ,\quad E[X(X-1)]=\lambda ^{2},\quad E[X(X-1)(X-2)]=\lambda ^{3},\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27f18d1393a99b37c525e21e568f340cf27889d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.464ex; height:3.176ex;" alt="{\displaystyle E[X]=\lambda ,\quad E[X(X-1)]=\lambda ^{2},\quad E[X(X-1)(X-2)]=\lambda ^{3},\cdots }">When the expected value is set to λ = 1, <a href="/wiki/Dobinski%27s_formula" class="mw-redirect" title="Dobinski's formula">Dobinski's formula</a> implies that the n‑th moment is equal to the number of <a href="/wiki/Partition_of_a_set" title="Partition of a set">partitions of a set</a> of size n. A simple upper bound is:<a href="#cite_note-20">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}=E[X^{k}]\leq \left({\frac {k}{\log(k/\lambda +1)}}\right)^{k}\leq \lambda ^{k}\exp \left({\frac {k^{2}}{2\lambda }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k}=E[X^{k}]\leq \left({\frac {k}{\log(k/\lambda +1)}}\right)^{k}\leq \lambda ^{k}\exp \left({\frac {k^{2}}{2\lambda }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d2997208e48609996f73dc03db70e997b01b29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.244ex; height:6.843ex;" alt="{\displaystyle m_{k}=E[X^{k}]\leq \left({\frac {k}{\log(k/\lambda +1)}}\right)^{k}\leq \lambda ^{k}\exp \left({\frac {k^{2}}{2\lambda }}\right).}"> <div class="mw-heading mw-heading3"><h3 id="Sums_of_Poisson-distributed_random_variables">Sums of Poisson-distributed random variables</h3></div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}\sim \operatorname {Pois} (\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}\sim \operatorname {Pois} (\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57094fb3ef6d2a578bb7d80a681849e56cffae13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle X_{i}\sim \operatorname {Pois} (\lambda _{i})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f2132430a61b900cf2c4380774394ca9f09c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}"> are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{n}X_{i}\sim \operatorname {Pois} \left(\sum _{i=1}^{n}\lambda _{i}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}X_{i}\sim \operatorname {Pois} \left(\sum _{i=1}^{n}\lambda _{i}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc45fecdb579f6f3fc2668ca72f6e5bd13f3ac02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.224ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}X_{i}\sim \operatorname {Pois} \left(\sum _{i=1}^{n}\lambda _{i}\right).}"><a href="#cite_note-Lehmann1986-21">[21]</a>: 65  A converse is <a href="/wiki/Raikov%27s_theorem" title="Raikov's theorem">Raikov's theorem</a>, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.<a href="#cite_note-Raikov1937-22">[22]</a><a href="#cite_note-vonMises1964-23">[23]</a> <div class="mw-heading mw-heading3"><h3 id="Maximum_entropy">Maximum entropy</h3></div> It is a <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum-entropy distribution</a> among the set of generalized binomial distributions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700c8c6b217dd619c9d04d59d6dd891f0195a619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.147ex; height:2.843ex;" alt="{\displaystyle B_{n}(\lambda )}"> with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }">,<a href="#cite_note-24">[24]</a> where a generalized binomial distribution is defined as a distribution of the sum of N independent but not identically distributed Bernoulli variables. <div class="mw-heading mw-heading3"><h3 id="Other_properties">Other properties</h3></div> <ul><li>The Poisson distributions are <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinitely divisible</a> probability distributions.<a href="#cite_note-Laha1979-25">[25]</a>: 233 <a href="#cite_note-Johnson2005-8">[8]</a>: 164 </li> <li>The directed <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475e008e214bee8d96d3f528d07bf3dffae9bd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.317ex; height:2.843ex;" alt="{\displaystyle P=\operatorname {Pois} (\lambda )}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=\operatorname {Pois} (\lambda _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=\operatorname {Pois} (\lambda _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36995280929cd439a91c5c05ac56230c1ded708b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.172ex; height:2.843ex;" alt="{\displaystyle P_{0}=\operatorname {Pois} (\lambda _{0})}"> is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} _{\text{KL}}(P\parallel P_{0})=\lambda _{0}-\lambda +\lambda \log {\frac {\lambda }{\lambda _{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∥</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <mo>+</mo> <mi>λ</mi> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {D} _{\text{KL}}(P\parallel P_{0})=\lambda _{0}-\lambda +\lambda \log {\frac {\lambda }{\lambda _{0}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28957e0866ada1c7781ed5f9af0b120d5e1fada" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.405ex; height:5.843ex;" alt="{\displaystyle \operatorname {D} _{\text{KL}}(P\parallel P_{0})=\lambda _{0}-\lambda +\lambda \log {\frac {\lambda }{\lambda _{0}}}.}"></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a37870accb1d7cb8912fd80b53a7e4c9bc61e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.616ex; height:2.343ex;" alt="{\displaystyle \lambda \geq 1}"> is an integer, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a75008c5e80fc6c742fc674dbcbcf9548917b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.345ex; height:2.843ex;" alt="{\displaystyle Y\sim \operatorname {Pois} (\lambda )}"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y\geq E[Y])\geq {\frac {1}{2}}}"> <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>E</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y\geq E[Y])\geq {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b13690a640b1826b92b683358e58098f957138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.115ex; height:5.176ex;" alt="{\displaystyle \Pr(Y\geq E[Y])\geq {\frac {1}{2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y\leq E[Y])\geq {\frac {1}{2}}.}"> <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>E</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y\leq E[Y])\geq {\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e3c38659a011ac0e4b6dd42cc920e61e92b936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.762ex; height:5.176ex;" alt="{\displaystyle \Pr(Y\leq E[Y])\geq {\frac {1}{2}}.}"><a href="#cite_note-26">[26]</a>[<a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">failed verification</a> – <a href="/wiki/Talk:Poisson_distribution#Other_Properties_-_Mitzenmacher" title="Talk:Poisson distribution">see discussion</a>]</li> <li>Bounds for the tail probabilities of a Poisson random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83722493a50f90c09c96abb65364b7605ab55469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.551ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> can be derived using a <a href="/wiki/Chernoff_bound" title="Chernoff bound">Chernoff bound</a> argument.<a href="#cite_note-Mitzenmacher2005-27">[27]</a>: 97-98  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X\geq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x>\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>></mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\geq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x>\lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9dbce798aa0c81ae4d37d2b507555564b39c74f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.282ex; height:5.843ex;" alt="{\displaystyle P(X\geq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x>\lambda ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X\leq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x<\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo><</mo> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\leq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x<\lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abdcf3977c0fc3686907d0fbefa0ad20d5a4061" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.282ex; height:5.843ex;" alt="{\displaystyle P(X\leq x)\leq {\frac {(e\lambda )^{x}e^{-\lambda }}{x^{x}}},{\text{ for }}x<\lambda .}"></li> <li>The upper tail probability can be tightened (by a factor of at least two) as follows:<a href="#cite_note-28">[28]</a></li></ul> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X\geq x)\leq {\frac {e^{-\operatorname {D} _{\text{KL}}(Q\parallel P)}}{\max {(2,{\sqrt {4\pi \operatorname {D} _{\text{KL}}(Q\parallel P)}}})}},{\text{ for }}x>\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mi>π</mi> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>></mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\geq x)\leq {\frac {e^{-\operatorname {D} _{\text{KL}}(Q\parallel P)}}{\max {(2,{\sqrt {4\pi \operatorname {D} _{\text{KL}}(Q\parallel P)}}})}},{\text{ for }}x>\lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5288c3fe36c7e33ba8b384b16b93d549ef72a745" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.392ex; height:7.009ex;" alt="{\displaystyle P(X\geq x)\leq {\frac {e^{-\operatorname {D} _{\text{KL}}(Q\parallel P)}}{\max {(2,{\sqrt {4\pi \operatorname {D} _{\text{KL}}(Q\parallel P)}}})}},{\text{ for }}x>\lambda ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} _{\text{KL}}(Q\parallel P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {D} _{\text{KL}}(Q\parallel P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963891e9c244ce24baa845ef4239bfc21f0c910f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.16ex; height:2.843ex;" alt="{\displaystyle \operatorname {D} _{\text{KL}}(Q\parallel P)}"> is the Kullback–Leibler divergence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\operatorname {Pois} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\operatorname {Pois} (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e075409d68cf73d611c649825f8ca7c4be6f85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.384ex; height:2.843ex;" alt="{\displaystyle Q=\operatorname {Pois} (x)}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475e008e214bee8d96d3f528d07bf3dffae9bd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.317ex; height:2.843ex;" alt="{\displaystyle P=\operatorname {Pois} (\lambda )}">. <ul><li>Inequalities that relate the distribution function of a Poisson random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83722493a50f90c09c96abb65364b7605ab55469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.551ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> to the <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">Standard normal distribution</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Phi (x)}"> are as follows:<a href="#cite_note-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \left(\operatorname {sign} (k-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}}\right)<P(X\leq k)<\Phi \left(\operatorname {sign} (k+1-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}}\right),{\text{ for }}k>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow> <mi>sign</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow> <mi>sign</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>k</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi \left(\operatorname {sign} (k-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}}\right)<P(X\leq k)<\Phi \left(\operatorname {sign} (k+1-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}}\right),{\text{ for }}k>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d508dff09baa73d69dcc23890be3009d2ce0201" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:98.769ex; height:4.843ex;" alt="{\displaystyle \Phi \left(\operatorname {sign} (k-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}}\right)<P(X\leq k)<\Phi \left(\operatorname {sign} (k+1-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}}\right),{\text{ for }}k>0,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03fb2deee9168d32f4fcf89ead75b9c7cf0dfe24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}"> is the Kullback–Leibler divergence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{-}=\operatorname {Pois} (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{-}=\operatorname {Pois} (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3936299a039702e3f36292e7bbe7b4df1d934f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.777ex; height:2.843ex;" alt="{\displaystyle Q_{-}=\operatorname {Pois} (k)}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475e008e214bee8d96d3f528d07bf3dffae9bd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.317ex; height:2.843ex;" alt="{\displaystyle P=\operatorname {Pois} (\lambda )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>∥</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee367cb9e271a607be983341a7e4784ea593adf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}"> is the Kullback–Leibler divergence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{+}=\operatorname {Pois} (k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{+}=\operatorname {Pois} (k+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b01f8a926dc19d8c84fec740eb863736a9ec517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.779ex; height:2.843ex;" alt="{\displaystyle Q_{+}=\operatorname {Pois} (k+1)}"> from <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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Poisson_races">Poisson races</h3></div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Pois} (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83722493a50f90c09c96abb65364b7605ab55469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.551ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Pois} (\lambda )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \operatorname {Pois} (\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \operatorname {Pois} (\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c17e18a900ce05dae1e372f8ee84cca95658e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.391ex; height:2.843ex;" alt="{\displaystyle Y\sim \operatorname {Pois} (\mu )}"> be independent random variables, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda <\mu ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo><</mo> <mi>μ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda <\mu ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c39745dc3c331494d2e3cf15cd5bc9f335e03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.502ex; height:2.676ex;" alt="{\displaystyle \lambda <\mu ,}"> then we have that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}{(\lambda +\mu )^{2}}}-{\frac {e^{-(\lambda +\mu )}}{2{\sqrt {\lambda \mu }}}}-{\frac {e^{-(\lambda +\mu )}}{4\lambda \mu }}\leq P(X-Y\geq 0)\leq e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> <mi>μ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mn>4</mn> <mi>λ</mi> <mi>μ</mi> </mrow> </mfrac> </mrow> <mo>≤</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>Y</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}{(\lambda +\mu )^{2}}}-{\frac {e^{-(\lambda +\mu )}}{2{\sqrt {\lambda \mu }}}}-{\frac {e^{-(\lambda +\mu )}}{4\lambda \mu }}\leq P(X-Y\geq 0)\leq e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511aa4ec49b20e83b83bcc2a24bef763dabe1d90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.341ex; height:7.509ex;" alt="{\displaystyle {\frac {e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}{(\lambda +\mu )^{2}}}-{\frac {e^{-(\lambda +\mu )}}{2{\sqrt {\lambda \mu }}}}-{\frac {e^{-(\lambda +\mu )}}{4\lambda \mu }}\leq P(X-Y\geq 0)\leq e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}"> The upper bound is proved using a standard Chernoff bound. The lower bound can be proved by noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X-Y\geq 0\mid X+Y=i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>Y</mi> <mo>≥</mo> <mn>0</mn> <mo>∣</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X-Y\geq 0\mid X+Y=i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d8a24c0406c0b31012b78f0522864108c355b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.841ex; height:2.843ex;" alt="{\displaystyle P(X-Y\geq 0\mid X+Y=i)}"> is the probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z\geq {\frac {i}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Z</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z\geq {\frac {i}{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514fc71347875ec4954d207a57d64ed7f3731687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.084ex; height:3.509ex;" alt="{\textstyle Z\geq {\frac {i}{2}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z\sim \operatorname {Bin} \left(i,{\frac {\lambda }{\lambda +\mu }}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Z</mi> <mo>∼</mo> <mi>Bin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z\sim \operatorname {Bin} \left(i,{\frac {\lambda }{\lambda +\mu }}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89453934127eadfc8418406e3724e9264c5e60c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.687ex; height:4.843ex;" alt="{\textstyle Z\sim \operatorname {Bin} \left(i,{\frac {\lambda }{\lambda +\mu }}\right),}"> which is bounded below by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{(i+1)^{2}}}e^{-iD\left(0.5\|{\frac {\lambda }{\lambda +\mu }}\right)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>D</mi> <mrow> <mo>(</mo> <mrow> <mn>0.5</mn> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{(i+1)^{2}}}e^{-iD\left(0.5\|{\frac {\lambda }{\lambda +\mu }}\right)},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b50e47160cf654b9dea04b316a10efd0cc3e206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.583ex; height:6.676ex;" alt="{\textstyle {\frac {1}{(i+1)^{2}}}e^{-iD\left(0.5\|{\frac {\lambda }{\lambda +\mu }}\right)},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">relative entropy</a> (See the entry on <a href="/wiki/Binomial_distribution#Tail_bounds" title="Binomial distribution">bounds on tails of binomial distributions</a> for details). Further noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+Y\sim \operatorname {Pois} (\lambda +\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+Y\sim \operatorname {Pois} (\lambda +\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ef4f27c243ff74934b3a02bb29348c83941c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.054ex; height:2.843ex;" alt="{\displaystyle X+Y\sim \operatorname {Pois} (\lambda +\mu ),}"> and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al.<a href="#cite_note-Kamath2015-30">[30]</a> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2></div> <div class="mw-heading mw-heading3"><h3 id="As_a_Binomial_distribution_with_infinitesimal_time-steps">As a Binomial distribution with infinitesimal time-steps</h3></div> The Poisson distribution can be derived as a limiting case to the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> as the number of trials goes to infinity and the <a href="/wiki/Expected_value" title="Expected value">expected</a> number of successes remains fixed — see <a href="#law_of_rare_events">law of rare events</a> below. Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and n p ≤ 10.<a href="#cite_note-NIST2006-31">[31]</a> Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {B} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64491e3f1723572954668fdd637c03106b30bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:2.509ex;" alt="{\displaystyle F_{\mathrm {B} }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {P} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {P} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d64a355eb5f1217b4f12b7f27230b2ff31b2e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle F_{\mathrm {P} }}"> be the respective <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative density functions</a> of the binomial and Poisson distributions, one has: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {B} }(k;n,p)\ \approx \ F_{\mathrm {P} }(k;\lambda =np).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>≈</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {B} }(k;n,p)\ \approx \ F_{\mathrm {P} }(k;\lambda =np).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e9f3365dbaa79aea54e1122722b66c6198237a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.368ex; height:2.843ex;" alt="{\displaystyle F_{\mathrm {B} }(k;n,p)\ \approx \ F_{\mathrm {P} }(k;\lambda =np).}">One derivation of this uses <a href="/wiki/Probability-generating_function" title="Probability-generating function">probability-generating functions</a>.<a href="#cite_note-32">[32]</a> Consider a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a> (coin-flip) whose probability of one success (or expected number of successes) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9556892967123042718936f92d8bd5762e6c0c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.616ex; height:2.343ex;" alt="{\displaystyle \lambda \leq 1}"> within a given interval. Split the interval into n parts, and perform a trial in each subinterval with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a950558663bc182414afa929ad713a28632abd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.509ex;" alt="{\displaystyle {\tfrac {\lambda }{n}}}">. The probability of k successes out of n trials over the entire interval is then given by the binomial distribution<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{k}^{(n)}={\binom {n}{k}}\left({\frac {\lambda }{n}}\right)^{\!k}\left(1{-}{\frac {\lambda }{n}}\right)^{\!n-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <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> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>k</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{k}^{(n)}={\binom {n}{k}}\left({\frac {\lambda }{n}}\right)^{\!k}\left(1{-}{\frac {\lambda }{n}}\right)^{\!n-k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4de733b4d846dc074b8bf5812f270c374644ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:30.261ex; height:6.676ex;" alt="{\displaystyle p_{k}^{(n)}={\binom {n}{k}}\left({\frac {\lambda }{n}}\right)^{\!k}\left(1{-}{\frac {\lambda }{n}}\right)^{\!n-k},}"> </blockquote>whose generating function is:<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{(n)}(x)=\sum _{k=0}^{n}p_{k}^{(n)}x^{k}=\left(1-{\frac {\lambda }{n}}+{\frac {\lambda }{n}}x\right)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <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>n</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{(n)}(x)=\sum _{k=0}^{n}p_{k}^{(n)}x^{k}=\left(1-{\frac {\lambda }{n}}+{\frac {\lambda }{n}}x\right)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f44d186774f4fb931e12cc1d9dbdb9dfb9d359c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.404ex; height:7.009ex;" alt="{\displaystyle P^{(n)}(x)=\sum _{k=0}^{n}p_{k}^{(n)}x^{k}=\left(1-{\frac {\lambda }{n}}+{\frac {\lambda }{n}}x\right)^{n}.}"></blockquote>Taking the limit as n increases to infinity (with x fixed) and applying the product limit definition of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, this reduces to the generating function of the Poisson distribution:<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }P^{(n)}(x)=\lim _{n\to \infty }\left(1{+}{\tfrac {\lambda (x-1)}{n}}\right)^{n}=e^{\lambda (x-1)}=\sum _{k=0}^{\infty }e^{-\lambda }{\frac {\lambda ^{k}}{k!}}x^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <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 mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }P^{(n)}(x)=\lim _{n\to \infty }\left(1{+}{\tfrac {\lambda (x-1)}{n}}\right)^{n}=e^{\lambda (x-1)}=\sum _{k=0}^{\infty }e^{-\lambda }{\frac {\lambda ^{k}}{k!}}x^{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c19b12a85cbf115d837d9dda86721eae1a4068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.386ex; height:7.009ex;" alt="{\displaystyle \lim _{n\to \infty }P^{(n)}(x)=\lim _{n\to \infty }\left(1{+}{\tfrac {\lambda (x-1)}{n}}\right)^{n}=e^{\lambda (x-1)}=\sum _{k=0}^{\infty }e^{-\lambda }{\frac {\lambda ^{k}}{k!}}x^{k}.}"></blockquote> <div class="mw-heading mw-heading3"><h3 id="General">General</h3></div> <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15d0eb82dbe950fd9a69fd735a9eb5854bb1516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.991ex; height:2.843ex;" alt="{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92230875c7140dbb0b8c9c4949fc710c330bdda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.991ex; height:2.843ex;" alt="{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}"> are independent, then the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X_{1}-X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X_{1}-X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627abb84b192e001722eb79e17a0489e26adbcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.669ex; height:2.509ex;" alt="{\displaystyle Y=X_{1}-X_{2}}"> follows a <a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam distribution</a>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15d0eb82dbe950fd9a69fd735a9eb5854bb1516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.991ex; height:2.843ex;" alt="{\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92230875c7140dbb0b8c9c4949fc710c330bdda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.991ex; height:2.843ex;" alt="{\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}"> are independent, then the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"> conditional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}+X_{2}}"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}+X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96d604ba472d9c8d3964bfb198de16b68e5f8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.797ex; height:2.509ex;" alt="{\displaystyle X_{1}+X_{2}}"> is a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>. <div class="paragraphbreak" style="margin-top:0.5em"></div> Specifically, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}+X_{2}=k,}"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}+X_{2}=k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5518d7a389f305366c8d983ca3f0ad18ba925a3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.754ex; height:2.509ex;" alt="{\displaystyle X_{1}+X_{2}=k,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}|X_{1}+X_{2}=k\sim \mathrm {Binom} (k,\lambda _{1}/(\lambda _{1}+\lambda _{2})).}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}|X_{1}+X_{2}=k\sim \mathrm {Binom} (k,\lambda _{1}/(\lambda _{1}+\lambda _{2})).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/596d3a787cb1f14db2692e0d74292a73fb6bad54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.256ex; height:2.843ex;" alt="{\displaystyle X_{1}|X_{1}+X_{2}=k\sim \mathrm {Binom} (k,\lambda _{1}/(\lambda _{1}+\lambda _{2})).}"> <div class="paragraphbreak" style="margin-top:0.5em"></div> More generally, if X1, X2, ..., Xn are independent Poisson random variables with parameters λ1, λ2, ..., λn then <dl><dd>given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{n}X_{j}=k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{n}X_{j}=k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0ed7c42d244e0ad66477b8276e926216da4819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.533ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{n}X_{j}=k,}"> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}{\Big |}\sum _{j=1}^{n}X_{j}=k\sim \mathrm {Binom} \left(k,{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}{\Big |}\sum _{j=1}^{n}X_{j}=k\sim \mathrm {Binom} \left(k,{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861dd5b85ea53c65668200ee34c63949f3a811f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.725ex; height:7.676ex;" alt="{\displaystyle X_{i}{\Big |}\sum _{j=1}^{n}X_{j}=k\sim \mathrm {Binom} \left(k,{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right).}"> In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{i}\}\sim \mathrm {Multinom} \left(k,\left\{{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right\}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{i}\}\sim \mathrm {Multinom} \left(k,\left\{{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right\}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/255ff85774dccd7aa239bacebe1b3c25fe46f3ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.206ex; height:7.509ex;" alt="{\displaystyle \{X_{i}\}\sim \mathrm {Multinom} \left(k,\left\{{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right\}\right).}"></dd></dl></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Pois} (\lambda )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Pois} (\lambda )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0823122073f11c547d1e8fcbd3d76415110ba320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.938ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Pois} (\lambda )\,}"> and the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> conditional on X = k is a <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\mid (X=k)\sim \mathrm {Binom} (k,p),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\mid (X=k)\sim \mathrm {Binom} (k,p),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef553b99963ef62014dee3714a938fa13037e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.462ex; height:2.843ex;" alt="{\displaystyle Y\mid (X=k)\sim \mathrm {Binom} (k,p),}"> then the distribution of Y follows a Poisson distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \mathrm {Pois} (\lambda \cdot p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>⋅</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {Pois} (\lambda \cdot p).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4cdebce8ebc73815fd41d64470af1da7c28358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.84ex; height:2.843ex;" alt="{\displaystyle Y\sim \mathrm {Pois} (\lambda \cdot p).}"> In fact, if, conditional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X=k\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X=k\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17be57fc8b36992200d2ccf2550315e033d18bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.261ex; height:2.843ex;" alt="{\displaystyle \{X=k\},}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Y_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{i}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e92004a07c56dd2b161c0647531c2893103b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.475ex; height:2.843ex;" alt="{\displaystyle \{Y_{i}\}}"> follows a <a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial distribution</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Y_{i}\}\mid (X=k)\sim \mathrm {Multinom} \left(k,p_{i}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{i}\}\mid (X=k)\sim \mathrm {Multinom} \left(k,p_{i}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc9796a889849a05d0ae236237823ccee6c4eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.067ex; height:2.843ex;" alt="{\displaystyle \{Y_{i}\}\mid (X=k)\sim \mathrm {Multinom} \left(k,p_{i}\right),}"> then each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}"> follows an independent Poisson distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}\sim \mathrm {Pois} (\lambda \cdot p_{i}),\rho (Y_{i},Y_{j})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}\sim \mathrm {Pois} (\lambda \cdot p_{i}),\rho (Y_{i},Y_{j})=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe251c0f62960e8d27c54fd1154174fdf8b34b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.767ex; height:3.009ex;" alt="{\displaystyle Y_{i}\sim \mathrm {Pois} (\lambda \cdot p_{i}),\rho (Y_{i},Y_{j})=0.}"></li> <li>The Poisson distribution is a <a href="/wiki/Special_case" title="Special case">special case</a> of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.<a href="#cite_note-Zhang2013-33">[33]</a><a href="#cite_note-Zhang2016-34">[34]</a> The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a <a href="/wiki/Compound_Poisson_distribution#Special_cases" title="Compound Poisson distribution">special case</a> of a <a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">compound Poisson distribution</a>.</li> <li>For sufficiently large values of λ, (say λ>1000), the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean λ and variance λ (standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95574539cdee5bead1cb65a182d4effd2800937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.291ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\lambda }}}">) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate <a href="/wiki/Continuity_correction" title="Continuity correction">continuity correction</a> is performed, i.e., if P(X ≤ x), where x is a non-negative integer, is replaced by P(X ≤ x + 0.5). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {Poisson} }(x;\lambda )\approx F_{\mathrm {normal} }(x;\mu =\lambda ,\sigma ^{2}=\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>=</mo> <mi>λ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {Poisson} }(x;\lambda )\approx F_{\mathrm {normal} }(x;\mu =\lambda ,\sigma ^{2}=\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2081e807e7da69a44d326395c39a1668a3f7fda3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.44ex; height:3.176ex;" alt="{\displaystyle F_{\mathrm {Poisson} }(x;\lambda )\approx F_{\mathrm {normal} }(x;\mu =\lambda ,\sigma ^{2}=\lambda )}"></li> <li><a href="/wiki/Variance-stabilizing_transformation" title="Variance-stabilizing transformation">Variance-stabilizing transformation</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {Pois} (\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Pois} (\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b515db49fba626cb9a58a3903002bcca7ed9f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.198ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Pois} (\lambda ),}"> then<a href="#cite_note-Johnson2005-8">[8]</a>: 168  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=2{\sqrt {X}}\approx {\mathcal {N}}(2{\sqrt {\lambda }};1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>X</mi> </msqrt> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <mo>;</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=2{\sqrt {X}}\approx {\mathcal {N}}(2{\sqrt {\lambda }};1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c733320f48ae2c177634a84d833578d27bfc4f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.43ex; height:3.176ex;" alt="{\displaystyle Y=2{\sqrt {X}}\approx {\mathcal {N}}(2{\sqrt {\lambda }};1),}"> and<a href="#cite_note-McCullagh1989-35">[35]</a>: 196  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y={\sqrt {X}}\approx {\mathcal {N}}({\sqrt {\lambda }};1/4).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>X</mi> </msqrt> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <mo>;</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y={\sqrt {X}}\approx {\mathcal {N}}({\sqrt {\lambda }};1/4).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd4704720188a745950690ed57b972446012d35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.43ex; height:3.176ex;" alt="{\displaystyle Y={\sqrt {X}}\approx {\mathcal {N}}({\sqrt {\lambda }};1/4).}"> Under this transformation, the convergence to normality (as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> increases) is far faster than the untransformed variable.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Other, slightly more complicated, variance stabilizing transformations are available,<a href="#cite_note-Johnson2005-8">[8]</a>: 168  one of which is <a href="/wiki/Anscombe_transform" title="Anscombe transform">Anscombe transform</a>.<a href="#cite_note-Anscombe1948-36">[36]</a> See <a href="/wiki/Data_transformation_(statistics)" title="Data transformation (statistics)">Data transformation (statistics)</a> for more general uses of transformations.</li> <li>If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean λt, then the sequence of inter-arrival times are independent and identically distributed <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a> random variables having mean 1/λ.<a href="#cite_note-Ross2010-37">[37]</a>: 317–319 </li> <li>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> of the Poisson and <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distributions</a> are related in the following ways:<a href="#cite_note-Johnson2005-8">[8]</a>: 167  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{Poisson}}(k;\lambda )=1-F_{\chi ^{2}}(2\lambda ;2(k+1))\quad \quad {\text{ integer }}k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Poisson</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>λ</mi> <mo>;</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> integer </mtext> </mrow> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{Poisson}}(k;\lambda )=1-F_{\chi ^{2}}(2\lambda ;2(k+1))\quad \quad {\text{ integer }}k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7df4d319ceaf2cac7a2ef625ae5ffd2c5fa9cdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.45ex; height:3.176ex;" alt="{\displaystyle F_{\text{Poisson}}(k;\lambda )=1-F_{\chi ^{2}}(2\lambda ;2(k+1))\quad \quad {\text{ integer }}k,}"> and<a href="#cite_note-Johnson2005-8">[8]</a>: 158  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=k)=F_{\chi ^{2}}(2\lambda ;2(k+1))-F_{\chi ^{2}}(2\lambda ;2k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>λ</mi> <mo>;</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>λ</mi> <mo>;</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=k)=F_{\chi ^{2}}(2\lambda ;2(k+1))-F_{\chi ^{2}}(2\lambda ;2k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afa8a79672290e8714ff5292485ab07267bc94e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.886ex; height:3.176ex;" alt="{\displaystyle P(X=k)=F_{\chi ^{2}}(2\lambda ;2(k+1))-F_{\chi ^{2}}(2\lambda ;2k).}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Poisson_approximation">Poisson approximation</h3></div> Assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\sim \operatorname {Pois} (\lambda _{1}),X_{2}\sim \operatorname {Pois} (\lambda _{2}),\dots ,X_{n}\sim \operatorname {Pois} (\lambda _{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> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mi>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\sim \operatorname {Pois} (\lambda _{1}),X_{2}\sim \operatorname {Pois} (\lambda _{2}),\dots ,X_{n}\sim \operatorname {Pois} (\lambda _{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc5feeea5be9edb9e578edc20b16e8ec84b8721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.353ex; height:2.843ex;" alt="{\displaystyle X_{1}\sim \operatorname {Pois} (\lambda _{1}),X_{2}\sim \operatorname {Pois} (\lambda _{2}),\dots ,X_{n}\sim \operatorname {Pois} (\lambda _{n})}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}+\lambda _{2}+\dots +\lambda _{n}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}+\lambda _{2}+\dots +\lambda _{n}=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44d5d58dee88c3f8ad224d8d0e04a880b4f8b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.545ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}+\lambda _{2}+\dots +\lambda _{n}=1,}"> then<a href="#cite_note-38">[38]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{1},X_{2},\dots ,X_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},X_{2},\dots ,X_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6e1cb7ae1719556a29f9dc023666490c5a29cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.121ex; height:2.843ex;" alt="{\displaystyle (X_{1},X_{2},\dots ,X_{n})}"> is <a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomially distributed</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Mult} (N,\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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>Mult</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Mult} (N,\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46baab90dfd7fd4c25a179f4f4e453b4a5e0d1b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.706ex; height:2.843ex;" alt="{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Mult} (N,\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}"> conditioned on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=X_{1}+X_{2}+\dots X_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=X_{1}+X_{2}+\dots X_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f48f264ff52d5db4884b2b4e8e34814920cf37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.7ex; height:2.509ex;" alt="{\displaystyle N=X_{1}+X_{2}+\dots X_{n}.}"> This means<a href="#cite_note-Mitzenmacher2005-27">[27]</a>: 101-102 , among other things, that for any nonnegative function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},x_{2},\dots ,x_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2},\dots ,x_{n}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da01e9b5b86096828bfbaa50301c203194310223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.263ex; height:2.843ex;" alt="{\displaystyle f(x_{1},x_{2},\dots ,x_{n}),}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y_{1},Y_{2},\dots ,Y_{n})\sim \operatorname {Mult} (m,\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>Mult</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{1},Y_{2},\dots ,Y_{n})\sim \operatorname {Mult} (m,\mathbf {p} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3b622ac92d19286cb8d573e25a819d50e1d1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.843ex; height:2.843ex;" alt="{\displaystyle (Y_{1},Y_{2},\dots ,Y_{n})\sim \operatorname {Mult} (m,\mathbf {p} )}"> is multinomially distributed, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [f(Y_{1},Y_{2},\dots ,Y_{n})]\leq e{\sqrt {m}}\operatorname {E} [f(X_{1},X_{2},\dots ,X_{n})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>≤</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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 stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [f(Y_{1},Y_{2},\dots ,Y_{n})]\leq e{\sqrt {m}}\operatorname {E} [f(X_{1},X_{2},\dots ,X_{n})]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d196f4817af3673334cad96f9aa090d2ae3cb7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.377ex; height:3.009ex;" alt="{\displaystyle \operatorname {E} [f(Y_{1},Y_{2},\dots ,Y_{n})]\leq e{\sqrt {m}}\operatorname {E} [f(X_{1},X_{2},\dots ,X_{n})]}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Pois} (\mathbf {p} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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>Pois</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Pois} (\mathbf {p} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0672e887adf63853e4828e901e15aaf2c7dd9882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.469ex; height:2.843ex;" alt="{\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Pois} (\mathbf {p} ).}"> The factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e{\sqrt {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e{\sqrt {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e2f9d15ce82286860f087bfff8be600abdd707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.06ex; height:3.009ex;" alt="{\displaystyle e{\sqrt {m}}}"> can be replaced by 2 if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is further assumed to be monotonically increasing or decreasing. <div class="mw-heading mw-heading3"><h3 id="Bivariate_Poisson_distribution">Bivariate Poisson distribution</h3></div> This distribution has been extended to the <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">bivariate</a> case.<a href="#cite_note-Loukas1986-39">[39]</a> The <a href="/wiki/Generating_function" title="Generating function">generating function</a> for this distribution is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(u,v)=\exp[(\theta _{1}-\theta _{12})(u-1)+(\theta _{2}-\theta _{12})(v-1)+\theta _{12}(uv-1)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mi>v</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(u,v)=\exp[(\theta _{1}-\theta _{12})(u-1)+(\theta _{2}-\theta _{12})(v-1)+\theta _{12}(uv-1)]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d994b2c4f3b36c80cfd0b97ed72fe289c0855d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.882ex; height:2.843ex;" alt="{\displaystyle g(u,v)=\exp[(\theta _{1}-\theta _{12})(u-1)+(\theta _{2}-\theta _{12})(v-1)+\theta _{12}(uv-1)]}"> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1},\theta _{2}>\theta _{12}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1},\theta _{2}>\theta _{12}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59173fc2edd20c7435e8352e8a027320c586d1f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.649ex; height:2.509ex;" alt="{\displaystyle \theta _{1},\theta _{2}>\theta _{12}>0}"> The marginal distributions are Poisson(θ1) and Poisson(θ2) and the correlation coefficient is limited to the range <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \rho \leq \min \left\{{\sqrt {\frac {\theta _{1}}{\theta _{2}}}},{\sqrt {\frac {\theta _{2}}{\theta _{1}}}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>ρ</mi> <mo>≤</mo> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \rho \leq \min \left\{{\sqrt {\frac {\theta _{1}}{\theta _{2}}}},{\sqrt {\frac {\theta _{2}}{\theta _{1}}}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a96cf8c37ebcf307871a8c94ef332df613bbb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.213ex; height:7.509ex;" alt="{\displaystyle 0\leq \rho \leq \min \left\{{\sqrt {\frac {\theta _{1}}{\theta _{2}}}},{\sqrt {\frac {\theta _{2}}{\theta _{1}}}}\right\}}"> A simple way to generate a bivariate Poisson distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}}"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6099d6fb3c34ad5e22fad9c79c40c4ebfee1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.991ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}}"> is to take three independent Poisson distributions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1},Y_{2},Y_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{1},Y_{2},Y_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db225a2d7020b587c310f70d465d81a5ff666ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.282ex; height:2.509ex;" alt="{\displaystyle Y_{1},Y_{2},Y_{3}}"> with means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b8c04bd1b2d2ea63bf44bf3ef1712baa2e7248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.296ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"> and then set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=Y_{1}+Y_{3},X_{2}=Y_{2}+Y_{3}.}"> <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> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=Y_{1}+Y_{3},X_{2}=Y_{2}+Y_{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2edfaef32686a54ff3d93cc8ddf1cbd1cebde812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.135ex; height:2.509ex;" alt="{\displaystyle X_{1}=Y_{1}+Y_{3},X_{2}=Y_{2}+Y_{3}.}"> The probability function of the bivariate Poisson distribution is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{1}=k_{1},X_{2}=k_{2})=\exp \left(-\lambda _{1}-\lambda _{2}-\lambda _{3}\right){\frac {\lambda _{1}^{k_{1}}}{k_{1}!}}{\frac {\lambda _{2}^{k_{2}}}{k_{2}!}}\sum _{k=0}^{\min(k_{1},k_{2})}{\binom {k_{1}}{k}}{\binom {k_{2}}{k}}k!\left({\frac {\lambda _{3}}{\lambda _{1}\lambda _{2}}}\right)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo 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"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>k</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{1}=k_{1},X_{2}=k_{2})=\exp \left(-\lambda _{1}-\lambda _{2}-\lambda _{3}\right){\frac {\lambda _{1}^{k_{1}}}{k_{1}!}}{\frac {\lambda _{2}^{k_{2}}}{k_{2}!}}\sum _{k=0}^{\min(k_{1},k_{2})}{\binom {k_{1}}{k}}{\binom {k_{2}}{k}}k!\left({\frac {\lambda _{3}}{\lambda _{1}\lambda _{2}}}\right)^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b00b19d41813c00b3f18480f2b375e0c3b96b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:85.447ex; height:7.843ex;" alt="{\displaystyle \Pr(X_{1}=k_{1},X_{2}=k_{2})=\exp \left(-\lambda _{1}-\lambda _{2}-\lambda _{3}\right){\frac {\lambda _{1}^{k_{1}}}{k_{1}!}}{\frac {\lambda _{2}^{k_{2}}}{k_{2}!}}\sum _{k=0}^{\min(k_{1},k_{2})}{\binom {k_{1}}{k}}{\binom {k_{2}}{k}}k!\left({\frac {\lambda _{3}}{\lambda _{1}\lambda _{2}}}\right)^{k}}"> <div class="mw-heading mw-heading3"><h3 id="Free_Poisson_distribution">Free Poisson distribution</h3></div> The free Poisson distribution<a href="#cite_note-40">[40]</a> with jump size <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 }"> and rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> arises in <a href="/wiki/Free_probability" title="Free probability">free probability</a> theory as the limit of repeated <a href="/wiki/Free_convolution" title="Free convolution">free convolution</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>N</mi> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊞</mo> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/347f43443ec8f033d16f84eebd4b79fdfa12a591" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.245ex; height:6.676ex;" alt="{\displaystyle \left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}}"> as N → ∞. In other words, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b8110b9148fb2ec3c86f96e3b72bf2b095ce81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle X_{N}}"> be random variables so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b8110b9148fb2ec3c86f96e3b72bf2b095ce81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle X_{N}}"> has value <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 }"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\lambda }{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\lambda }{N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8c6dd7c0dd3faa1d620596f105d0f77b84dda6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.295ex; height:3.676ex;" alt="{\textstyle {\frac {\lambda }{N}}}"> and value 0 with the remaining probability. Assume also that the family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots }"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869cccabc3ea7b90d40e40158d740fc88e07889a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots }"> are <a href="/wiki/Free_independence" title="Free independence">freely independent</a>. Then the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23159ea0d291e21c5709a6dd7486bed7f18febe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\to \infty }"> of the law of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}+\cdots +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}+\cdots +X_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9197405e63e2962856f275d4e008a079cc153507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.998ex; height:2.509ex;" alt="{\displaystyle X_{1}+\cdots +X_{N}}"> is given by the Free Poisson law with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,\alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>,</mo> <mi>α</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ,\alpha .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/566a4f5b368707891c0532108f72d7a2b077e8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:2.509ex;" alt="{\displaystyle \lambda ,\alpha .}"> This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process. The measure associated to the free Poisson law is given by<a href="#cite_note-41">[41]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ={\begin{cases}(1-\lambda )\delta _{0}+\nu ,&{\text{if }}0\leq \lambda \leq 1\\\nu ,&{\text{if }}\lambda >1,\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ν</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>ν</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\begin{cases}(1-\lambda )\delta _{0}+\nu ,&{\text{if }}0\leq \lambda \leq 1\\\nu ,&{\text{if }}\lambda >1,\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ede5882d0f32fa5d0ddb7080d12312a755132d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.107ex; height:6.176ex;" alt="{\displaystyle \mu ={\begin{cases}(1-\lambda )\delta _{0}+\nu ,&{\text{if }}0\leq \lambda \leq 1\\\nu ,&{\text{if }}\lambda >1,\end{cases}}}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>α</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mi>λ</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6efca6804113d51fd4d767acd0e9b69f4af5955c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.853ex; height:5.176ex;" alt="{\displaystyle \nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}\,dt}"> and has support <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e632a5959c5dd7a328d80a02e2d9134178a173e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.265ex; height:3.176ex;" alt="{\displaystyle [\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}].}"> This law also arises in <a href="/wiki/Random_matrix" title="Random matrix">random matrix</a> theory as the <a href="/wiki/Marchenko%E2%80%93Pastur_law" class="mw-redirect" title="Marchenko–Pastur law">Marchenko–Pastur law</a>. Its <a href="/wiki/Cumulant#Free_cumulants" title="Cumulant">free cumulants</a> are equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{n}=\lambda \alpha ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{n}=\lambda \alpha ^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b422dcc3a689aa0100f254a5cafda73542e2709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.364ex; height:2.676ex;" alt="{\displaystyle \kappa _{n}=\lambda \alpha ^{n}.}"> <div class="mw-heading mw-heading4"><h4 id="Some_transforms_of_this_law">Some transforms of this law</h4></div> We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher<a href="#cite_note-42">[42]</a> The R-transform of the free Poisson law is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(z)={\frac {\lambda \alpha }{1-\alpha z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mi>α</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(z)={\frac {\lambda \alpha }{1-\alpha z}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2538f0ee816527e86beff9fa47634e9db8e68e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.821ex; height:5.509ex;" alt="{\displaystyle R(z)={\frac {\lambda \alpha }{1-\alpha z}}.}"> The Cauchy transform (which is the negative of the <a href="/wiki/Stieltjes_transformation" title="Stieltjes transformation">Stieltjes transformation</a>) is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda ))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>α</mi> <mo>−</mo> <mi>λ</mi> <mi>α</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>λ</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>α</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda ))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eddab052ca3fa8b01262cb9072033c37e8217873" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.269ex; height:6.176ex;" alt="{\displaystyle G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda ))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}}"> The S-transform is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(z)={\frac {1}{z+\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mo>+</mo> <mi>λ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(z)={\frac {1}{z+\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31ba028fa580921b77ba78f25fda1fd553564b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.615ex; height:5.509ex;" alt="{\displaystyle S(z)={\frac {1}{z+\lambda }}}"> in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add6db39940aae24215bb0437f5bb12d63fcbfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.395ex; height:2.176ex;" alt="{\displaystyle \alpha =1.}"> <div class="mw-heading mw-heading3"><h3 id="Weibull_and_stable_count">Weibull and stable count</h3></div> Poisson's probability mass function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814f588da9821bfdc85e401bdd842df42e71599b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.688ex; height:2.843ex;" alt="{\displaystyle f(k;\lambda )}"> can be expressed in a form similar to the product distribution of a <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a> and a variant form of the <a href="/wiki/Stable_count_distribution" title="Stable count distribution">stable count distribution</a>. The variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9f13644a6be482d7ddb19a6e0c706564773085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}"> can be regarded as inverse of Lévy's stability parameter in the stable count distribution: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k;\lambda )=\int _{0}^{\infty }{\frac {1}{u}}\,W_{k+1}\left({\frac {\lambda }{u}}\right)\left[(k+1)u^{k}\,{\mathfrak {N}}_{\frac {1}{k+1}}(u^{k+1})\right]\,du,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>u</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k;\lambda )=\int _{0}^{\infty }{\frac {1}{u}}\,W_{k+1}\left({\frac {\lambda }{u}}\right)\left[(k+1)u^{k}\,{\mathfrak {N}}_{\frac {1}{k+1}}(u^{k+1})\right]\,du,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2646e3d03684519bf4726333b17d41b116396efa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.817ex; height:6.176ex;" alt="{\displaystyle f(k;\lambda )=\int _{0}^{\infty }{\frac {1}{u}}\,W_{k+1}\left({\frac {\lambda }{u}}\right)\left[(k+1)u^{k}\,{\mathfrak {N}}_{\frac {1}{k+1}}(u^{k+1})\right]\,du,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16503339cce78afe1b7b86dcf6d064fb7f34b979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}"> is a standard stable count distribution of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/(k+1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1/(k+1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1c3ebaf0f78bc957548104915203bcb8835b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.581ex; height:2.843ex;" alt="{\displaystyle \alpha =1/(k+1),}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k+1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{k+1}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02ff89392318889f448267367c534b517bba472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.522ex; height:2.843ex;" alt="{\displaystyle W_{k+1}(x)}"> is a standard Weibull distribution of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k+1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5b9a64aeb4f9176797360fce8f523764ac79e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.861ex; height:2.343ex;" alt="{\displaystyle k+1.}"> <div class="mw-heading mw-heading2"><h2 id="Statistical_inference">Statistical inference</h2></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">See also: <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regression</a></div> <div class="mw-heading mw-heading3"><h3 id="Parameter_estimation">Parameter estimation</h3></div> Given a sample of n measured values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}\in \{0,1,\dots \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}\in \{0,1,\dots \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99452d212e85d679221bbd9768c0ac42ea84366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.939ex; height:2.843ex;" alt="{\displaystyle k_{i}\in \{0,1,\dots \},}"> for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimate is<a href="#cite_note-43">[43]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }={\frac {1}{n}}\sum _{i=1}^{n}k_{i}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }={\frac {1}{n}}\sum _{i=1}^{n}k_{i}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2d34749b1c93976303296e6b87b64740d69f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.938ex; height:6.843ex;" alt="{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }={\frac {1}{n}}\sum _{i=1}^{n}k_{i}\ .}"></dd></dl> Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an <a href="/wiki/Unbiased_estimator" class="mw-redirect" title="Unbiased estimator">unbiased estimator</a> of λ. It is also an efficient estimator since its variance achieves the <a href="/wiki/Cram%C3%A9r%E2%80%93Rao_lower_bound" class="mw-redirect" title="Cramér–Rao lower bound">Cramér–Rao lower bound</a> (CRLB).<a href="#cite_note-44">[44]</a> Hence it is <a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum-variance unbiased</a>. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ. To prove sufficiency we may use the <a href="/wiki/Sufficient_statistic" title="Sufficient statistic">factorization theorem</a>. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }">, called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(\mathbf {x} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ade5d2635b53f734929e6ad7f25bfcf1437c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle h(\mathbf {x} )}">, and one that depends on the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> and the sample <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> only through the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {x} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {x} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca30ed81563926023bc46e53f5a624207ba0e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {x} ).}"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {x} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a3259cedd58b92e1140db52ab63ae096baa16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {x} )}"> is a sufficient statistic for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb9c58e3f6b2de892e10ef516f96f07da0423e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\mathbf {x} )=\prod _{i=1}^{n}{\frac {\lambda ^{x_{i}}e^{-\lambda }}{x_{i}!}}={\frac {1}{\prod _{i=1}^{n}x_{i}!}}\times \lambda ^{\sum _{i=1}^{n}x_{i}}e^{-n\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mi>λ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\mathbf {x} )=\prod _{i=1}^{n}{\frac {\lambda ^{x_{i}}e^{-\lambda }}{x_{i}!}}={\frac {1}{\prod _{i=1}^{n}x_{i}!}}\times \lambda ^{\sum _{i=1}^{n}x_{i}}e^{-n\lambda }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/935b182bf975ed1c368e4fc741d515f297ce66ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.138ex; height:6.843ex;" alt="{\displaystyle P(\mathbf {x} )=\prod _{i=1}^{n}{\frac {\lambda ^{x_{i}}e^{-\lambda }}{x_{i}!}}={\frac {1}{\prod _{i=1}^{n}x_{i}!}}\times \lambda ^{\sum _{i=1}^{n}x_{i}}e^{-n\lambda }}"></dd></dl> The first term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(\mathbf {x} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ade5d2635b53f734929e6ad7f25bfcf1437c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle h(\mathbf {x} )}"> depends only on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }">. The second term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(T(\mathbf {x} )|\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(T(\mathbf {x} )|\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e569d1063c1dc4b137dfe9d9f6e6c14a3de12e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.784ex; height:2.843ex;" alt="{\displaystyle g(T(\mathbf {x} )|\lambda )}"> depends on the sample only through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}x_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbbe249485e9a50ad73da79e651548b90b05f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.472ex; height:3.176ex;" alt="{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}x_{i}.}"> Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {x} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a3259cedd58b92e1140db52ab63ae096baa16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {x} )}"> is sufficient. To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ell (\lambda )&=\ln \prod _{i=1}^{n}f(k_{i}\mid \lambda )\\&=\sum _{i=1}^{n}\ln \!\left({\frac {e^{-\lambda }\lambda ^{k_{i}}}{k_{i}!}}\right)\\&=-n\lambda +\left(\sum _{i=1}^{n}k_{i}\right)\ln(\lambda )-\sum _{i=1}^{n}\ln(k_{i}!).\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>ℓ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mi>λ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell (\lambda )&=\ln \prod _{i=1}^{n}f(k_{i}\mid \lambda )\\&=\sum _{i=1}^{n}\ln \!\left({\frac {e^{-\lambda }\lambda ^{k_{i}}}{k_{i}!}}\right)\\&=-n\lambda +\left(\sum _{i=1}^{n}k_{i}\right)\ln(\lambda )-\sum _{i=1}^{n}\ln(k_{i}!).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7dd9f64c691cf995378c5076ef2ee1fc40d824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:43.944ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\ell (\lambda )&=\ln \prod _{i=1}^{n}f(k_{i}\mid \lambda )\\&=\sum _{i=1}^{n}\ln \!\left({\frac {e^{-\lambda }\lambda ^{k_{i}}}{k_{i}!}}\right)\\&=-n\lambda +\left(\sum _{i=1}^{n}k_{i}\right)\ln(\lambda )-\sum _{i=1}^{n}\ln(k_{i}!).\end{aligned}}}"></dd></dl> We take the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> with respect to λ and compare it to zero: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ell (\lambda )=0\iff -n+\left(\sum _{i=1}^{n}k_{i}\right){\frac {1}{\lambda }}=0.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>λ</mi> </mrow> </mfrac> </mrow> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ell (\lambda )=0\iff -n+\left(\sum _{i=1}^{n}k_{i}\right){\frac {1}{\lambda }}=0.\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6330dd6ba39f01a7d9496099abadca2afaa244ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.204ex; width:41.556ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ell (\lambda )=0\iff -n+\left(\sum _{i=1}^{n}k_{i}\right){\frac {1}{\lambda }}=0.\!}"></dd></dl> Solving for λ gives a stationary point. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {\sum _{i=1}^{n}k_{i}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {\sum _{i=1}^{n}k_{i}}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1112ab9fcb4209853c6d052470a90f943a9966e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.042ex; height:5.843ex;" alt="{\displaystyle \lambda ={\frac {\sum _{i=1}^{n}k_{i}}{n}}}"></dd></dl> So λ is the average of the ki values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-\lambda ^{-2}\sum _{i=1}^{n}k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ℓ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-\lambda ^{-2}\sum _{i=1}^{n}k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654ea47fda7c1ad93809d44e976db5c35194d54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.298ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-\lambda ^{-2}\sum _{i=1}^{n}k_{i}}"></dd></dl> Evaluating the second derivative at the stationary point gives: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-{\frac {n^{2}}{\sum _{i=1}^{n}k_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ℓ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-{\frac {n^{2}}{\sum _{i=1}^{n}k_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8b29d81a13cf3d78f6a1bc3cc471b8901b014e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.058ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-{\frac {n^{2}}{\sum _{i=1}^{n}k_{i}}}}"></dd></dl> which is the negative of n times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function. For <a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">completeness</a>, a family of distributions is said to be complete if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(g(T))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(g(T))=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c6007bca4ce4b24cf02815548f1e055ad72f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.407ex; height:2.843ex;" alt="{\displaystyle E(g(T))=0}"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\lambda }(g(T)=0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\lambda }(g(T)=0)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10615fff67b9f789f12132b9a9f484a150f0347d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.576ex; height:2.843ex;" alt="{\displaystyle P_{\lambda }(g(T)=0)=1}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb9c58e3f6b2de892e10ef516f96f07da0423e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}"> If the individual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> are iid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Po} (\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Po} (\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a250ddea76d3c64a0b6620d448d7a2defc7606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.557ex; height:2.843ex;" alt="{\displaystyle \mathrm {Po} (\lambda ),}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}X_{i}\sim \mathrm {Po} (n\lambda ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}X_{i}\sim \mathrm {Po} (n\lambda ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0340a1347b82398d5a884332b50cc040aead69e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.47ex; height:3.176ex;" alt="{\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}X_{i}\sim \mathrm {Po} (n\lambda ).}"> Knowing the distribution we want to investigate, it is easy to see that the statistic is complete. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(g(T))=\sum _{t=0}^{\infty }g(t){\frac {(n\lambda )^{t}e^{-n\lambda }}{t!}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>t</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(g(T))=\sum _{t=0}^{\infty }g(t){\frac {(n\lambda )^{t}e^{-n\lambda }}{t!}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4684b4947377fcbaeda1a4e5d70d3c67ea456ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.773ex; height:7.009ex;" alt="{\displaystyle E(g(T))=\sum _{t=0}^{\infty }g(t){\frac {(n\lambda )^{t}e^{-n\lambda }}{t!}}=0}"></dd></dl> For this equality to hold, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f700860ee7af27797d11ddfad3d185eb7af0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}"> must be 0. This follows from the fact that none of the other terms will be 0 for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> in the sum and for all possible values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb9c58e3f6b2de892e10ef516f96f07da0423e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}"> Hence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(g(T))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(g(T))=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c6007bca4ce4b24cf02815548f1e055ad72f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.407ex; height:2.843ex;" alt="{\displaystyle E(g(T))=0}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\lambda }(g(T)=0)=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\lambda }(g(T)=0)=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2735b990a3da082f7e8fe46a09bfc9386bac4a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.222ex; height:2.843ex;" alt="{\displaystyle P_{\lambda }(g(T)=0)=1,}"> and the statistic has been shown to be complete. <div class="mw-heading mw-heading3"><h3 id="Confidence_interval">Confidence interval</h3></div> The <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distributions</a>. The chi-squared distribution is itself closely related to the <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a>, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level 1 – α is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\chi ^{2}(\alpha /2;2k)\leq \mu \leq {\tfrac {1}{2}}\chi ^{2}(1-\alpha /2;2k+2),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>μ</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\chi ^{2}(\alpha /2;2k)\leq \mu \leq {\tfrac {1}{2}}\chi ^{2}(1-\alpha /2;2k+2),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476c9f37c5d5609256cc62365f918d8ff05f0af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.645ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\chi ^{2}(\alpha /2;2k)\leq \mu \leq {\tfrac {1}{2}}\chi ^{2}(1-\alpha /2;2k+2),}"></dd></dl> or equivalently, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(\alpha /2;k,1)\leq \mu \leq F^{-1}(1-\alpha /2;k+1,1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> <mi>k</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>μ</mi> <mo>≤</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{-1}(\alpha /2;k,1)\leq \mu \leq F^{-1}(1-\alpha /2;k+1,1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d082f70606fe3e7eb468e0747fc4f68e3d4645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.673ex; height:3.176ex;" alt="{\displaystyle F^{-1}(\alpha /2;k,1)\leq \mu \leq F^{-1}(1-\alpha /2;k+1,1),}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{2}(p;n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}(p;n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acf3cab1e0b39e1ab70dae512aba74f84bf9793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:3.176ex;" alt="{\displaystyle \chi ^{2}(p;n)}"> is the <a href="/wiki/Quantile_function" title="Quantile function">quantile function</a> (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(p;n,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{-1}(p;n,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1decb5d0b1ee42a9fb4f495e03a8df1bfe2fc659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.751ex; height:3.176ex;" alt="{\displaystyle F^{-1}(p;n,1)}"> is the quantile function of a <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a> with shape parameter n and scale parameter 1.<a href="#cite_note-Johnson2005-8">[8]</a>: 176-178 <a href="#cite_note-Garwood1936-45">[45]</a> This interval is '<a href="/wiki/Exact_statistics" title="Exact statistics">exact</a>' in the sense that its <a href="/wiki/Coverage_probability" title="Coverage probability">coverage probability</a> is never less than the nominal 1 – α. When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the <a href="/wiki/Wilson%E2%80%93Hilferty_transformation" class="mw-redirect" title="Wilson–Hilferty transformation">Wilson–Hilferty transformation</a>):<a href="#cite_note-Breslow1987-46">[46]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left(1-{\frac {1}{9k}}-{\frac {z_{\alpha /2}}{3{\sqrt {k}}}}\right)^{3}\leq \mu \leq (k+1)\left(1-{\frac {1}{9(k+1)}}+{\frac {z_{\alpha /2}}{3{\sqrt {k+1}}}}\right)^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>9</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≤</mo> <mi>μ</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>9</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left(1-{\frac {1}{9k}}-{\frac {z_{\alpha /2}}{3{\sqrt {k}}}}\right)^{3}\leq \mu \leq (k+1)\left(1-{\frac {1}{9(k+1)}}+{\frac {z_{\alpha /2}}{3{\sqrt {k+1}}}}\right)^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e11987e9faefbe440c6466ae8bb785b24284d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:65.643ex; height:6.843ex;" alt="{\displaystyle k\left(1-{\frac {1}{9k}}-{\frac {z_{\alpha /2}}{3{\sqrt {k}}}}\right)^{3}\leq \mu \leq (k+1)\left(1-{\frac {1}{9(k+1)}}+{\frac {z_{\alpha /2}}{3{\sqrt {k+1}}}}\right)^{3},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{\alpha /2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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_{\alpha /2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d12f94696aa58f31c6fd0149affc0ccd8271442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle z_{\alpha /2}}"> denotes the <a href="/wiki/Standard_normal_deviate" title="Standard normal deviate">standard normal deviate</a> with upper tail area α / 2. For application of these formulae in the same context as above (given a sample of n measured values ki each drawn from a Poisson distribution with mean λ), one would set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\sum _{i=1}^{n}k_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\sum _{i=1}^{n}k_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9280aa512dedfbdc5c9e230b3f391a2cb8a13641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.71ex; height:6.843ex;" alt="{\displaystyle k=\sum _{i=1}^{n}k_{i},}"></dd></dl> calculate an interval for μ = n λ , and then derive the interval for λ. <div class="mw-heading mw-heading3"><h3 id="Bayesian_inference">Bayesian inference</h3></div> In <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for the rate parameter λ of the Poisson distribution is the <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a>.<a href="#cite_note-Fink1976-47">[47]</a> Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \sim \mathrm {Gamma} (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \sim \mathrm {Gamma} (\alpha ,\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6269ce405144ac3c4624d83be0468bfce0a5b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.138ex; height:2.843ex;" alt="{\displaystyle \lambda \sim \mathrm {Gamma} (\alpha ,\beta )}"></dd></dl> denote that λ is distributed according to the gamma <a href="/wiki/Probability_density_function" title="Probability density function">density</a> g parameterized in terms of a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> α and an inverse <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> β: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\lambda \mid \alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\;\lambda ^{\alpha -1}\;e^{-\beta \,\lambda }\qquad {\text{ for }}\lambda >0\,\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>∣</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thickmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mspace width="thinmathspace" /> <mi>λ</mi> </mrow> </msup> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\lambda \mid \alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\;\lambda ^{\alpha -1}\;e^{-\beta \,\lambda }\qquad {\text{ for }}\lambda >0\,\!.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122e1dcf0a3b4dd6b4789bac2c3c53b3e7cf651f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.523ex; height:6.176ex;" alt="{\displaystyle g(\lambda \mid \alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\;\lambda ^{\alpha -1}\;e^{-\beta \,\lambda }\qquad {\text{ for }}\lambda >0\,\!.}"></dd></dl> Then, given the same sample of n measured values ki <a href="#Maximum_likelihood">as before</a>, and a prior of Gamma(α, β), the posterior distribution is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \sim \mathrm {Gamma} \left(\alpha +\sum _{i=1}^{n}k_{i},\beta +n\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>β</mi> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \sim \mathrm {Gamma} \left(\alpha +\sum _{i=1}^{n}k_{i},\beta +n\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8559dea969ec3ef330ffbcf7b6a5122b402500a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.259ex; height:7.509ex;" alt="{\displaystyle \lambda \sim \mathrm {Gamma} \left(\alpha +\sum _{i=1}^{n}k_{i},\beta +n\right).}"></dd></dl> Note that the posterior mean is linear and is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[\lambda \mid k_{1},\ldots ,k_{n}]={\frac {\alpha +\sum _{i=1}^{n}k_{i}}{\beta +n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>λ</mi> <mo>∣</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>β</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\lambda \mid k_{1},\ldots ,k_{n}]={\frac {\alpha +\sum _{i=1}^{n}k_{i}}{\beta +n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dc8f117cfc4b31656f10b620be9392cfe96f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.897ex; height:6.343ex;" alt="{\displaystyle E[\lambda \mid k_{1},\ldots ,k_{n}]={\frac {\alpha +\sum _{i=1}^{n}k_{i}}{\beta +n}}.}"></dd></dl> It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean. Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"> distance than the prior distribution of λ must be close to gamma distribution in <a href="/wiki/L%C3%A9vy_metric" title="Lévy metric">Levy distance</a>.<a href="#cite_note-48">[48]</a> The posterior mean E[λ] approaches the maximum likelihood estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f567cbf61632b01e5a165312085022841b84ceab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.241ex; height:3.176ex;" alt="{\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }}"> in the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \to 0,\beta \to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> <mi>β</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \to 0,\beta \to 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2a49af16aacf6c1547a3cf5d152b4bace20eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.053ex; height:2.509ex;" alt="{\displaystyle \alpha \to 0,\beta \to 0,}"> which follows immediately from the general expression of the mean of the <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a>. The <a href="/wiki/Posterior_predictive_distribution" title="Posterior predictive distribution">posterior predictive distribution</a> for a single additional observation is a <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a>,<a href="#cite_note-Gelman2003-49">[49]</a>: 53  sometimes called a gamma–Poisson distribution. <div class="mw-heading mw-heading3"><h3 id="Simultaneous_estimation_of_multiple_Poisson_means">Simultaneous estimation of multiple Poisson means</h3></div> Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots ,X_{p}}"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots ,X_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada35ba220996e5473a2cdba9cf268c39622a52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.153ex; height:2.843ex;" alt="{\displaystyle X_{1},X_{2},\dots ,X_{p}}"> is a set of independent random variables from a set of <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}"> Poisson distributions, each with a parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04732c5e1c10d90f7b95c9f06e461f2f5d797f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.802ex; height:2.509ex;" alt="{\displaystyle \lambda _{i},}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b674d21e32f8698da555e2c3df44e423e71e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.058ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,p,}"> and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L(\lambda ,{\hat {\lambda }})=\sum _{i=1}^{p}\lambda _{i}^{-1}({\hat {\lambda }}_{i}-\lambda _{i})^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L(\lambda ,{\hat {\lambda }})=\sum _{i=1}^{p}\lambda _{i}^{-1}({\hat {\lambda }}_{i}-\lambda _{i})^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac90ce6d5cc2924dc5ce689f4a435c4d31275ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.325ex; height:3.509ex;" alt="{\textstyle L(\lambda ,{\hat {\lambda }})=\sum _{i=1}^{p}\lambda _{i}^{-1}({\hat {\lambda }}_{i}-\lambda _{i})^{2},}"> when <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> <mo>,</mo> </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/4b18ad57c656bab68c7bd45e2bf4596d09de7c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>1,}"> then, similar as in <a href="/wiki/Stein%27s_example" title="Stein's example">Stein's example</a> for the Normal means, the MLE estimator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\lambda }}_{i}=X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\lambda }}_{i}=X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9002c76d7a2565ccbb29f606e857a7ef4739616e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.977ex; height:3.176ex;" alt="{\displaystyle {\hat {\lambda }}_{i}=X_{i}}"> is <a href="/wiki/Admissible_decision_rule" title="Admissible decision rule">inadmissible</a>. <a href="#cite_note-Clevenson1975-50">[50]</a> In this case, a family of <a href="/wiki/Minimax_estimator" title="Minimax estimator">minimax estimators</a> is given for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<c\leq 2(p-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>c</mi> <mo>≤</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<c\leq 2(p-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708c4212d5b7e46eacc6dfcc02598f33417c9367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.51ex; height:2.843ex;" alt="{\displaystyle 0<c\leq 2(p-1)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\geq (p-2+p^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq (p-2+p^{-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539b782a1c3d2eee117b40d94d8f861d564cf7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.42ex; height:3.176ex;" alt="{\displaystyle b\geq (p-2+p^{-1})}"> as<a href="#cite_note-Berger1985-51">[51]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\lambda }}_{i}=\left(1-{\frac {c}{b+\sum _{i=1}^{p}X_{i}}}\right)X_{i},\qquad i=1,\dots ,p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>b</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\lambda }}_{i}=\left(1-{\frac {c}{b+\sum _{i=1}^{p}X_{i}}}\right)X_{i},\qquad i=1,\dots ,p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2819abefca0978a228d37a2a0ede6457eaf51ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.925ex; height:7.509ex;" alt="{\displaystyle {\hat {\lambda }}_{i}=\left(1-{\frac {c}{b+\sum _{i=1}^{p}X_{i}}}\right)X_{i},\qquad i=1,\dots ,p.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This article needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>. Please help <a href="/wiki/Special:EditPage/Poisson_distribution" title="Special:EditPage/Poisson distribution">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed. Find sources: <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Poisson+distribution%22">"Poisson distribution"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Poisson+distribution%22+-wikipedia&tbs=ar:1">news</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Poisson+distribution%22&tbs=bkt:s&tbm=bks">newspapers</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Poisson+distribution%22+-wikipedia">books</a> · <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Poisson+distribution%22">scholar</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Poisson+distribution%22&acc=on&wc=on">JSTOR</a> (December 2019) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> Some applications of the Poisson distribution to <a href="/wiki/Count_data" title="Count data">count data</a> (number of events):<a href="#cite_note-Rasch1963-52">[52]</a> <ul><li><a href="/wiki/Telecommunication" class="mw-redirect" title="Telecommunication">telecommunication</a>: telephone calls arriving in a system,</li> <li><a href="/wiki/Astronomy" title="Astronomy">astronomy</a>: photons arriving at a telescope,</li> <li><a href="/wiki/Chemistry" title="Chemistry">chemistry</a>: the <a href="/wiki/Molar_mass_distribution" title="Molar mass distribution">molar mass distribution</a> of a <a href="/wiki/Living_polymerization" title="Living polymerization">living polymerization</a>,<a href="#cite_note-Flory1940-53">[53]</a></li> <li><a href="/wiki/Biology" title="Biology">biology</a>: the number of mutations on a strand of <a href="/wiki/DNA" title="DNA">DNA</a> per unit length,</li> <li><a href="/wiki/Management" title="Management">management</a>: customers arriving at a counter or call centre,</li> <li><a href="/wiki/Finance_and_insurance" class="mw-redirect" title="Finance and insurance">finance and insurance</a>: number of losses or claims occurring in a given period of time,</li> <li><a href="/wiki/Earthquake_seismology" class="mw-redirect" title="Earthquake seismology">seismology</a>: asymptotic Poisson model of risk for large earthquakes,<a href="#cite_note-Lomnitz1994-54">[54]</a></li> <li><a href="/wiki/Radioactivity" class="mw-redirect" title="Radioactivity">radioactivity</a>: decays in a given time interval in a radioactive sample,</li> <li><a href="/wiki/Optics" title="Optics">optics</a>: number of photons emitted in a single laser pulse (a major vulnerability of <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> protocols, known as photon number splitting).</li></ul> More examples of counting events that may be modelled as Poisson processes include: <ul><li>soldiers killed by horse-kicks each year in each corps in the <a href="/wiki/Prussia" title="Prussia">Prussian</a> cavalry. This example was used in a book by <a href="/wiki/Ladislaus_Bortkiewicz" title="Ladislaus Bortkiewicz">Ladislaus Bortkiewicz</a> (1868–1931),<a href="#cite_note-vonBortkiewitsch1898-12">[12]</a>: 23-25 </li> <li>yeast cells used when brewing <a href="/wiki/Guinness" title="Guinness">Guinness</a> beer. This example was used by <a href="/wiki/William_Sealy_Gosset" title="William Sealy Gosset">William Sealy Gosset</a> (1876–1937),<a href="#cite_note-Student1907-55">[55]</a><a href="#cite_note-Boland1984-56">[56]</a></li> <li>phone calls arriving at a <a href="/wiki/Call_centre" title="Call centre">call centre</a> within a minute. This example was described by <a href="/wiki/Agner_Krarup_Erlang" title="Agner Krarup Erlang">A.K. Erlang</a> (1878–1929),<a href="#cite_note-Erlang1909-57">[57]</a></li> <li>goals in sports involving two competing teams,<a href="#cite_note-Hornby2014-58">[58]</a></li> <li>deaths per year in a given age group,</li> <li>jumps in a stock price in a given time interval,</li> <li>times a <a href="/wiki/Web_server" title="Web server">web server</a> is accessed per minute (under an assumption of <a href="/wiki/Poisson_process#Homogeneous" class="mw-redirect" title="Poisson process">homogeneity</a>),</li> <li><a href="/wiki/Mutation" title="Mutation">mutations</a> in a given stretch of <a href="/wiki/DNA" title="DNA">DNA</a> after a certain amount of radiation,</li> <li><a href="/wiki/Cells_(biology)" class="mw-redirect" title="Cells (biology)">cells</a> infected at a given <a href="/wiki/Multiplicity_of_infection" title="Multiplicity of infection">multiplicity of infection</a>,</li> <li>bacteria in a certain amount of liquid,<a href="#cite_note-Koyama2016-59">[59]</a></li> <li><a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a> arriving on a pixel circuit at a given illumination over a given time period,</li> <li>landing of <a href="/wiki/V-1_flying_bomb" title="V-1 flying bomb">V-1 flying bombs</a> on London during World War II, investigated by R. D. Clarke in 1946.<a href="#cite_note-Clarke1946-60">[60]</a></li></ul> In <a href="/wiki/Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, <a href="/wiki/Patrick_X._Gallagher" title="Patrick X. Gallagher">Gallagher</a> showed in 1976 that, if a certain version of the unproved <a href="/wiki/Second_Hardy%E2%80%93Littlewood_conjecture" title="Second Hardy–Littlewood conjecture">prime r-tuple conjecture</a> holds,<a href="#cite_note-Hardy1923-61">[61]</a> then the counts of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> in short intervals would obey a Poisson distribution.<a href="#cite_note-Gallagher1976-62">[62]</a> <div class="mw-heading mw-heading3"><h3 id="Law_of_rare_events">Law of rare events</h3></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/Poisson_limit_theorem" title="Poisson limit theorem">Poisson limit theorem</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binomial_versus_poisson.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Binomial_versus_poisson.svg/330px-Binomial_versus_poisson.svg.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Binomial_versus_poisson.svg/495px-Binomial_versus_poisson.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Binomial_versus_poisson.svg/660px-Binomial_versus_poisson.svg.png 2x" data-file-width="360" data-file-height="360" /></a><figcaption>Comparison of the Poisson distribution (black lines) and the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events k. As n gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.</figcaption></figure> The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the binomial one, given only the information of expected number of total events in the whole interval. Let the total number of events in the whole interval be denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb9c58e3f6b2de892e10ef516f96f07da0423e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}"> Divide the whole interval into <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}"> subintervals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1},\dots ,I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1},\dots ,I_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900ddceee4944ca19a2d730f7dc5b38ca0ff3b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.497ex; height:2.509ex;" alt="{\displaystyle I_{1},\dots ,I_{n}}"> of equal size, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>\lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b423624ae8ae4ef73b6146e2b2ad520253a079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle n>\lambda }"> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in each of the n subintervals is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda /n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda /n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05745c37bfaf2b7dbe0e4e2b917a3de82da969a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle \lambda /n.}"> Now we assume that the occurrence of an event in the whole interval can be seen as a sequence of n <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a>, where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-th <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a> corresponds to looking whether an event happens at the subinterval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d896cad5ad71ce7719175ca29d4d109333d4c320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.823ex; height:2.509ex;" alt="{\displaystyle I_{i}}"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda /n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda /n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05745c37bfaf2b7dbe0e4e2b917a3de82da969a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle \lambda /n.}"> The expected number of total events in <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}"> such trials would be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00aebb041f4a569408e310294efcc29e0eded7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.002ex; height:2.509ex;" alt="{\displaystyle \lambda ,}"> the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textrm {B}}(n,\lambda /n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {B}}(n,\lambda /n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071c92f12249a65f6db4ee6171b7594d67618f67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.443ex; height:2.843ex;" alt="{\displaystyle {\textrm {B}}(n,\lambda /n).}"> As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <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}"> goes to infinity. In this case the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> converges to what is known as the Poisson distribution by the <a href="/wiki/Poisson_limit_theorem" title="Poisson limit theorem">Poisson limit theorem</a>. In several of the above examples — such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, that is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\textrm {B}}(n,p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\textrm {B}}(n,p).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268f66779d5b1b445f9be227b268f01cc8810376" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.778ex; height:2.843ex;" alt="{\displaystyle X\sim {\textrm {B}}(n,p).}"> In such cases n is very large and p is very small (and so the expectation n p is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\textrm {Pois}}(np).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pois</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\textrm {Pois}}(np).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ade4f98ec7a470e3712bf22be9029eafea52a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.407ex; height:2.843ex;" alt="{\displaystyle X\sim {\textrm {Pois}}(np).}"> This approximation is sometimes known as the law of rare events,<a href="#cite_note-Cameron1998-63">[63]</a>: 5  since each of the n individual <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli events</a> rarely occurs. The name "law of rare events" may be misleading because the total count of success events in a Poisson process need not be rare if the parameter n p is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour. The variance of the binomial distribution is 1 − p times that of the Poisson distribution, so almost equal when p is very small. The word law is sometimes used as a synonym of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.<a href="#cite_note-vonBortkiewitsch1898-12">[12]</a><a href="#cite_note-Edgeworth1913-64">[64]</a> <div class="mw-heading mw-heading3"><h3 id="Poisson_point_process">Poisson point process</h3></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/Poisson_point_process" title="Poisson point process">Poisson point process</a></div> The Poisson distribution arises as the number of points of a <a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson point process</a> located in some finite region. More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if N(D) denotes the number of points in D, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(N(D)=k)={\frac {(\lambda |D|)^{k}e^{-\lambda |D|}}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(N(D)=k)={\frac {(\lambda |D|)^{k}e^{-\lambda |D|}}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1f8614f8f1be080a93b37789b1c1cf7b3d55c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.542ex; height:6.176ex;" alt="{\displaystyle P(N(D)=k)={\frac {(\lambda |D|)^{k}e^{-\lambda |D|}}{k!}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Poisson_regression_and_negative_binomial_regression">Poisson regression and negative binomial regression</h3></div> <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regression</a> and <a href="/wiki/Negative_binomial" class="mw-redirect" title="Negative binomial">negative binomial</a> regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ... ) of the number of events or occurrences in an interval. <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3></div> The <a href="/wiki/Luria%E2%80%93Delbr%C3%BCck_experiment" title="Luria–Delbrück experiment">Luria–Delbrück experiment</a> tested against the hypothesis of Lamarckian evolution, which should result in a Poisson distribution. Katz and Miledi measured the <a href="/wiki/Membrane_potential" title="Membrane potential">membrane potential</a> with and without the presence of <a href="/wiki/Acetylcholine" title="Acetylcholine">acetylcholine</a> (ACh).<a href="#cite_note-65">[65]</a> When ACh is present, <a href="/wiki/Ion_channel" title="Ion channel">ion channels</a> on the membrane would be open randomly at a small fraction of the time. As there are a large number of ion channels each open for a small fraction of the time, the total number of ion channels open at any moment is Poisson distributed. When ACh is not present, effectively no ion channels are open. The membrane potential is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=N_{\text{open}}V_{\text{ion}}+V_{0}+V_{\text{noise}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>open</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ion</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>noise</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=N_{\text{open}}V_{\text{ion}}+V_{0}+V_{\text{noise}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8b1ef4733f61005ad220b73eb5d24888937a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.394ex; height:2.843ex;" alt="{\displaystyle V=N_{\text{open}}V_{\text{ion}}+V_{0}+V_{\text{noise}}}">. Subtracting the effect of noise, Katz and Miledi found the mean and variance of membrane potential to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8.5\times 10^{-3}\;\mathrm {V} ,(29.2\times 10^{-6}\;\mathrm {V} )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <mn>29.2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8.5\times 10^{-3}\;\mathrm {V} ,(29.2\times 10^{-6}\;\mathrm {V} )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79e5d856fdc389dcd6aed1df4ce84f76d36f41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.776ex; height:3.176ex;" alt="{\displaystyle 8.5\times 10^{-3}\;\mathrm {V} ,(29.2\times 10^{-6}\;\mathrm {V} )^{2}}">, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{ion}}=10^{-7}\;\mathrm {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ion</mtext> </mrow> </msub> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{ion}}=10^{-7}\;\mathrm {V} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90da5a36afc87e92f7fa59bf2267d0fb01e329e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.925ex; height:3.009ex;" alt="{\displaystyle V_{\text{ion}}=10^{-7}\;\mathrm {V} }">. (pp. 94-95 <a href="#cite_note-:0-66">[66]</a>) During each cellular replication event, the number of mutations is roughly Poisson distributed.<a href="#cite_note-67">[67]</a> For example, the HIV virus has 10,000 base pairs, and has a mutation rate of about 1 per 30,000 base pairs, meaning the number of mutations per replication event is distributed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Pois} (1/3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Pois} (1/3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2d7c79e179b1e533463faf0764ab89b3f64dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.605ex; height:2.843ex;" alt="{\displaystyle \mathrm {Pois} (1/3)}">. (p. 64 <a href="#cite_note-:0-66">[66]</a>) <div class="mw-heading mw-heading3"><h3 id="Other_applications_in_science">Other applications in science</h3></div> In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}={\sqrt {\lambda }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>λ</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{k}={\sqrt {\lambda }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912807fd1b959cacf647a52a0438b99e8dbcb2be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.453ex; height:3.009ex;" alt="{\displaystyle \sigma _{k}={\sqrt {\lambda }}.}"> These fluctuations are denoted as Poisson noise or (particularly in electronics) as <a href="/wiki/Shot_noise" title="Shot noise">shot noise</a>. The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an <a href="/wiki/Electric_current" title="Electric current">electric current</a> with its <a href="/wiki/Shot_noise" title="Shot noise">shot noise</a>. If N electrons pass a point in a given time t on the average, the <a href="/wiki/Mean" title="Mean">mean</a> <a href="/wiki/Electric_current" title="Electric current">current</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=eN/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>e</mi> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=eN/t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aecab312802db893e6116bf164aeaee799ca0a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.419ex; height:2.843ex;" alt="{\displaystyle I=eN/t}">; since the current fluctuations should be of the order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{I}=e{\sqrt {N}}/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{I}=e{\sqrt {N}}/t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13f1b2bab7890e46bbe7c0d7f33335a13bf2e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.572ex; height:3.176ex;" alt="{\displaystyle \sigma _{I}=e{\sqrt {N}}/t}"> (i.e., the standard deviation of the <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>), the charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> can be estimated from the ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\sigma _{I}^{2}/I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\sigma _{I}^{2}/I.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335820841eea0aec675990249b61cc6f2502e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.209ex; height:3.176ex;" alt="{\displaystyle t\sigma _{I}^{2}/I.}">[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced <a href="/wiki/Silver" title="Silver">silver</a> grains, not to the individual grains themselves. By <a href="/wiki/Correlation" title="Correlation">correlating</a> the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In <a href="/wiki/Causal_set" class="mw-redirect" title="Causal set">causal set</a> theory the discrete elements of spacetime follow a Poisson distribution in the volume. The Poisson distribution also appears in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, especially <a href="/wiki/Quantum_optics" title="Quantum optics">quantum optics</a>. Namely, for a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> system in a <a href="/wiki/Coherent_state" title="Coherent state">coherent state</a>, the probability of measuring a particular energy level has a Poisson distribution. <div class="mw-heading mw-heading2"><h2 id="Computational_methods">Computational methods</h2></div> The Poisson distribution poses two different tasks for dedicated software libraries: evaluating the distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k;\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k;\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5cf99bf30825eb98616f88bdc970f476290a77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.155ex; height:2.843ex;" alt="{\displaystyle P(k;\lambda )}">, and drawing random numbers according to that distribution. <div class="mw-heading mw-heading3"><h3 id="Evaluating_the_Poisson_distribution">Evaluating the Poisson distribution</h3></div> Computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k;\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k;\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5cf99bf30825eb98616f88bdc970f476290a77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.155ex; height:2.843ex;" alt="{\displaystyle P(k;\lambda )}"> for given <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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> is a trivial task that can be accomplished by using the standard definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k;\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k;\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5cf99bf30825eb98616f88bdc970f476290a77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.155ex; height:2.843ex;" alt="{\displaystyle P(k;\lambda )}"> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λk and k!. The fraction of λk to k! can also produce a rounding error that is very large compared to e−λ, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \!f(k;\lambda )=\exp \left[k\ln \lambda -\lambda -\ln \Gamma (k+1)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>;</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi>λ</mi> <mo>−</mo> <mi>λ</mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <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;\lambda )=\exp \left[k\ln \lambda -\lambda -\ln \Gamma (k+1)\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a8a374612dfba5c3e699450970a5b2330925c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.387ex; width:38.398ex; height:2.843ex;" alt="{\displaystyle \!f(k;\lambda )=\exp \left[k\ln \lambda -\lambda -\ln \Gamma (k+1)\right],}"></dd></dl> which is mathematically equivalent but numerically stable. The natural logarithm of the <a href="/wiki/Gamma_function" title="Gamma function">Gamma function</a> can be obtained using the <code>lgamma</code> function in the <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> standard library (C99 version) or <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>, the <code>gammaln</code> function in <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> or <a href="/wiki/SciPy" title="SciPy">SciPy</a>, or the <code>log_gamma</code> function in <a href="/wiki/Fortran" title="Fortran">Fortran</a> 2008 and later. Some computing languages provide built-in functions to evaluate the Poisson distribution, namely <ul><li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>: function <code>dpois(x, lambda)</code>;</li> <li><a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Excel</a>: function <code>POISSON( x, mean, cumulative)</code>, with a flag to specify the cumulative distribution;</li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>: univariate Poisson distribution as <code>PoissonDistribution[<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">]</code>,<a href="#cite_note-WLPoissonRefPage-68">[68]</a> bivariate Poisson distribution as <code>MultivariatePoissonDistribution[<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{12},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{12},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f26a79fe87c901c3bb88846895b3599960cbb777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.614ex; height:2.509ex;" alt="{\displaystyle \theta _{12},}">{ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1}-\theta _{12},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1}-\theta _{12},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff99713bc6dc41d48bd97c5df8289b6395eef474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.599ex; height:2.509ex;" alt="{\displaystyle \theta _{1}-\theta _{12},}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}-\theta _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}-\theta _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c483cd5cb89c4f51726b45ce6f01b5f3eb256df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.952ex; height:2.509ex;" alt="{\displaystyle \theta _{2}-\theta _{12}}">}]</code>,.<a href="#cite_note-WLMvPoissonRefPage-69">[69]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Random_variate_generation">Random variate generation</h3></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/Non-uniform_random_variate_generation" title="Non-uniform random variate generation">Non-uniform random variate generation</a></div> The less trivial task is to draw integer <a href="/wiki/Random_variate" title="Random variate">random variate</a> from the Poisson distribution with given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb9c58e3f6b2de892e10ef516f96f07da0423e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \lambda .}"> Solutions are provided by: <ul><li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>: function <code>rpois(n, lambda)</code>;</li> <li><a href="/wiki/GNU_Scientific_Library" title="GNU Scientific Library">GNU Scientific Library</a> (GSL): function <a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/doc/html/randist.html#the-poisson-distribution">gsl_ran_poisson</a></li></ul> A simple algorithm to generate random Poisson-distributed numbers (<a href="/wiki/Pseudo-random_number_sampling" class="mw-redirect" title="Pseudo-random number sampling">pseudo-random number sampling</a>) has been given by <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth</a>:<a href="#cite_note-Knuth1997-70">[70]</a>: 137-138  <pre>algorithm poisson random number (Knuth): init: Let L ← e−λ, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in [0,1] and let p ← p × u. while p > L. return k − 1. </pre> The complexity is linear in the returned value k, which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see <a href="#References">§ References</a> below. For large values of λ, the value of L = e−λ may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e−STEP does not underflow: [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <pre>algorithm poisson random number (Junhao, based on Knuth): init: Let λLeft ← λ, k ← 0 and p ← 1. do: k ← k + 1. Generate uniform random number u in (0,1) and let p ← p × u. while p < 1 and λLeft > 0: if λLeft > STEP: p ← p × eSTEP λLeft ← λLeft − STEP else: p ← p × eλLeft λLeft ← 0 while p > 1. return k − 1. </pre> The choice of STEP depends on the threshold of overflow. For double precision floating point format the threshold is near e700, so 500 should be a safe STEP. Other solutions for large values of λ include <a href="/wiki/Rejection_sampling" title="Rejection sampling">rejection sampling</a> and using Gaussian approximation. <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">Inverse transform sampling</a> is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u. <pre>algorithm Poisson generator based upon the inversion by sequential search:<a href="#cite_note-Devroye1986-71">[71]</a>: 505  init: Let x ← 0, p ← e−λ, s ← p. Generate uniform random number u in [0,1]. while u > s do: x ← x + 1. p ← p × λ / x. s ← s + p. return x. </pre> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 15em;"> <ul><li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson distribution</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson distribution</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma distribution</a></li> <li><a href="/wiki/Hermite_distribution" title="Hermite distribution">Hermite distribution</a></li> <li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial distribution</a></li> <li><a href="/wiki/Poisson_clumping" title="Poisson clumping">Poisson clumping</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson point process</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regression</a></li> <li><a href="/wiki/Poisson_sampling" title="Poisson sampling">Poisson sampling</a></li> <li><a href="/wiki/Poisson_wavelet" title="Poisson wavelet">Poisson wavelet</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Robbins_lemma" title="Robbins lemma">Robbins lemma</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam distribution</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie distribution</a></li> <li><a href="/wiki/Zero-inflated_model" title="Zero-inflated model">Zero-inflated model</a></li> <li><a href="/wiki/Zero-truncated_Poisson_distribution" title="Zero-truncated Poisson distribution">Zero-truncated Poisson distribution</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Haight1967-1">^ <a href="#cite_ref-Haight1967_1-0">a</a> <a href="#cite_ref-Haight1967_1-1">b</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 .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .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 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Osborn. pp. 190–219.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Of+the+Laws+of+Chance&rft.btitle=The+Philosophical+Transactions+from+the+Year+MDCC+%28where+Mr.+Lowthorp+Ends%29+to+the+Year+MDCCXX.+Abridg%27d%2C+and+Dispos%27d+Under+General+Heads&rft.place=London%2C+Great+Britain&rft.pages=%3Cspan+class%3D%22nowrap%22%3E190-%3C%2Fspan%3E219&rft.pub=R.+Wilkin%2C+R.+Robinson%2C+S.+Ballard%2C+W.+and+J.+Innys%2C+and+J.+Osborn&rft.date=1721&rft.aulast=de+Moivre&rft.aufirst=Abraham&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Johnson2005-8">^ <a href="#cite_ref-Johnson2005_8-0">a</a> <a href="#cite_ref-Johnson2005_8-1">b</a> <a href="#cite_ref-Johnson2005_8-2">c</a> <a href="#cite_ref-Johnson2005_8-3">d</a> <a href="#cite_ref-Johnson2005_8-4">e</a> <a href="#cite_ref-Johnson2005_8-5">f</a> <a href="#cite_ref-Johnson2005_8-6">g</a> <a href="#cite_ref-Johnson2005_8-7">h</a> <a href="#cite_ref-Johnson2005_8-8">i</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonKempKotz2005" class="citation book cs1">Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel (2005). 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New York, NY, US: John Wiley & Sons, Inc. pp. 156–207. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F0471715816">10.1002/0471715816</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-27246-5" title="Special:BookSources/978-0-471-27246-5"><bdi>978-0-471-27246-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Poisson+Distribution&rft.btitle=Univariate+Discrete+Distributions&rft.place=New+York%2C+NY%2C+US&rft.pages=%3Cspan+class%3D%22nowrap%22%3E156-%3C%2Fspan%3E207&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2005&rft_id=info%3Adoi%2F10.1002%2F0471715816&rft.isbn=978-0-471-27246-5&rft.aulast=Johnson&rft.aufirst=Norman+L.&rft.au=Kemp%2C+Adrienne+W.&rft.au=Kotz%2C+Samuel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Stigler1982-9"><a href="#cite_ref-Stigler1982_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler1982" class="citation journal cs1">Stigler, Stephen M. (1982). "Poisson on the Poisson Distribution". Statistics & Probability Letters. 1 (1): 33–35. <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%2882%2990010-4">10.1016/0167-7152(82)90010-4</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=Poisson+on+the+Poisson+Distribution&rft.volume=1&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E33-%3C%2Fspan%3E35&rft.date=1982&rft_id=info%3Adoi%2F10.1016%2F0167-7152%2882%2990010-4&rft.aulast=Stigler&rft.aufirst=Stephen+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Hald1984-10"><a href="#cite_ref-Hald1984_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaldde_MoivreMcClintock1984" class="citation journal cs1">Hald, Anders; de Moivre, Abraham; McClintock, Bruce (1984). "A. de Moivre: 'De Mensura Sortis' or 'On the Measurement of Chance'". International Statistical Review / Revue Internationale de Statistique. 52 (3): 229–262. <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%2F1403045">10.2307/1403045</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/1403045">1403045</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review+%2F+Revue+Internationale+de+Statistique&rft.atitle=A.+de+Moivre%3A+%27De+Mensura+Sortis%27+or+%27On+the+Measurement+of+Chance%27&rft.volume=52&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E262&rft.date=1984&rft_id=info%3Adoi%2F10.2307%2F1403045&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1403045%23id-name%3DJSTOR&rft.aulast=Hald&rft.aufirst=Anders&rft.au=de+Moivre%2C+Abraham&rft.au=McClintock%2C+Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Newcomb1860-11"><a href="#cite_ref-Newcomb1860_11-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewcomb1860" class="citation journal cs1">Newcomb, Simon (1860). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075590&seq=150">"Notes on the theory of probabilities"</a>. The Mathematical Monthly. 2 (4): 134–140.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Monthly&rft.atitle=Notes+on+the+theory+of+probabilities&rft.volume=2&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E134-%3C%2Fspan%3E140&rft.date=1860&rft.aulast=Newcomb&rft.aufirst=Simon&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dnyp.33433069075590%26seq%3D150&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-vonBortkiewitsch1898-12">^ <a href="#cite_ref-vonBortkiewitsch1898_12-0">a</a> <a href="#cite_ref-vonBortkiewitsch1898_12-1">b</a> <a href="#cite_ref-vonBortkiewitsch1898_12-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Bortkiewitsch1898" class="citation book cs1 cs1-prop-foreign-lang-source">von Bortkiewitsch, Ladislaus (1898). Das Gesetz der kleinen Zahlen [The law of small numbers] (in German). Leipzig, Germany: B.G. Teubner. pp. 1, 23–25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Das+Gesetz+der+kleinen+Zahlen&rft.place=Leipzig%2C+Germany&rft.pages=1%2C+%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E25&rft.pub=B.G.+Teubner&rft.date=1898&rft.aulast=von+Bortkiewitsch&rft.aufirst=Ladislaus&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> <dl><dd>On <a rel="nofollow" class="external text" href="https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/13">page 1</a>, Bortkiewicz presents the Poisson distribution.</dd> <dd>On <a rel="nofollow" class="external text" href="https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/35">pages 23–25</a>, Bortkiewitsch presents his analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten." [4. Example: Those killed in the Prussian army by a horse's kick.]</dd></dl> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> For the proof, see: <a href="https://proofwiki.org/wiki/Expectation_of_Poisson_Distribution" class="extiw" title="proofwiki:Expectation of Poisson Distribution">Proof wiki: expectation</a> and <a href="https://proofwiki.org/wiki/Variance_of_Poisson_Distribution" class="extiw" title="proofwiki:Variance of Poisson Distribution">Proof wiki: variance</a> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKardar2007" class="citation book cs1"><a href="/wiki/Mehran_Kardar" title="Mehran Kardar">Kardar, Mehran</a> (2007). <a href="/wiki/Statistical_Physics_of_Particles" title="Statistical Physics of Particles">Statistical Physics of Particles</a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 42. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87342-0" title="Special:BookSources/978-0-521-87342-0"><bdi>978-0-521-87342-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/860391091">860391091</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Physics+of+Particles&rft.pages=42&rft.pub=Cambridge+University+Press&rft.date=2007&rft_id=info%3Aoclcnum%2F860391091&rft.isbn=978-0-521-87342-0&rft.aulast=Kardar&rft.aufirst=Mehran&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekkingKraaikampLopuhaäMeester2005" class="citation book cs1">Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/1-84628-168-7">A Modern Introduction to Probability and Statistics</a>. Springer Texts in Statistics. p. 167. <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%2F1-84628-168-7">10.1007/1-84628-168-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-896-1" title="Special:BookSources/978-1-85233-896-1"><bdi>978-1-85233-896-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Introduction+to+Probability+and+Statistics&rft.series=Springer+Texts+in+Statistics&rft.pages=167&rft.date=2005&rft_id=info%3Adoi%2F10.1007%2F1-84628-168-7&rft.isbn=978-1-85233-896-1&rft.aulast=Dekking&rft.aufirst=Frederik+Michel&rft.au=Kraaikamp%2C+Cornelis&rft.au=Lopuha%C3%A4%2C+Hendrik+Paul&rft.au=Meester%2C+Ludolf+Erwin&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2F1-84628-168-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Ugarte2016-16"><a href="#cite_ref-Ugarte2016_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUgarteMilitinoArnholt2016" class="citation book cs1"><a href="/wiki/Mar%C3%ADa_Dolores_Ugarte" title="María Dolores Ugarte">Ugarte, M.D.</a>; <a href="/wiki/Ana_Fern%C3%A1ndez_Militino" title="Ana Fernández Militino">Militino, A.F.</a>; Arnholt, A.T. (2016). Probability and Statistics with R (2nd ed.). Boca Raton, FL, US: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-0439-4" title="Special:BookSources/978-1-4665-0439-4"><bdi>978-1-4665-0439-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Statistics+with+R&rft.place=Boca+Raton%2C+FL%2C+US&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2016&rft.isbn=978-1-4665-0439-4&rft.aulast=Ugarte&rft.aufirst=M.D.&rft.au=Militino%2C+A.F.&rft.au=Arnholt%2C+A.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Helske2017-17"><a href="#cite_ref-Helske2017_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelske2017" class="citation journal cs1">Helske, Jouni (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v078.i10">"KFAS: Exponential Family State Space Models in R"</a>. Journal of Statistical Software. 78 (10). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.01907">1612.01907</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v078.i10">10.18637/jss.v078.i10</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:14379617">14379617</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+Statistical+Software&rft.atitle=KFAS%3A+Exponential+Family+State+Space+Models+in+R&rft.volume=78&rft.issue=10&rft.date=2017&rft_id=info%3Aarxiv%2F1612.01907&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14379617%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.18637%2Fjss.v078.i10&rft.aulast=Helske&rft.aufirst=Jouni&rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v078.i10&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Choi1994-18"><a href="#cite_ref-Choi1994_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChoi1994" class="citation journal cs1">Choi, Kwok P. (1994). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2160389">"On the medians of gamma distributions and an equation of Ramanujan"</a>. Proceedings of the American Mathematical Society. 121 (1): 245–251. <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%2F2160389">10.2307/2160389</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/2160389">2160389</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=On+the+medians+of+gamma+distributions+and+an+equation+of+Ramanujan&rft.volume=121&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E245-%3C%2Fspan%3E251&rft.date=1994&rft_id=info%3Adoi%2F10.2307%2F2160389&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2160389%23id-name%3DJSTOR&rft.aulast=Choi&rft.aufirst=Kwok+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F2160389&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Riordan1937-19"><a href="#cite_ref-Riordan1937_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiordan1937" class="citation journal cs1">Riordan, John (1937). <a rel="nofollow" class="external text" href="https://projecteuclid.org/download/pdf_1/euclid.aoms/1177732430">"Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions"</a> (PDF). Annals of Mathematical Statistics. 8 (2): 103–111. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177732430">10.1214/aoms/1177732430</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/2957598">2957598</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Moment+Recurrence+Relations+for+Binomial%2C+Poisson+and+Hypergeometric+Frequency+Distributions&rft.volume=8&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E103-%3C%2Fspan%3E111&rft.date=1937&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177732430&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2957598%23id-name%3DJSTOR&rft.aulast=Riordan&rft.aufirst=John&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.aoms%2F1177732430&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._Ahle2022" class="citation journal cs1">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%3APoisson+distribution" class="Z3988"> </li> <li id="cite_note-Lehmann1986-21"><a href="#cite_ref-Lehmann1986_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehmann1986" class="citation book cs1">Lehmann, Erich Leo (1986). 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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 href="/wiki/Binomial_distribution" title="Binomial distribution">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 class="mw-selflink selflink">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/Hartman%E2%80%93Watson_distribution" title="Hartman–Watson distribution">Hartman–Watson</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="" 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Poisson distribution - Wikipedia