Pareto distribution

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id="toc-Cumulative_distribution_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cumulative_distribution_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Cumulative distribution function</span> </div> </a> <ul id="toc-Cumulative_distribution_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_density_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_density_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Probability density function</span> </div> </a> <ul id="toc-Probability_density_function-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"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Moments_and_characteristic_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moments_and_characteristic_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Moments and characteristic function</span> </div> </a> <ul id="toc-Moments_and_characteristic_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditional_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Conditional distributions</span> </div> </a> <ul id="toc-Conditional_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_characterization_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_characterization_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>A characterization theorem</span> </div> </a> <ul id="toc-A_characterization_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_mean" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_mean"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Geometric mean</span> </div> </a> <ul id="toc-Geometric_mean-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_mean" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_mean"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Harmonic mean</span> </div> </a> <ul id="toc-Harmonic_mean-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graphical_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graphical_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Graphical representation</span> </div> </a> <ul id="toc-Graphical_representation-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"> <span class="vector-toc-numb">3</span> <span>Related distributions</span> </div> </a> <button aria-controls="toc-Related_distributions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related distributions subsection</span> </button> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> <li id="toc-Generalized_Pareto_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_Pareto_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Generalized Pareto distributions</span> </div> </a> <ul id="toc-Generalized_Pareto_distributions-sublist" class="vector-toc-list"> <li id="toc-Pareto_types_I–IV" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pareto_types_I–IV"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Pareto types I–IV</span> </div> </a> <ul id="toc-Pareto_types_I–IV-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feller–Pareto_distribution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Feller–Pareto_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Feller–Pareto distribution</span> </div> </a> <ul id="toc-Feller–Pareto_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inverse-Pareto_Distribution_/_Power_Distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse-Pareto_Distribution_/_Power_Distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Inverse-Pareto Distribution / Power Distribution</span> </div> </a> <ul id="toc-Inverse-Pareto_Distribution_/_Power_Distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_exponential_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_exponential_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Relation to the exponential distribution</span> </div> </a> <ul id="toc-Relation_to_the_exponential_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_log-normal_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_log-normal_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Relation to the log-normal distribution</span> </div> </a> <ul id="toc-Relation_to_the_log-normal_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_generalized_Pareto_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_generalized_Pareto_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Relation to the generalized Pareto distribution</span> </div> </a> <ul id="toc-Relation_to_the_generalized_Pareto_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bounded_Pareto_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bounded_Pareto_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Bounded Pareto distribution</span> </div> </a> <ul id="toc-Bounded_Pareto_distribution-sublist" class="vector-toc-list"> <li id="toc-Generating_bounded_Pareto_random_variables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Generating_bounded_Pareto_random_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.1</span> <span>Generating bounded Pareto random variables</span> </div> </a> <ul id="toc-Generating_bounded_Pareto_random_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symmetric_Pareto_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetric_Pareto_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Symmetric Pareto distribution</span> </div> </a> <ul id="toc-Symmetric_Pareto_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multivariate_Pareto_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multivariate_Pareto_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Multivariate Pareto distribution</span> </div> </a> <ul id="toc-Multivariate_Pareto_distribution-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"> <span class="vector-toc-numb">4</span> <span>Statistical inference</span> </div> </a> <button aria-controls="toc-Statistical_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Statistical inference subsection</span> </button> <ul id="toc-Statistical_inference-sublist" class="vector-toc-list"> <li id="toc-Estimation_of_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation_of_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Estimation of parameters</span> </div> </a> <ul id="toc-Estimation_of_parameters-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"> <span class="vector-toc-numb">5</span> <span>Occurrence and applications</span> </div> </a> <button aria-controls="toc-Occurrence_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Occurrence and applications subsection</span> </button> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> <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"> <span class="vector-toc-numb">5.1</span> <span>General</span> </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Zipf's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_Zipf's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Relation to Zipf's law</span> </div> </a> <ul id="toc-Relation_to_Zipf's_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_"Pareto_principle"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_"Pareto_principle""> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Relation to the "Pareto principle"</span> </div> </a> <ul id="toc-Relation_to_the_"Pareto_principle"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Price's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_Price's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Relation to Price's law</span> </div> </a> <ul id="toc-Relation_to_Price's_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorenz_curve_and_Gini_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorenz_curve_and_Gini_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Lorenz curve and Gini coefficient</span> </div> </a> <ul id="toc-Lorenz_curve_and_Gini_coefficient-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Random_variate_generation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Random_variate_generation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Random variate generation</span> </div> </a> <ul id="toc-Random_variate_generation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </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"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Pareto distribution</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 28 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-28" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">28 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A8%D8%A7%D8%B1%D9%8A%D8%AA%D9%88" title="توزيع باريتو – Arabic" lang="ar" hreflang="ar" data-title="توزيع باريتو" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_de_Pareto" title="Distribución de Pareto – Asturian" lang="ast" hreflang="ast" data-title="Distribución de Pareto" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_de_Pareto" title="Distribució de Pareto – Catalan" lang="ca" hreflang="ca" data-title="Distribució de Pareto" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Paretovo_rozd%C4%9Blen%C3%AD" title="Paretovo rozdělení – Czech" lang="cs" hreflang="cs" data-title="Paretovo rozdělení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pareto-fordeling" title="Pareto-fordeling – Danish" lang="da" hreflang="da" data-title="Pareto-fordeling" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pareto-Verteilung" title="Pareto-Verteilung – German" lang="de" hreflang="de" data-title="Pareto-Verteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Pareto" title="Distribución de Pareto – Spanish" lang="es" hreflang="es" data-title="Distribución de Pareto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Pareto_banaketa" title="Pareto banaketa – Basque" lang="eu" hreflang="eu" data-title="Pareto banaketa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D9%BE%D8%A7%D8%B1%D8%AA%D9%88" title="توزیع پارتو – Persian" lang="fa" hreflang="fa" data-title="توزیع پارتو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_de_Pareto" title="Loi de Pareto – French" lang="fr" hreflang="fr" data-title="Loi de Pareto" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_de_Pareto" title="Distribución de Pareto – Galician" lang="gl" hreflang="gl" data-title="Distribución de Pareto" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8C%8C%EB%A0%88%ED%86%A0_%EB%B6%84%ED%8F%AC" title="파레토 분포 – Korean" lang="ko" hreflang="ko" data-title="파레토 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_paretiana" title="Distribuzione paretiana – Italian" lang="it" hreflang="it" data-title="Distribuzione paretiana" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%A4%D7%90%D7%A8%D7%98%D7%95" title="התפלגות פארטו – Hebrew" lang="he" hreflang="he" data-title="התפלגות פארטו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pareto-eloszl%C3%A1s" title="Pareto-eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Pareto-eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Paretoverdeling" title="Paretoverdeling – Dutch" lang="nl" hreflang="nl" data-title="Paretoverdeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%91%E3%83%AC%E3%83%BC%E3%83%88%E5%88%86%E5%B8%83" title="パレート分布 – Japanese" lang="ja" hreflang="ja" data-title="パレート分布" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_Pareta" title="Rozkład Pareta – Polish" lang="pl" hreflang="pl" data-title="Rozkład Pareta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9F%D0%B0%D1%80%D0%B5%D1%82%D0%BE" title="Распределение Парето – Russian" lang="ru" hreflang="ru" data-title="Распределение Парето" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_Pareto" title="Shpërndarja Pareto – Albanian" lang="sq" hreflang="sq" data-title="Shpërndarja Pareto" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Pareto_distribution" title="Pareto distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Pareto distribution" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Paretova_porazdelitev" title="Paretova porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Paretova porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Paretova_raspodela" title="Paretova raspodela – Serbian" lang="sr" hreflang="sr" data-title="Paretova raspodela" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pareto-jakauma" title="Pareto-jakauma – Finnish" lang="fi" hreflang="fi" data-title="Pareto-jakauma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Pareto_da%C4%9F%C4%B1l%C4%B1m%C4%B1" title="Pareto dağılımı – Turkish" lang="tr" hreflang="tr" data-title="Pareto dağılımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BF%D0%BE%D0%B4%D1%96%D0%BB_%D0%9F%D0%B0%D1%80%D0%B5%D1%82%D0%BE" title="Розподіл Парето – Ukrainian" lang="uk" hreflang="uk" data-title="Розподіл Парето" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue 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searchaux" style="display:none">Probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1247679731">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Pareto Type I</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability density function</div><span typeof="mw:File"><a href="/wiki/File:Probability_density_function_of_Pareto_distribution.svg" class="mw-file-description" title="Pareto Type I probability density functions for various α"><img alt="Pareto Type I probability density functions for various α" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Probability_density_function_of_Pareto_distribution.svg/325px-Probability_density_function_of_Pareto_distribution.svg.png" decoding="async" width="325" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Probability_density_function_of_Pareto_distribution.svg/488px-Probability_density_function_of_Pareto_distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Probability_density_function_of_Pareto_distribution.svg/650px-Probability_density_function_of_Pareto_distribution.svg.png 2x" data-file-width="368" data-file-height="248" /></a></span><br />Pareto Type I probability density functions for various <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <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></span><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 }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6ad0977f3df766a42740e59fad636bdfc0012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.839ex; height:2.509ex;" alt="{\displaystyle x_{\mathrm {m} }=1.}"></span> As <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \rightarrow \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \rightarrow \infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/125a9380c5931a9d42cf8f91ebf14b36b12b07b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.072ex; height:2.176ex;" alt="{\displaystyle \alpha \rightarrow \infty ,}"></span> the distribution approaches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x-x_{\mathrm {m} }),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x-x_{\mathrm {m} }),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe7a36122e176f46ffd096aa9fd6ac7d670681b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.606ex; height:2.843ex;" alt="{\displaystyle \delta (x-x_{\mathrm {m} }),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>.</td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><span typeof="mw:File"><a href="/wiki/File:Cumulative_distribution_function_of_Pareto_distribution.svg" class="mw-file-description" title="Pareto Type I cumulative distribution functions for various α"><img alt="Pareto Type I cumulative distribution functions for various α" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Cumulative_distribution_function_of_Pareto_distribution.svg/325px-Cumulative_distribution_function_of_Pareto_distribution.svg.png" decoding="async" width="325" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Cumulative_distribution_function_of_Pareto_distribution.svg/488px-Cumulative_distribution_function_of_Pareto_distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Cumulative_distribution_function_of_Pareto_distribution.svg/650px-Cumulative_distribution_function_of_Pareto_distribution.svg.png 2x" data-file-width="368" data-file-height="246" /></a></span><br />Pareto Type I cumulative distribution functions for various <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <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></span><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 }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6ad0977f3df766a42740e59fad636bdfc0012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.839ex; height:2.509ex;" alt="{\displaystyle x_{\mathrm {m} }=1.}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1cdd03fec030c005fa686571788130f9c25a9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.192ex; height:2.509ex;" alt="{\displaystyle x_{\mathrm {m} }>0}"></span> <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a> (<a href="/wiki/Real_number" title="Real number">real</a>)<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (real)</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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [x_{\mathrm {m} },\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [x_{\mathrm {m} },\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1353d7fdc761ba14872a4b4225faa0583e3f1c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.01ex; height:2.843ex;" alt="{\displaystyle x\in [x_{\mathrm {m} },\infty )}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9540472e6411a94f070944a6e8cfc7978801ae36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.551ex; height:5.509ex;" alt="{\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f34628cd3a4b8221565660d2b1d4c41cabe8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.829ex; height:4.843ex;" alt="{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Quantile_function" title="Quantile function">Quantile</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0df339f9fe83fd0de4aff256f9eec3b45c98db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.113ex; height:4.009ex;" alt="{\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9948b5592808afb534fe7f331ba307f8ef4355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.006ex; margin-bottom: -0.331ex; width:20.389ex; height:7.843ex;" alt="{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1a9e02a1d60cf9cd611b13188b078509904bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.029ex; height:3.009ex;" alt="{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddce2d6463b7cc6de9d022a861b92e450434fda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.009ex;" alt="{\displaystyle x_{\mathrm {m} }}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>α</mi> <mo>></mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf518cc574a4fc79a8ac942a6a75a9b19ab9a778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.745ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> <mi>α</mi> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>α</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539b213d7dbe7ca0c2b864a03c02a25103556001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.644ex; height:6.176ex;" alt="{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>α</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>α</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02731e4b801e81407754b790466d88c32b7c96d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.921ex; height:6.676ex;" alt="{\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2581cbc359c3d0775610440ab0afeb1ef2f3819c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.014ex; height:6.176ex;" alt="{\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}"></span></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"> does not exist</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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>α</mi> <mo>,</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a900a005ddef054f1c72031b63109a286a1c2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.935ex; height:2.843ex;" alt="{\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e9daf9821878294e8d09162cad9f3623b5edfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.264ex; margin-left: -0.069ex; margin-bottom: -0.241ex; width:25.241ex; height:12.176ex;" alt="{\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_shortfall" title="Expected shortfall">Expected shortfall</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad93d3c3ad13850d92181ac088ff257239263e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.04ex; height:6.343ex;" alt="{\displaystyle {\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}}"></span><sup id="cite_ref-norton_1-0" class="reference"><a href="#cite_note-norton-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr></tbody></table> <p>The <b>Pareto distribution</b>, named after the Italian <a href="/wiki/Civil_engineer" title="Civil engineer">civil engineer</a>, <a href="/wiki/Economist" title="Economist">economist</a>, and <a href="/wiki/Sociologist" class="mw-redirect" title="Sociologist">sociologist</a> <a href="/wiki/Vilfredo_Pareto" title="Vilfredo Pareto">Vilfredo Pareto</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Power-law" class="mw-redirect" title="Power-law">power-law</a> <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> that is used in description of <a href="/wiki/Social" title="Social">social</a>, <a href="/wiki/Quality_control" title="Quality control">quality control</a>, <a href="/wiki/Scientific" class="mw-redirect" title="Scientific">scientific</a>, <a href="/wiki/Geophysical" class="mw-redirect" title="Geophysical">geophysical</a>, <a href="/wiki/Actuarial_science" title="Actuarial science">actuarial</a>, and many other types of observable phenomena; the principle originally applied to describing the <a href="/wiki/Distribution_of_wealth" title="Distribution of wealth">distribution of wealth</a> in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_4-0" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Pareto_principle" title="Pareto principle">Pareto principle</a> or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (<span class="texhtml"><i>α</i></span>) of log<sub>4</sub>5 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and human activities.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>X</i> is a <a href="/wiki/Random_variable" title="Random variable">random variable</a> with a Pareto (Type I) distribution,<sup id="cite_ref-arnold_8-0" class="reference"><a href="#cite_note-arnold-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> then the probability that <i>X</i> is greater than some number <i>x</i>, i.e., the <a href="/wiki/Survival_function" title="Survival function">survival function</a> (also called tail function), is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>x</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65040da59877a56f4968a63d15ab16f65f807d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.353ex; height:6.176ex;" alt="{\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}}"></span></dd></dl> <p>where <i>x</i><sub>m</sub> is the (necessarily positive) minimum possible value of <i>X</i>, and <i>α</i> is a positive parameter. The type I Pareto distribution is characterized by a <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> <i>x</i><sub>m</sub> and a <a href="/wiki/Shape_parameter" title="Shape parameter">shape parameter</a> <i>α</i>, which is known as the <i>tail index</i>. If this distribution is used to model the distribution of wealth, then the parameter <i>α</i> is called the <a href="/wiki/Pareto_index" title="Pareto index">Pareto index</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Cumulative_distribution_function">Cumulative distribution function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=2" title="Edit section: Cumulative distribution function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the definition, the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a Pareto random variable with parameters <i>α</i> and <i>x</i><sub>m</sub> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc54bf848bcd80222c2f04609175b027f7b5fbf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.656ex; height:6.176ex;" alt="{\displaystyle F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Probability_density_function">Probability density function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=3" title="Edit section: Probability density function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It follows (by <a href="/wiki/Derivative" title="Derivative">differentiation</a>) that the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae9303fffd60858ddcc5e697dec847dc49849ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.462ex; height:7.509ex;" alt="{\displaystyle f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}"></span></dd></dl> <p>When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes <a href="/wiki/Asymptotically" class="mw-redirect" title="Asymptotically">asymptotically</a>. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a <a href="/wiki/Log%E2%80%93log_plot" title="Log–log plot">log–log plot</a>, the distribution is represented by a straight line. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Moments_and_characteristic_function">Moments and characteristic function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=5" title="Edit section: Moments and characteristic function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> following a Pareto distribution is</li></ul> <dl><dd> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mi>α</mi> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>α</mi> <mo>></mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231bd7e111557666fc015df16ff19e4686073368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.787ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} (X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}}"></span></dd></dl></dd></dl> <ul><li>The <a href="/wiki/Variance" title="Variance">variance</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> following a Pareto distribution is</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>α</mi> <mo>></mo> <mn>2.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda6ae1a69ab2c130545abd2053226a4d6510558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:37.344ex; height:8.176ex;" alt="{\displaystyle \operatorname {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}}"></span></dd></dl></dd></dl> <dl><dd>(If <i>α</i> ≤ 1, the variance does not exist.)</dd></dl> <ul><li>The raw <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> are</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}'={\begin{cases}\infty &\alpha \leq n,\\{\frac {\alpha x_{\mathrm {m} }^{n}}{\alpha -n}}&\alpha >n.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi mathvariant="normal">∞</mi> </mtd> <mtd> <mi>α</mi> <mo>≤</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>α</mi> <mo>></mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}'={\begin{cases}\infty &\alpha \leq n,\\{\frac {\alpha x_{\mathrm {m} }^{n}}{\alpha -n}}&\alpha >n.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950b5c45974152256f828be1bd1911d187e50fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.446ex; height:7.509ex;" alt="{\displaystyle \mu _{n}'={\begin{cases}\infty &\alpha \leq n,\\{\frac {\alpha x_{\mathrm {m} }^{n}}{\alpha -n}}&\alpha >n.\end{cases}}}"></span></dd></dl></dd></dl> <ul><li>The <a href="/wiki/Moment-generating_function" title="Moment-generating function">moment generating function</a> is only defined for non-positive values <i>t</i> ≤ 0 as</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\left(t;\alpha ,x_{\mathrm {m} }\right)=\operatorname {E} \left[e^{tX}\right]=\alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>α</mi> <mo>,</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\left(t;\alpha ,x_{\mathrm {m} }\right)=\operatorname {E} \left[e^{tX}\right]=\alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b03963721b9c85e5030aa7a26056af4ef07a4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.323ex; height:3.343ex;" alt="{\displaystyle M\left(t;\alpha ,x_{\mathrm {m} }\right)=\operatorname {E} \left[e^{tX}\right]=\alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\left(0,\alpha ,x_{\mathrm {m} }\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\left(0,\alpha ,x_{\mathrm {m} }\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086d18efe758d63b9c0d3e6a138303549b399a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.195ex; height:2.843ex;" alt="{\displaystyle M\left(0,\alpha ,x_{\mathrm {m} }\right)=1.}"></span></dd></dl></dd></dl> <p>Thus, since the expectation does not converge on an <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span> we say that the moment generating function does not exist. </p> <ul><li>The <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> is given by</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t;\alpha ,x_{\mathrm {m} })=\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>α</mi> <mo>,</mo> <mo>−</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t;\alpha ,x_{\mathrm {m} })=\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c5e5d13be57c7355aaebd480bf62c5f3fd9763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.336ex; height:2.843ex;" alt="{\displaystyle \varphi (t;\alpha ,x_{\mathrm {m} })=\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t),}"></span></dd></dl></dd></dl> <dl><dd>where Γ(<i>a</i>, <i>x</i>) is the <a href="/wiki/Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>.</dd></dl> <p>The parameters may be solved for using the <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Conditional_distributions">Conditional distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=6" title="Edit section: Conditional distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Conditional_probability_distribution" title="Conditional probability distribution">conditional probability distribution</a> of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> exceeding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46d897493a7617e16afb4749af96db61c9e6ce23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.009ex;" alt="{\displaystyle x_{\text{m}}}"></span>, is a Pareto distribution with the same Pareto index <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <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></span><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 }"></span> but with minimum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46d897493a7617e16afb4749af96db61c9e6ce23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.009ex;" alt="{\displaystyle x_{\text{m}}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Pr}}(X\geq x|X\geq x_{1})={\begin{cases}\left({\frac {x_{1}}{x}}\right)^{\alpha }&x\geq x_{1},\\1&x<x_{1}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>x</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Pr}}(X\geq x|X\geq x_{1})={\begin{cases}\left({\frac {x_{1}}{x}}\right)^{\alpha }&x\geq x_{1},\\1&x<x_{1}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767cf654d1f52218b506146df2765ebb2f4d5792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.218ex; height:6.176ex;" alt="{\displaystyle {\text{Pr}}(X\geq x|X\geq x_{1})={\begin{cases}\left({\frac {x_{1}}{x}}\right)^{\alpha }&x\geq x_{1},\\1&x<x_{1}.\end{cases}}}"></span></dd></dl> <p>This implies that the conditional expected value (if it is finite, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}"></span>) is proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{E}}(X|X\geq x_{1})\propto x_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>≥</mo> <msub> <mi>x</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>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{E}}(X|X\geq x_{1})\propto x_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deab24ffeec9728f19f4b83192c726bdedb87dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.611ex; height:2.843ex;" alt="{\displaystyle {\text{E}}(X|X\geq x_{1})\propto x_{1}.}"></span></dd></dl> <p>In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the <a href="/wiki/Lindy_effect" title="Lindy effect">Lindy effect</a> or Lindy's Law.<sup id="cite_ref-:02_10-0" class="reference"><a href="#cite_note-:02-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="A_characterization_theorem">A characterization theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=7" title="Edit section: A characterization theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},\dotsc }"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},\dotsc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48c0e8e6476ca91c7ba5a2d637d9567024c5aa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.761ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},\dotsc }"></span> are <a href="/wiki/Independent_identically_distributed" class="mw-redirect" title="Independent identically distributed">independent identically distributed</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a> whose probability distribution is supported on the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{\text{m}},\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{\text{m}},\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4317eca53c65def10efc4051784a7396a81be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.84ex; height:2.843ex;" alt="{\displaystyle [x_{\text{m}},\infty )}"></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{m}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{m}}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4d17f55b7a3113b04a88d752f00ca2b88cd38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.192ex; height:2.509ex;" alt="{\displaystyle x_{\text{m}}>0}"></span>. Suppose that for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>, the two random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min\{X_{1},\dotsc ,X_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min\{X_{1},\dotsc ,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686d0ebdbb5ffa3b85d5219a3e2ffe43bb32889b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.5ex; height:2.843ex;" alt="{\displaystyle \min\{X_{1},\dotsc ,X_{n}\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{1}+\dotsb +X_{n})/\min\{X_{1},\dotsc ,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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1}+\dotsb +X_{n})/\min\{X_{1},\dotsc ,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3eceb99d05af60457eda52cd1b22e2b28d1f440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.384ex; height:2.843ex;" alt="{\displaystyle (X_{1}+\dotsb +X_{n})/\min\{X_{1},\dotsc ,X_{n}\}}"></span> are independent. Then the common distribution is a Pareto distribution.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2012)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_mean">Geometric mean</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=8" title="Edit section: Geometric mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> (<i>G</i>) is<sup id="cite_ref-Johnson1994_11-0" class="reference"><a href="#cite_note-Johnson1994-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e12d30fda3fbc7f61f3ff6a67c683c7716a62c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.187ex; height:6.176ex;" alt="{\displaystyle G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Harmonic_mean">Harmonic mean</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=9" title="Edit section: Harmonic mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> (<i>H</i>) is<sup id="cite_ref-Johnson1994_11-1" class="reference"><a href="#cite_note-Johnson1994-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada4cb3f40bcd25b5b20cb631ef803ca3820b9ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.262ex; height:6.176ex;" alt="{\displaystyle H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Graphical_representation">Graphical representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=10" title="Edit section: Graphical representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The characteristic curved '<a href="/wiki/Long_tail" title="Long tail">long tail</a>' distribution, when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a <a href="/wiki/Log-log_graph" class="mw-redirect" title="Log-log graph">log-log graph</a>, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for <i>x</i> ≥ <i>x</i><sub>m</sub>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be20919e1495ace4bc5ef99562cc9440c5aff165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.959ex; height:6.176ex;" alt="{\displaystyle \log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.}"></span></dd></dl> <p>Since <i>α</i> is positive, the gradient −(<i>α</i> + 1) is negative. </p> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=11" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Generalized_Pareto_distributions">Generalized Pareto distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=12" title="Edit section: Generalized Pareto distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto distribution</a></div> <p>There is a hierarchy <sup id="cite_ref-arnold_8-1" class="reference"><a href="#cite_note-arnold-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jkb94_12-0" class="reference"><a href="#cite_note-jkb94-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.<sup id="cite_ref-arnold_8-2" class="reference"><a href="#cite_note-arnold-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jkb94_12-1" class="reference"><a href="#cite_note-jkb94-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kk03_13-0" class="reference"><a href="#cite_note-kk03-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto<sup id="cite_ref-jkb94_12-2" class="reference"><a href="#cite_note-jkb94-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feller_14-0" class="reference"><a href="#cite_note-feller-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> distribution generalizes Pareto Type IV. </p> <div class="mw-heading mw-heading4"><h4 id="Pareto_types_I–IV"><span id="Pareto_types_I.E2.80.93IV"></span>Pareto types I–IV</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=13" title="Edit section: Pareto types I–IV"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pareto distribution hierarchy is summarized in the next table comparing the <a href="/wiki/Survival_function" title="Survival function">survival functions</a> (complementary CDF). </p><p>When <i>μ</i> = 0, the Pareto distribution Type II is also known as the <a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax distribution</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In this section, the symbol <i>x</i><sub>m</sub>, used before to indicate the minimum value of <i>x</i>, is replaced by <i>σ</i>. </p> <table class="wikitable"> <caption>Pareto distributions </caption> <tbody><tr> <th></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}(x)=1-F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {F}}(x)=1-F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1941dc5171830203a01848eab8b8fba7fe36503c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.136ex; height:3.509ex;" alt="{\displaystyle {\overline {F}}(x)=1-F(x)}"></span></th> <th>Support</th> <th>Parameters </th></tr> <tr> <td>Type I </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {x}{\sigma }}\right]^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {x}{\sigma }}\right]^{-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/debc11c1d4259755203a2e95e5171e4b2c28b695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.923ex; height:5.009ex;" alt="{\displaystyle \left[{\frac {x}{\sigma }}\right]^{-\alpha }}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d398298ad0c7e50e54ea431fd12e79f95af3dcca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.758ex; height:2.176ex;" alt="{\displaystyle x\geq \sigma }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0,\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0,\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754f7860537646aecb5a1dd34fd99dfb792c98ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.112ex; height:2.509ex;" alt="{\displaystyle \sigma >0,\alpha }"></span> </td></tr> <tr> <td>Type II </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c05d4c866664355381925ebc7f1d6854a8b4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.428ex; height:6.509ex;" alt="{\displaystyle \left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f293c50cd836cfc8ad1e95a44c12538c61f0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.509ex;" alt="{\displaystyle x\geq \mu }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {R} ,\sigma >0,\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {R} ,\sigma >0,\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d133c8542cb10d35880958772047afe9e286c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.067ex; height:2.676ex;" alt="{\displaystyle \mu \in \mathbb {R} ,\sigma >0,\alpha }"></span> </td></tr> <tr> <td>Lomax </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1+{\frac {x}{\sigma }}\right]^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>σ</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1+{\frac {x}{\sigma }}\right]^{-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f6d8660cc815594ad3f6fbbba08e57eaa4bf12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.926ex; height:5.009ex;" alt="{\displaystyle \left[1+{\frac {x}{\sigma }}\right]^{-\alpha }}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2608e2b392b079f5b763f27bf52883dbee3b64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0,\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0,\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754f7860537646aecb5a1dd34fd99dfb792c98ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.112ex; height:2.509ex;" alt="{\displaystyle \sigma >0,\alpha }"></span> </td></tr> <tr> <td>Type III </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d45a24039951a4a164feb7f48ee05c3b852a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.644ex; height:8.009ex;" alt="{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f293c50cd836cfc8ad1e95a44c12538c61f0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.509ex;" alt="{\displaystyle x\geq \mu }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dbf5d82227cde44d7f9244c41e02971516e1be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.841ex; height:2.676ex;" alt="{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0}"></span> </td></tr> <tr> <td>Type IV </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95750fc2c1674af87b4f4d3115af6dbf9728743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.874ex; height:7.843ex;" alt="{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f293c50cd836cfc8ad1e95a44c12538c61f0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.509ex;" alt="{\displaystyle x\geq \mu }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0,\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0,\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e6c204db2658745897265a373f0dac44364442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.363ex; height:2.676ex;" alt="{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0,\alpha }"></span> </td></tr> </tbody></table> <p>The shape parameter <i>α</i> is the <a href="/w/index.php?title=Tail_index&action=edit&redlink=1" class="new" title="Tail index (page does not exist)">tail index</a>, <i>μ</i> is location, <i>σ</i> is scale, <i>γ</i> is an inequality parameter. Some special cases of Pareto Type (IV) are </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6f8e8c88565426048eecb1d4833c2985591c01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.867ex; height:2.843ex;" alt="{\displaystyle P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf04f1cc9787e5cecbfda3de3c94ad6fad7217e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.546ex; height:2.843ex;" alt="{\displaystyle P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>I</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a6f01233a01e6c58c63f6631bcaed18c59955f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.267ex; height:2.843ex;" alt="{\displaystyle P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).}"></span></dd></dl></dd></dl> <p>The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index <i>α</i> (inequality index <i>γ</i>). In particular, fractional <i>δ</i>-moments are finite for some <i>δ</i> > 0, as shown in the table below, where <i>δ</i> is not necessarily an integer. </p> <table class="wikitable"> <caption>Moments of Pareto I–IV distributions (case <i>μ</i> = 0) </caption> <tbody><tr> <th></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd294aa33c0865f58e2b1bdaf44ebe911dbf93" 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 \operatorname {E} [X]}"></span></th> <th>Condition</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{\delta }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{\delta }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab8f72a2621c18717c6afbb3a3772ca30a36b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.847ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [X^{\delta }]}"></span></th> <th>Condition </th></tr> <tr> <td>Type I </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma \alpha }{\alpha -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi>α</mi> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma \alpha }{\alpha -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d32c41c1d0acafe91654d1ac0de3d30e0fd9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.327ex; height:4.843ex;" alt="{\displaystyle {\frac {\sigma \alpha }{\alpha -1}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mi>α</mi> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192cdddb9d6a1f4708602a176d88ad0c0ed1a084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:6.213ex; height:6.009ex;" alt="{\displaystyle {\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta <\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo><</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta <\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff464b659e69361101e12a17e33adfebaaef928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.635ex; height:2.343ex;" alt="{\displaystyle \delta <\alpha }"></span> </td></tr> <tr> <td>Type II </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma }{\alpha -1}}+\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma }{\alpha -1}}+\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a789af488c5b59422e7ba14153c07fbbf13464ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.569ex; height:4.843ex;" alt="{\displaystyle {\frac {\sigma }{\alpha -1}}+\mu }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1036e13ad1abd3ee5c8a7d62fa8f763dbafd68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.093ex; height:6.676ex;" alt="{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\delta <\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\delta <\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349c54d9b5ad1fb5b258a69b29eca6bcab3c94e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.896ex; height:2.343ex;" alt="{\displaystyle 0<\delta <\alpha }"></span> </td></tr> <tr> <td>Type III </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \Gamma (1-\gamma )\Gamma (1+\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \Gamma (1-\gamma )\Gamma (1+\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce8fb0ab6c11b2c96fc12a7771276222d8a3bb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.384ex; height:2.843ex;" alt="{\displaystyle \sigma \Gamma (1-\gamma )\Gamma (1+\gamma )}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1<\gamma <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<\gamma <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae7089418e9361dc307c11afcaf870a6b4dc123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.592ex; height:2.676ex;" alt="{\displaystyle -1<\gamma <1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>γ</mi> <mi>δ</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>γ</mi> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a98539534219af6acc798fd7b16173a62ab7cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.456ex; height:3.176ex;" alt="{\displaystyle \sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\gamma ^{-1}<\delta <\gamma ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mi>δ</mi> <mo><</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\gamma ^{-1}<\delta <\gamma ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2f31720059235e37557ac2b36a692ed5b4926a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.279ex; height:3.176ex;" alt="{\displaystyle -\gamma ^{-1}<\delta <\gamma ^{-1}}"></span> </td></tr> <tr> <td>Type IV </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83da7cc0d107d5c2632cc25317bc0376f319c1fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.545ex; height:6.509ex;" alt="{\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1<\gamma <\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<\gamma <\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ed2c3350bb317a56a945d0735ac73a35e68b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.917ex; height:2.676ex;" alt="{\displaystyle -1<\gamma <\alpha }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \delta )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>γ</mi> <mi>δ</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>γ</mi> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \delta )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d340350a6baaba39c7b354a2799a15ce2226669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.617ex; height:6.676ex;" alt="{\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \delta )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mi>δ</mi> <mo><</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8c76025e050d97abca370bfb945210ab2f9521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.579ex; height:3.176ex;" alt="{\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }"></span> </td></tr> </tbody></table> <div class="mw-heading mw-heading4"><h4 id="Feller–Pareto_distribution"><span id="Feller.E2.80.93Pareto_distribution"></span>Feller–Pareto distribution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=14" title="Edit section: Feller–Pareto distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Feller<sup id="cite_ref-jkb94_12-3" class="reference"><a href="#cite_note-jkb94-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-feller_14-1" class="reference"><a href="#cite_note-feller-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> defines a Pareto variable by transformation <i>U</i> = <i>Y</i><sup>−1</sup> − 1 of a <a href="/wiki/Beta_distribution" title="Beta distribution">beta random variable</a> ,<i>Y</i>, whose probability density function is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mn>1</mn> <mo>;</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d70f10dfc751c4eee45d93dc34a2f1213a8fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.187ex; height:6.676ex;" alt="{\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,}"></span></dd></dl> <p>where <i>B</i>( ) is the <a href="/wiki/Beta_function" title="Beta function">beta function</a>. If </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69852671d4ccae45fdaf5dc49789e66493e400be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.749ex; height:3.176ex;" alt="{\displaystyle W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,}"></span></dd></dl> <p>then <i>W</i> has a Feller–Pareto distribution FP(<i>μ</i>, <i>σ</i>, <i>γ</i>, <i>γ</i><sub>1</sub>, <i>γ</i><sub>2</sub>).<sup id="cite_ref-arnold_8-3" class="reference"><a href="#cite_note-arnold-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}\sim \Gamma (\delta _{1},1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}\sim \Gamma (\delta _{1},1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/172691f90639bceb6c262e4fcc0d000dac2fc90e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.285ex; height:2.843ex;" alt="{\displaystyle U_{1}\sim \Gamma (\delta _{1},1)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}\sim \Gamma (\delta _{2},1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}\sim \Gamma (\delta _{2},1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da46054d777edf76dd296d45a8c0c0a8ed4e8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.285ex; height:2.843ex;" alt="{\displaystyle U_{2}\sim \Gamma (\delta _{2},1)}"></span> are independent <a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma variables</a>, another construction of a Feller–Pareto (FP) variable is<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b8c0164d5c0a96f9d38d79ed9ded2d7327bb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.129ex; height:6.176ex;" alt="{\displaystyle W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }}"></span></dd></dl> <p>and we write <i>W</i> ~ FP(<i>μ</i>, <i>σ</i>, <i>γ</i>, <i>δ</i><sub>1</sub>, <i>δ</i><sub>2</sub>). Special cases of the Feller–Pareto distribution are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39a8516aed87400bcdb8c4644efc57f931239fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.389ex; height:2.843ex;" alt="{\displaystyle FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1937b4858c22a1e262f0126423439ebfeeee9a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.068ex; height:2.843ex;" alt="{\displaystyle FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>I</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948db77463b7875e4f0aa5fcd5a65a3096025eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.789ex; height:2.843ex;" alt="{\displaystyle FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a7ceb9854dcffa48a10f4e4ee5fd2eac1af9cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.726ex; height:2.843ex;" alt="{\displaystyle FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inverse-Pareto_Distribution_/_Power_Distribution"><span id="Inverse-Pareto_Distribution_.2F_Power_Distribution"></span>Inverse-Pareto Distribution / Power Distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=15" title="Edit section: Inverse-Pareto Distribution / Power Distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> follows a pareto distribution, then its inverse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=1/Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=1/Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d773a191ecdfc20eb046188961951321111f20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.177ex; height:2.843ex;" alt="{\displaystyle X=1/Y}"></span> follows an <a href="/w/index.php?title=Power_distribution_(Statistics)&action=edit&redlink=1" class="new" title="Power distribution (Statistics) (page does not exist)">Inverse Pareto distribution</a>. <a href="/w/index.php?title=Power_distribution_(Statistics)&action=edit&redlink=1" class="new" title="Power distribution (Statistics) (page does not exist)">Inverse Pareto distribution</a> is equivalent to a <a href="/w/index.php?title=Power_distribution_(Statistics)&action=edit&redlink=1" class="new" title="Power distribution (Statistics) (page does not exist)">Power distribution</a><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \mathrm {Pa} (\alpha ,x_{m})={\frac {\alpha x_{m}^{\alpha }}{y^{\alpha +1}}}\quad (y\geq x_{m})\quad \Leftrightarrow \quad X\sim \mathrm {iPa} (\alpha ,x_{m})=\mathrm {Power} (x_{m}^{-1},\alpha )={\frac {\alpha x^{\alpha -1}}{(x_{m}^{-1})^{\alpha }}}\quad (0<x\leq x_{m}^{-1})}"> <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">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>y</mi> <mo>≥</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">⇔</mo> <mspace width="1em" /> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>≤</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {Pa} (\alpha ,x_{m})={\frac {\alpha x_{m}^{\alpha }}{y^{\alpha +1}}}\quad (y\geq x_{m})\quad \Leftrightarrow \quad X\sim \mathrm {iPa} (\alpha ,x_{m})=\mathrm {Power} (x_{m}^{-1},\alpha )={\frac {\alpha x^{\alpha -1}}{(x_{m}^{-1})^{\alpha }}}\quad (0<x\leq x_{m}^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29393851ccca878bb4fdaef5e3178cade4093deb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:103.521ex; height:6.676ex;" alt="{\displaystyle Y\sim \mathrm {Pa} (\alpha ,x_{m})={\frac {\alpha x_{m}^{\alpha }}{y^{\alpha +1}}}\quad (y\geq x_{m})\quad \Leftrightarrow \quad X\sim \mathrm {iPa} (\alpha ,x_{m})=\mathrm {Power} (x_{m}^{-1},\alpha )={\frac {\alpha x^{\alpha -1}}{(x_{m}^{-1})^{\alpha }}}\quad (0<x\leq x_{m}^{-1})}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_exponential_distribution">Relation to the exponential distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=16" title="Edit section: Relation to the exponential distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pareto distribution is related to the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> as follows. If <i>X</i> is Pareto-distributed with minimum <i>x</i><sub>m</sub> and index <i>α</i>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726fa2844e2560b5f2b89b285459993b161b45c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.032ex; height:6.176ex;" alt="{\displaystyle Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)}"></span></dd></dl> <p>is <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponentially distributed</a> with rate parameter <i>α</i>. Equivalently, if <i>Y</i> is exponentially distributed with rate <i>α</i>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {m} }e^{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {m} }e^{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68bf8fc1a6e204f6fe0e3667ef8a2d3199308cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.5ex; height:3.009ex;" alt="{\displaystyle x_{\mathrm {m} }e^{Y}}"></span></dd></dl> <p>is Pareto-distributed with minimum <i>x</i><sub>m</sub> and index <i>α</i>. </p><p>This can be shown using the standard change-of-variable techniques: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}.\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 movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>y</mi> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d021765f90d4a42a1cb0e95769743c9d8e731d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:58.545ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}.\end{aligned}}}"></span></dd></dl> <p>The last expression is the cumulative distribution function of an exponential distribution with rate <i>α</i>. </p><p>Pareto distribution can be constructed by hierarchical exponential distributions.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi |a\sim {\text{Exp}}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Exp</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi |a\sim {\text{Exp}}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc7b1a207b847cf0e0d0f9ade8d126562676c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.503ex; height:2.843ex;" alt="{\displaystyle \phi |a\sim {\text{Exp}}(a)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta |\phi \sim {\text{Exp}}(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Exp</mtext> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta |\phi \sim {\text{Exp}}(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68408ce44329a178ce928b47104d200ffac82cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.598ex; height:2.843ex;" alt="{\displaystyle \eta |\phi \sim {\text{Exp}}(\phi )}"></span>. Then we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\eta |a)={\frac {a}{(a+\eta )^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>η</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\eta |a)={\frac {a}{(a+\eta )^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cf3333906901884561ae38aae2a481021040e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:18.152ex; height:5.509ex;" alt="{\displaystyle p(\eta |a)={\frac {a}{(a+\eta )^{2}}}}"></span> and, as a result, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+\eta \sim {\text{Pareto}}(a,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>η</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pareto</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+\eta \sim {\text{Pareto}}(a,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2250b1344544d2fa460cda013c9a5256da9367a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.33ex; height:2.843ex;" alt="{\displaystyle a+\eta \sim {\text{Pareto}}(a,1)}"></span>. </p><p>More in general, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \sim {\text{Gamma}}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </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 {\text{Gamma}}(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3273dac15072128319f02863f115da55124a39" 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 {\text{Gamma}}(\alpha ,\beta )}"></span> (shape-rate parametrization) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta |\lambda \sim {\text{Exp}}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Exp</mtext> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta |\lambda \sim {\text{Exp}}(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23277b9162e4fbf302a10a8c2208f6e6f88638a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.537ex; height:2.843ex;" alt="{\displaystyle \eta |\lambda \sim {\text{Exp}}(\lambda )}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta +\eta \sim {\text{Pareto}}(\beta ,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>+</mo> <mi>η</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pareto</mtext> </mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta +\eta \sim {\text{Pareto}}(\beta ,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ab7be595364b8c7e5bdbd98444ddea24f30524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.86ex; height:2.843ex;" alt="{\displaystyle \beta +\eta \sim {\text{Pareto}}(\beta ,\alpha )}"></span>. </p><p>Equivalently, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim {\text{Gamma}}(\alpha ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gamma</mtext> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\text{Gamma}}(\alpha ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47d5f4d52bff6492627e54fba861cfb0b00232d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.386ex; height:2.843ex;" alt="{\displaystyle Y\sim {\text{Gamma}}(\alpha ,1)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\text{Exp}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Exp</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\text{Exp}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbc4d1f377fea40500e325fafc0d298a66c9643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.153ex; height:2.843ex;" alt="{\displaystyle X\sim {\text{Exp}}(1)}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{m}}\!\left(1+{\frac {X}{Y}}\right)\sim {\text{Pareto}}(x_{\text{m}},\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mi>Y</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pareto</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{m}}\!\left(1+{\frac {X}{Y}}\right)\sim {\text{Pareto}}(x_{\text{m}},\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b41f81dc3660ef391c8b3ec0faf7f14d31b40e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.288ex; height:6.176ex;" alt="{\displaystyle x_{\text{m}}\!\left(1+{\frac {X}{Y}}\right)\sim {\text{Pareto}}(x_{\text{m}},\alpha )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_log-normal_distribution">Relation to the log-normal distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=17" title="Edit section: Relation to the log-normal distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pareto distribution and <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a> are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> and <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. (See <a href="#Relation_to_the_exponential_distribution">the previous section</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_generalized_Pareto_distribution">Relation to the generalized Pareto distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=18" title="Edit section: Relation to the generalized Pareto distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pareto distribution is a special case of the <a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">generalized Pareto distribution</a>, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the <a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax distribution</a> as a special case. This family also contains both the unshifted and shifted <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distributions</a>. </p><p>The Pareto distribution with scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12100e1dc5769ced8c9806b219abc06ab321d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.009ex;" alt="{\displaystyle x_{m}}"></span> and shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <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></span><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 }"></span> is equivalent to the generalized Pareto distribution with location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =x_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37fe77116256d8d7f6867374b99069fff176c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.505ex; height:2.176ex;" alt="{\displaystyle \mu =x_{m}}"></span>, scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =x_{m}/\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =x_{m}/\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc7073a745d90b54fa3f51a9aa53143c3e0cbde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.083ex; height:2.843ex;" alt="{\displaystyle \sigma =x_{m}/\alpha }"></span> and shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =1/\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =1/\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc92183b5a91fbcfe2c01bc669eb26f5cf5e9258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.941ex; height:2.843ex;" alt="{\displaystyle \xi =1/\alpha }"></span> and, conversely, one can get the Pareto distribution from the GPD by taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{m}=\sigma /\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{m}=\sigma /\xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311ed3eed099f294a50110ec5f391d37fe3c414c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.625ex; height:2.843ex;" alt="{\displaystyle x_{m}=\sigma /\xi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/\xi }"> <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> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1/\xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9324a54d0416568a26857cbc546b5c3babce73a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.941ex; height:2.843ex;" alt="{\displaystyle \alpha =1/\xi }"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973be61089b52fdb5da82587ce45d0febc1aedac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.291ex; height:2.509ex;" alt="{\displaystyle \xi >0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Bounded_Pareto_distribution">Bounded Pareto distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=19" title="Edit section: Bounded Pareto distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Truncated_distribution" title="Truncated distribution">Truncated distribution</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1247679731"><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Bounded Pareto</caption><tbody><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d06bfbe00ff463870f868c958b37cbe46ea3a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L>0}"></span> <a href="/wiki/Location_parameter" title="Location parameter">location</a> (<a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a>)<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H>L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>></mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H>L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c246e8f436ce8b8571861564938abd98a3d0bbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.745ex; height:2.176ex;" alt="{\displaystyle H>L}"></span> <a href="/wiki/Location_parameter" title="Location parameter">location</a> (<a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a>)<br /> </p> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (real)</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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\leqslant x\leqslant H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\leqslant x\leqslant H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c41208ac35c270cfd8a92c4f9bcad22ad21958fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.173ex; height:2.343ex;" alt="{\displaystyle L\leqslant x\leqslant H}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7860008161c8fb1bd2c7576a579f8229b5f61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:11.194ex; height:8.343ex;" alt="{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"></span></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"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99a14e3797831adc5db9fa752397c1c310f68b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:11.598ex; height:8.176ex;" alt="{\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"></span></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"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>α</mi> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e69ceac315699d8199a9415463da077b94ec284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:49.859ex; height:8.176ex;" alt="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1}"></span><br /> </p> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},\alpha =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>H</mi> <mi>L</mi> </mfrac> </mrow> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},\alpha =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d91eddf791a693b56d2bc421ae0cf97a0b4694c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.719ex; height:5.343ex;" alt="{\displaystyle {\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},\alpha =1}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0b4dc625b406d001834c3db2e46f0909cb6bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.623ex; height:7.343ex;" alt="{\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>α</mi> <mo>≠</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d7f4051ffe97935d76de6180bb258549971902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:49.859ex; height:8.176ex;" alt="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},\alpha =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>H</mi> <mi>L</mi> </mfrac> </mrow> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},\alpha =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4880d6a2322c4e434bb988bbda2e32d0001e126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.867ex; height:6.343ex;" alt="{\displaystyle {\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},\alpha =2}"></span> </p> (this is the second raw moment, not the variance)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mi>α</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4abfc3c314ce6bddf9f3dac8cb13543893b949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.949ex; height:8.509ex;" alt="{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}"></span> </p> (this is the kth raw moment, not the skewness)</td></tr></tbody></table> <p>The bounded (or truncated) Pareto distribution has three parameters: <i>α</i>, <i>L</i> and <i>H</i>. As in the standard Pareto distribution <i>α</i> determines the shape. <i>L</i> denotes the minimal value, and <i>H</i> denotes the maximal value. </p><p>The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7860008161c8fb1bd2c7576a579f8229b5f61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:11.194ex; height:8.343ex;" alt="{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}"></span>,</dd></dl> <p>where <i>L</i> ≤ <i>x</i> ≤ <i>H</i>, and <i>α</i> > 0. </p> <div class="mw-heading mw-heading4"><h4 id="Generating_bounded_Pareto_random_variables">Generating bounded Pareto random variables</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=20" title="Edit section: Generating bounded Pareto random variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>U</i> is <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> on (0, 1), then applying inverse-transform method <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>H</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9a62bb8003c3eb6c82998607d2e9efd3bfe0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.479ex; height:7.176ex;" alt="{\displaystyle U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <mi>U</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94488b1c0d80a676b6e243bc1fb94977a026544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.583ex; height:7.176ex;" alt="{\displaystyle x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}}"></span></dd></dl> <p>is a bounded Pareto-distributed. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Symmetric_Pareto_distribution">Symmetric Pareto distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=21" title="Edit section: Symmetric Pareto distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.<sup id="cite_ref-:0_20-0" class="reference"><a href="#cite_note-:0-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:<sup id="cite_ref-:0_20-1" class="reference"><a href="#cite_note-:0-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over 2b-X})^{a}&X<b\\1-{\tfrac {1}{2}}({\tfrac {b}{X}})^{a}&X\geq b\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>b</mi> <mo>−</mo> <mi>X</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mtd> <mtd> <mi>X</mi> <mo><</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mi>X</mi> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mtd> <mtd> <mi>X</mi> <mo>≥</mo> <mi>b</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over 2b-X})^{a}&X<b\\1-{\tfrac {1}{2}}({\tfrac {b}{X}})^{a}&X\geq b\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b029e0cbc7a0839aac5e47ff636282ff237f3594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.918ex; margin-bottom: -0.254ex; width:43.521ex; height:7.509ex;" alt="{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over 2b-X})^{a}&X<b\\1-{\tfrac {1}{2}}({\tfrac {b}{X}})^{a}&X\geq b\end{cases}}}"></span> </p><p>The corresponding probability density function (PDF) is:<sup id="cite_ref-:0_20-2" class="reference"><a href="#cite_note-:0-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x-b\right\vert )^{a+1}},X\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>b</mi> </mrow> <mo>|</mo> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mi>X</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x-b\right\vert )^{a+1}},X\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ebb3ddc6525757514b27f0f7a792b71bf573a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:32.424ex; height:6.176ex;" alt="{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x-b\right\vert )^{a+1}},X\in R}"></span> </p><p>This distribution has two parameters: a and b. It is symmetric about b. Then the mathematic expectation is b. When, it has variance as following: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E((x-b)^{2})=\int _{-\infty }^{\infty }(x-b)^{2}p(x)dx={2b^{2} \over (a-2)(a-1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E((x-b)^{2})=\int _{-\infty }^{\infty }(x-b)^{2}p(x)dx={2b^{2} \over (a-2)(a-1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03b1534a2e4b52127e47a95ab73aba2804f7cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.065ex; height:6.509ex;" alt="{\displaystyle E((x-b)^{2})=\int _{-\infty }^{\infty }(x-b)^{2}p(x)dx={2b^{2} \over (a-2)(a-1)}}"></span> </p><p>The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over b-X})^{a}&X<0\\1-{\tfrac {1}{2}}({\tfrac {b}{b+X}})^{a}&X\geq 0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mi>b</mi> <mo>−</mo> <mi>X</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mtd> <mtd> <mi>X</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mrow> <mi>b</mi> <mo>+</mo> <mi>X</mi> </mrow> </mfrac> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mtd> <mtd> <mi>X</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over b-X})^{a}&X<0\\1-{\tfrac {1}{2}}({\tfrac {b}{b+X}})^{a}&X\geq 0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2300ab591ba3a47b4d095b239d6a75cc8a340ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.67ex; height:7.509ex;" alt="{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over b-X})^{a}&X<0\\1-{\tfrac {1}{2}}({\tfrac {b}{b+X}})^{a}&X\geq 0\end{cases}}}"></span> </p><p>The corresponding PDF is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x\right\vert )^{a+1}},X\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mi>X</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x\right\vert )^{a+1}},X\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc175af34ac9bf33e964c1f72008f7d260d3b97f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:28.586ex; height:6.176ex;" alt="{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x\right\vert )^{a+1}},X\in R}"></span> </p><p>This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.<sup id="cite_ref-:0_20-3" class="reference"><a href="#cite_note-:0-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multivariate_Pareto_distribution">Multivariate Pareto distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=22" title="Edit section: Multivariate Pareto distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The univariate Pareto distribution has been extended to a <a href="/wiki/Multivariate_Pareto_distribution" title="Multivariate Pareto distribution">multivariate Pareto distribution</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_inference">Statistical inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=23" title="Edit section: Statistical inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_parameters">Estimation of parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=24" title="Edit section: Estimation of parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> for the Pareto distribution parameters <i>α</i> and <i>x</i><sub>m</sub>, given an independent <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> <i>x</i> = (<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x<sub>n</sub></i>), is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\alpha ,x_{\mathrm {m} })=\prod _{i=1}^{n}\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x_{i}^{\alpha +1}}}=\alpha ^{n}x_{\mathrm {m} }^{n\alpha }\prod _{i=1}^{n}{\frac {1}{x_{i}^{\alpha +1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <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>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>α</mi> </mrow> </msubsup> <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> <mn>1</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\alpha ,x_{\mathrm {m} })=\prod _{i=1}^{n}\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x_{i}^{\alpha +1}}}=\alpha ^{n}x_{\mathrm {m} }^{n\alpha }\prod _{i=1}^{n}{\frac {1}{x_{i}^{\alpha +1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ad835631eed70acb507fdf67df62837d6acc8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.683ex; height:6.843ex;" alt="{\displaystyle L(\alpha ,x_{\mathrm {m} })=\prod _{i=1}^{n}\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x_{i}^{\alpha +1}}}=\alpha ^{n}x_{\mathrm {m} }^{n\alpha }\prod _{i=1}^{n}{\frac {1}{x_{i}^{\alpha +1}}}.}"></span></dd></dl> <p>Therefore, the logarithmic likelihood function is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (\alpha ,x_{\mathrm {m} })=n\ln \alpha +n\alpha \ln x_{\mathrm {m} }-(\alpha +1)\sum _{i=1}^{n}\ln x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>n</mi> <mi>α</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (\alpha ,x_{\mathrm {m} })=n\ln \alpha +n\alpha \ln x_{\mathrm {m} }-(\alpha +1)\sum _{i=1}^{n}\ln x_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f60b2d0e1de34f54094106fcac7a8b4e0cc73d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.665ex; height:6.843ex;" alt="{\displaystyle \ell (\alpha ,x_{\mathrm {m} })=n\ln \alpha +n\alpha \ln x_{\mathrm {m} }-(\alpha +1)\sum _{i=1}^{n}\ln x_{i}.}"></span></dd></dl> <p>It can be seen that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (\alpha ,x_{\mathrm {m} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (\alpha ,x_{\mathrm {m} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e622f9154add9c8fbc51730f878da89422b48ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.231ex; height:2.843ex;" alt="{\displaystyle \ell (\alpha ,x_{\mathrm {m} })}"></span> is monotonically increasing with <i>x</i><sub>m</sub>, that is, the greater the value of <i>x</i><sub>m</sub>, the greater the value of the likelihood function. Hence, since <i>x</i> ≥ <i>x</i><sub>m</sub>, we conclude that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {x}}_{\mathrm {m} }=\min _{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>x</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {x}}_{\mathrm {m} }=\min _{i}{x_{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96930a722359f7c5d36d877a4adc605569f9e340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.114ex; height:4.009ex;" alt="{\displaystyle {\widehat {x}}_{\mathrm {m} }=\min _{i}{x_{i}}.}"></span></dd></dl> <p>To find the <a href="/wiki/Estimator" title="Estimator">estimator</a> for <i>α</i>, we compute the corresponding partial derivative and determine where it is zero: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ell }{\partial \alpha }}={\frac {n}{\alpha }}+n\ln x_{\mathrm {m} }-\sum _{i=1}^{n}\ln x_{i}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ℓ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>α</mi> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ell }{\partial \alpha }}={\frac {n}{\alpha }}+n\ln x_{\mathrm {m} }-\sum _{i=1}^{n}\ln x_{i}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027852454b456586ab5621da9bcdb8f26cf8c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.89ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial \ell }{\partial \alpha }}={\frac {n}{\alpha }}+n\ln x_{\mathrm {m} }-\sum _{i=1}^{n}\ln x_{i}=0.}"></span></dd></dl> <p>Thus the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimator for <i>α</i> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\alpha }}={\frac {n}{\sum _{i}\ln(x_{i}/{\widehat {x}}_{\mathrm {m} })}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\alpha }}={\frac {n}{\sum _{i}\ln(x_{i}/{\widehat {x}}_{\mathrm {m} })}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b07b674e237ce413f68d7bb0fe2fab172f2c109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.727ex; height:5.509ex;" alt="{\displaystyle {\widehat {\alpha }}={\frac {n}{\sum _{i}\ln(x_{i}/{\widehat {x}}_{\mathrm {m} })}}.}"></span></dd></dl> <p>The expected statistical error is:<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>^</mo> </mover> </mrow> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc5da6be0e1ae91ce0af9cae1e421eea591b8f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.242ex; height:6.343ex;" alt="{\displaystyle \sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.}"></span></dd></dl> <p>Malik (1970)<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> gives the exact joint distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {x}}_{\mathrm {m} },{\hat {\alpha }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {x}}_{\mathrm {m} },{\hat {\alpha }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66fd9ab68f0e76fe050bb771b0b43d608c0baaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.262ex; height:2.843ex;" alt="{\displaystyle ({\hat {x}}_{\mathrm {m} },{\hat {\alpha }})}"></span>. In particular, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c1943e4e0797ca4d7b876351d59b56104c36803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}_{\mathrm {m} }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682d943d1947245b587f282aba6c88f0870fb302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.176ex;" alt="{\displaystyle {\hat {\alpha }}}"></span> are <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c1943e4e0797ca4d7b876351d59b56104c36803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}_{\mathrm {m} }}"></span> is Pareto with scale parameter <i>x</i><sub>m</sub> and shape parameter <i>nα</i>, whereas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682d943d1947245b587f282aba6c88f0870fb302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.176ex;" alt="{\displaystyle {\hat {\alpha }}}"></span> has an <a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse-gamma distribution</a> with shape and scale parameters <i>n</i> − 1 and <i>nα</i>, respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=25" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=26" title="Edit section: General"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Vilfredo_Pareto" title="Vilfredo Pareto">Vilfredo Pareto</a> originally used this distribution to describe the <a href="/wiki/Distribution_of_wealth" title="Distribution of wealth">allocation of wealth</a> among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.<sup id="cite_ref-:1_4-1" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This idea is sometimes expressed more simply as the <a href="/wiki/Pareto_principle" title="Pareto principle">Pareto principle</a> or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> As Michael Hudson points out (<i>The Collapse of Antiquity</i> [2023] p. 85 & n.7) "a mathematical corollary [is] that 10% would have 65% of the wealth, and 5% would have half the national wealth.” However, the 80-20 rule corresponds to a particular value of <i>α</i>, and in fact, Pareto's data on British income taxes in his <i>Cours d'économie politique</i> indicates that about 30% of the population had about 70% of the income.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2019)">citation needed</span></a></i>]</sup> The <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, <a href="/wiki/Net_worth" title="Net worth">net worth</a> may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed: </p> <ul><li>All four variables of the household's budget constraint: consumption, labor income, capital income, and wealth.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>The sizes of human settlements (few cities, many hamlets/villages)<sup id="cite_ref-Reed_26-0" class="reference"><a href="#cite_note-Reed-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Reed2002_27-0" class="reference"><a href="#cite_note-Reed2002-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li>File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<sup id="cite_ref-Reed_26-1" class="reference"><a href="#cite_note-Reed-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Hard_disk_drive" title="Hard disk drive">Hard disk drive</a> error rates<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li>Clusters of <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a> near <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a><sup id="cite_ref-Simon_29-0" class="reference"><a href="#cite_note-Simon-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:FitParetoDistr.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/FitParetoDistr.tif/lossy-page1-250px-FitParetoDistr.tif.jpg" decoding="async" width="250" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/FitParetoDistr.tif/lossy-page1-375px-FitParetoDistr.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/FitParetoDistr.tif/lossy-page1-500px-FitParetoDistr.tif.jpg 2x" data-file-width="623" data-file-height="465" /></a><figcaption>Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using <a href="/wiki/CumFreq" title="CumFreq">CumFreq</a>, see also <a href="/wiki/Distribution_fitting" class="mw-redirect" title="Distribution fitting">distribution fitting</a> </figcaption></figure> <ul><li>The values of <a href="/wiki/Oil_reserves" class="mw-redirect" title="Oil reserves">oil reserves</a> in oil fields (a few <a href="/wiki/Giant_oil_and_gas_fields" title="Giant oil and gas fields">large fields</a>, many <a href="/wiki/Stripper_well" title="Stripper well">small fields</a>)<sup id="cite_ref-Reed_26-2" class="reference"><a href="#cite_note-Reed-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li>The standardized price returns on individual stocks <sup id="cite_ref-Reed_26-3" class="reference"><a href="#cite_note-Reed-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>Sizes of sand particles <sup id="cite_ref-Reed_26-4" class="reference"><a href="#cite_note-Reed-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>The size of meteorites</li> <li>Severity of large <a href="/wiki/Casualty_(person)" title="Casualty (person)">casualty</a> losses for certain lines of business such as general liability, commercial auto, and workers compensation.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li> <li>Amount of time a user on <a href="/wiki/Steam_(service)" title="Steam (service)">Steam</a> will spend playing different games. (Some games get played a lot, but most get played almost never.) <a rel="nofollow" class="external autonumber" href="https://docs.google.com/spreadsheets/d/1AjtfgTQc1T84NCyJWGcCPN4jrVsOpX0bp0jgPZJEW6A/edit#gid=0">[2]</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research"><span title="The material near this tag possibly contains original research. (December 2020)">original research?</span></a></i>]</sup></li> <li>In <a href="/wiki/Hydrology" title="Hydrology">hydrology</a> the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% <a href="/wiki/Confidence_belt" class="mw-redirect" title="Confidence belt">confidence belt</a> based on the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>. The rainfall data are represented by <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting positions</a> as part of the <a href="/wiki/Cumulative_frequency_analysis" title="Cumulative frequency analysis">cumulative frequency analysis</a>.</li> <li>In Electric Utility Distribution Reliability (80% of the Customer Minutes Interrupted occur on approximately 20% of the days in a given year).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Relation_to_Zipf's_law"><span id="Relation_to_Zipf.27s_law"></span>Relation to Zipf's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=27" title="Edit section: Relation to Zipf's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pareto distribution is a continuous probability distribution. <a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf's law</a>, also sometimes called the <a href="/wiki/Zeta_distribution" title="Zeta distribution">zeta distribution</a>, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> values (incomes) are binned into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12100e1dc5769ced8c9806b219abc06ab321d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.009ex;" alt="{\displaystyle x_{m}}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha x_{\mathrm {m} }^{\alpha }={\frac {1}{H(N,\alpha -1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha x_{\mathrm {m} }^{\alpha }={\frac {1}{H(N,\alpha -1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/827b6d6c147eae9901834ff5e2da5aa7295ad8ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.814ex; height:6.009ex;" alt="{\displaystyle \alpha x_{\mathrm {m} }^{\alpha }={\frac {1}{H(N,\alpha -1)}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(N,\alpha -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(N,\alpha -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c09cee5e3a1ea2da6d535e41178edf5d2e813fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.461ex; height:2.843ex;" alt="{\displaystyle H(N,\alpha -1)}"></span> is the <a href="/wiki/Harmonic_number#Generalized_harmonic_numbers" title="Harmonic number">generalized harmonic number</a>. This makes Zipf's probability density function derivable from Pareto's. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}={\frac {1}{x^{s}H(N,s)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mi>H</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}={\frac {1}{x^{s}H(N,s)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea20ae981b40a3f4e811c6b7848e313e47ddcdd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.395ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}={\frac {1}{x^{s}H(N,s)}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\alpha -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\alpha -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd94e2c1591882e58c2ea0817120ca8c20e3095c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.679ex; height:2.343ex;" alt="{\displaystyle s=\alpha -1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> probability of ranking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_"Pareto_principle""><span id="Relation_to_the_.22Pareto_principle.22"></span>Relation to the "Pareto principle"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=28" title="Edit section: Relation to the "Pareto principle""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "<a href="/wiki/Pareto_principle" title="Pareto principle">80–20 law</a>", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⁡</mo> <mn>5</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>5</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>4</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>1.161</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb24809a871cf1835da7e93b8c8e1d4c15b14e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.889ex; height:7.176ex;" alt="{\displaystyle \alpha =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161}"></span>. This result can be derived from the <a href="/wiki/Lorenz_curve" title="Lorenz curve">Lorenz curve</a> formula given below. Moreover, the following have been shown<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> to be mathematically equivalent: </p> <ul><li>Income is distributed according to a Pareto distribution with index <i>α</i> > 1.</li> <li>There is some number 0 ≤ <i>p</i> ≤ 1/2 such that 100<i>p</i> % of all people receive 100(1 − <i>p</i>)% of all income, and similarly for every real (not necessarily integer) <i>n</i> > 0, 100<i>p<sup>n</sup></i> % of all people receive 100(1 − <i>p</i>)<sup><i>n</i></sup> percentage of all income. <i>α</i> and <i>p</i> are related by</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\frac {1}{\alpha }}={\frac {\ln(1-p)}{\ln(p)}}={\frac {\ln((1-p)^{n})}{\ln(p^{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{\alpha }}={\frac {\ln(1-p)}{\ln(p)}}={\frac {\ln((1-p)^{n})}{\ln(p^{n})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127997244e8a14876787e7792c4e1973cd41e9ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.065ex; height:6.509ex;" alt="{\displaystyle 1-{\frac {1}{\alpha }}={\frac {\ln(1-p)}{\ln(p)}}={\frac {\ln((1-p)^{n})}{\ln(p^{n})}}}"></span></dd></dl></dd></dl> <p>This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution. </p><p>This excludes Pareto distributions in which 0 < <i>α</i> ≤ 1, which, as noted above, have an infinite expected value, and so cannot reasonably model income distribution. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_Price's_law"><span id="Relation_to_Price.27s_law"></span>Relation to Price's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=29" title="Edit section: Relation to Price's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Derek_J._de_Solla_Price#Scientific_contributions" title="Derek J. de Solla Price">Price's square root law</a> is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d67a45a44be8b8f15e99b7def2b0cf0aba1717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =1}"></span>. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule. </p> <div class="mw-heading mw-heading3"><h3 id="Lorenz_curve_and_Gini_coefficient">Lorenz curve and Gini coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=30" title="Edit section: Lorenz curve and Gini coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:ParetoLorenzSVG.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/ParetoLorenzSVG.svg/325px-ParetoLorenzSVG.svg.png" decoding="async" width="325" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/ParetoLorenzSVG.svg/488px-ParetoLorenzSVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/ParetoLorenzSVG.svg/650px-ParetoLorenzSVG.svg.png 2x" data-file-width="540" data-file-height="360" /></a><figcaption>Lorenz curves for a number of Pareto distributions. The case <i>α</i> = ∞ corresponds to perfectly equal distribution (<i>G</i> = 0) and the <i>α</i> = 1 line corresponds to complete inequality (<i>G</i> = 1)</figcaption></figure> <p>The <a href="/wiki/Lorenz_curve" title="Lorenz curve">Lorenz curve</a> is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve <i>L</i>(<i>F</i>) is written in terms of the PDF <i>f</i> or the CDF <i>F</i> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msubsup> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c89d96d4b6db1a7717f6fd76acec8188221e464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.371ex; height:8.509ex;" alt="{\displaystyle L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}}"></span></dd></dl> <p>where <i>x</i>(<i>F</i>) is the inverse of the CDF. For the Pareto distribution, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136721c225f8a4fa4abaeb9bca36673aa4d22984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:18.29ex; height:6.343ex;" alt="{\displaystyle x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}}"></span></dd></dl> <p>and the Lorenz curve is calculated to be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/164df53c4e987a1e2344d1a9b2d7bbafa3fde7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.457ex; height:3.843ex;" alt="{\displaystyle L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},}"></span></dd></dl> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\alpha \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\alpha \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f3744f24aac421ebcecd035fe6d84f7d152740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.009ex; height:2.343ex;" alt="{\displaystyle 0<\alpha \leq 1}"></span> the denominator is infinite, yielding <i>L</i>=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right. </p><p>According to <a href="/wiki/Oxfam" title="Oxfam">Oxfam</a> (2016) the richest 62 people have as much wealth as the poorest half of the world's population.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> We can estimate the Pareto index that would apply to this situation. Letting ε equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 62/(7\times 10^{9})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>62</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 62/(7\times 10^{9})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b87260ff52963ae27ec69ed0a9ece4803b28a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.679ex; height:3.176ex;" alt="{\displaystyle 62/(7\times 10^{9})}"></span> we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(1/2)=1-L(1-\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(1/2)=1-L(1-\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bc240f30ca7373619accb5668cb5639cecdbef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.459ex; height:2.843ex;" alt="{\displaystyle L(1/2)=1-L(1-\varepsilon )}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-(1/2)^{1-{\frac {1}{\alpha }}}=\varepsilon ^{1-{\frac {1}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-(1/2)^{1-{\frac {1}{\alpha }}}=\varepsilon ^{1-{\frac {1}{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9888f124cca5685de15166af61a950e0378155a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.527ex; height:3.843ex;" alt="{\displaystyle 1-(1/2)^{1-{\frac {1}{\alpha }}}=\varepsilon ^{1-{\frac {1}{\alpha }}}}"></span></dd></dl> <p>The solution is that <i>α</i> equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Gini_coefficient" title="Gini coefficient">Gini coefficient</a> is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (<i>α</i> = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \geq 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 \geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810dcd1d13dc817a9a6e5dc25ed3ceca99b457d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.749ex; height:2.343ex;" alt="{\displaystyle \alpha \geq 1}"></span>) to be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1-2\left(\int _{0}^{1}L(F)\,dF\right)={\frac {1}{2\alpha -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>F</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1-2\left(\int _{0}^{1}L(F)\,dF\right)={\frac {1}{2\alpha -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8e828aee2c0585a221dce690550ba7dd412563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.868ex; height:6.343ex;" alt="{\displaystyle G=1-2\left(\int _{0}^{1}L(F)\,dF\right)={\frac {1}{2\alpha -1}}}"></span></dd></dl> <p>(see Aaberge 2005). </p> <div class="mw-heading mw-heading2"><h2 id="Random_variate_generation">Random variate generation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=31" title="Edit section: Random variate generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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> <p>Random samples can be generated using <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>. Given a random variate <i>U</i> drawn from the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> on the unit interval [0, 1], the variate <i>T</i> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {x_{\mathrm {m} }}{U^{1/\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {x_{\mathrm {m} }}{U^{1/\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21ed6bd51060be52a5788f8d8e78ef76e7f162e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.34ex; height:5.176ex;" alt="{\displaystyle T={\frac {x_{\mathrm {m} }}{U^{1/\alpha }}}}"></span></dd></dl> <p>is Pareto-distributed.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=32" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bradford%27s_law" title="Bradford's law">Bradford's law</a> – Pattern of references in science journals</li> <li><a href="/wiki/Gutenberg%E2%80%93Richter_law" title="Gutenberg–Richter law">Gutenberg–Richter law</a> – Law in seismology describing earthquake frequency and magnitude</li> <li><a href="/wiki/Matthew_effect" title="Matthew effect">Matthew effect</a> – The rich get richer and the poor get poorer</li> <li><a href="/wiki/Pareto_analysis" class="mw-redirect" title="Pareto analysis">Pareto analysis</a> – Statistical principle about ratio of effects to causes<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a> – Weakly optimal allocation of resources</li> <li><a href="/wiki/Pareto_interpolation" title="Pareto interpolation">Pareto interpolation</a> – Method of estimating the median of a population</li> <li><a href="/wiki/Power_law#Power-law_probability_distributions" title="Power law">Power law probability distributions</a> – Functional relationship between two quantities</li> <li><a href="/wiki/Sturgeon%27s_law" title="Sturgeon's law">Sturgeon's law</a> – "Ninety percent of everything is crap"</li> <li><a href="/wiki/Traffic_generation_model" title="Traffic generation model">Traffic generation model</a> – simulated flow of data in a communications network<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf's law</a> – Probability distribution</li> <li><a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">Heavy-tailed distribution</a> – Probability distribution</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=33" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-norton-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-norton_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-norton_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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 .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFNortonKhokhlovUryasev2019" class="citation journal cs1">Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). <a rel="nofollow" class="external text" href="http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf">"Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Operations Research</i>. <b>299</b> (1–2). Springer: 1281–1315. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1811.11301">1811.11301</a></span>. <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%2Fs10479-019-03373-1">10.1007/s10479-019-03373-1</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:254231768">254231768</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Operations+Research&rft.atitle=Calculating+CVaR+and+bPOE+for+common+probability+distributions+with+application+to+portfolio+optimization+and+density+estimation&rft.volume=299&rft.issue=1%E2%80%932&rft.pages=1281-1315&rft.date=2019&rft_id=info%3Aarxiv%2F1811.11301&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A254231768%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10479-019-03373-1&rft.aulast=Norton&rft.aufirst=Matthew&rft.au=Khokhlov%2C+Valentyn&rft.au=Uryasev%2C+Stan&rft_id=http%3A%2F%2Furyasev.ams.stonybrook.edu%2Fwp-content%2Fuploads%2F2019%2F10%2FNorton2019_CVaR_bPOE.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmoroso1938" class="citation journal cs1">Amoroso, Luigi (1938). 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(1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02626668609491037">"The Generalized Pareto distribution applied to rainfall depths"</a>. <i>Hydrological Sciences Journal</i>. <b>31</b> (2): 151–162. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986HydSJ..31..151V">1986HydSJ..31..151V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02626668609491037">10.1080/02626668609491037</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Hydrological+Sciences+Journal&rft.atitle=The+Generalized+Pareto+distribution+applied+to+rainfall+depths&rft.volume=31&rft.issue=2&rft.pages=151-162&rft.date=1986&rft_id=info%3Adoi%2F10.1080%2F02626668609491037&rft_id=info%3Abibcode%2F1986HydSJ..31..151V&rft.aulast=VAN+MONTFORT&rft.aufirst=M.A.J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%252F02626668609491037&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOancea2017" class="citation journal cs1">Oancea, Bogdan (2017). 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Bhatti (2018). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/322758024_Parameter_estimation_of_Pareto_distribution_Some_modified_moment_estimators">Parameter estimation of Pareto distribution: Some modified moment estimators</a>. <i>Maejo International Journal of Science and Technology</i> 12(1):11-27.</span> </li> <li id="cite_note-:02-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-:02_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEliazar2017" class="citation journal cs1">Eliazar, Iddo (November 2017). "Lindy's Law". <i>Physica A: Statistical Mechanics and Its Applications</i>. <b>486</b>: 797–805. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017PhyA..486..797E">2017PhyA..486..797E</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.physa.2017.05.077">10.1016/j.physa.2017.05.077</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:125349686">125349686</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A%3A+Statistical+Mechanics+and+Its+Applications&rft.atitle=Lindy%27s+Law&rft.volume=486&rft.pages=797-805&rft.date=2017-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125349686%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physa.2017.05.077&rft_id=info%3Abibcode%2F2017PhyA..486..797E&rft.aulast=Eliazar&rft.aufirst=Iddo&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-Johnson1994-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Johnson1994_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Johnson1994_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</span> </li> <li id="cite_note-jkb94-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-jkb94_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-jkb94_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-jkb94_12-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-jkb94_12-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Johnson, Kotz, and Balakrishnan (1994), (20.4).</span> </li> <li id="cite_note-kk03-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-kk03_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristian_KleiberSamuel_Kotz2003" class="citation book cs1">Christian Kleiber & Samuel Kotz (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7wLGjyB128IC"><i>Statistical Size Distributions in Economics and Actuarial Sciences</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-15064-0" title="Special:BookSources/978-0-471-15064-0"><bdi>978-0-471-15064-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Size+Distributions+in+Economics+and+Actuarial+Sciences&rft.pub=Wiley&rft.date=2003&rft.isbn=978-0-471-15064-0&rft.au=Christian+Kleiber&rft.au=Samuel+Kotz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7wLGjyB128IC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-feller-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-feller_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-feller_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller1971" class="citation book cs1">Feller, W. (1971). <i>An Introduction to Probability Theory and its Applications</i>. Vol. II (2nd ed.). New York: Wiley. p. 50.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Probability+Theory+and+its+Applications&rft.place=New+York&rft.pages=50&rft.edition=2nd&rft.pub=Wiley&rft.date=1971&rft.aulast=Feller&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span> "The densities (4.3) are sometimes called after the economist <i>Pareto</i>. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ <i>Ax</i><sup>−<i>α</i></sup> as <i>x</i> → ∞".</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLomax1954" class="citation journal cs1">Lomax, K. S. (1954). "Business failures. 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Springer. pp. 121–22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387727967" title="Special:BookSources/9780387727967"><bdi>9780387727967</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+7%3A+Pareto+and+Generalized+Pareto+Distributions&rft.btitle=Modeling+Income+Distributions+and+Lorenz+Curves&rft.pages=121-22&rft.pub=Springer&rft.date=2008-09-16&rft.isbn=9780387727967&rft.aulast=Chotikapanich&rft.aufirst=Duangkamon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfUJZZLj1kbwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Dallas, A. C. "Characterizing the Pareto and power distributions." Annals of the Institute of Statistical Mathematics 28.1 (1976): 491-497.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhite2006" class="citation thesis cs1">White, Gentry (2006). <a rel="nofollow" class="external text" href="https://mospace.umsystem.edu/xmlui/handle/10355/4450"><i>Bayesian semiparametric spatial and joint spatio-temporal modeling</i></a> (Thesis thesis). University of Missouri--Columbia.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Bayesian+semiparametric+spatial+and+joint+spatio-temporal+modeling&rft.degree=Thesis&rft.inst=University+of+Missouri--Columbia&rft.date=2006&rft.aulast=White&rft.aufirst=Gentry&rft_id=https%3A%2F%2Fmospace.umsystem.edu%2Fxmlui%2Fhandle%2F10355%2F4450&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span> section 5.3.1.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120117042753/http://www.cs.bgu.ac.il/~mps042/invtransnote.htm">"Inverse Transform Method"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.cs.bgu.ac.il/~mps042/invtransnote.htm">the original</a> on 2012-01-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-08-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Inverse+Transform+Method&rft_id=http%3A%2F%2Fwww.cs.bgu.ac.il%2F~mps042%2Finvtransnote.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-:0-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_20-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_20-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuang2004" class="citation journal cs1">Huang, Xiao-dong (2004). 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Retrieved <span class="nowrap">2010-09-10</span></span>. <q>We experimented with 5 different distributions (Geometric, Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ<sup>2</sup> statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=8th+Usenix+Conference+on+File+and+Storage+Technologies+%28FAST+2010%29&rft.atitle=Understanding+latent+sector+error+and+how+to+protect+against+them&rft.date=2010-02-24&rft.aulast=Schroeder&rft.aufirst=Bianca&rft.au=Damouras%2C+Sotirios&rft.au=Gill%2C+Phillipa&rft_id=http%3A%2F%2Fwww.usenix.org%2Fevent%2Ffast10%2Ftech%2Ffull_papers%2Fschroeder.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-Simon-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Simon_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYuji_IjiriSimon1975" class="citation journal cs1">Yuji Ijiri; Simon, Herbert A. 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USA</i>. <b>72</b> (5): 1654–57. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975PNAS...72.1654I">1975PNAS...72.1654I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.72.5.1654">10.1073/pnas.72.5.1654</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC432601">432601</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578724">16578724</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Natl.+Acad.+Sci.+USA&rft.atitle=Some+Distributions+Associated+with+Bose%E2%80%93Einstein+Statistics&rft.volume=72&rft.issue=5&rft.pages=1654-57&rft.date=1975-05&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC432601%23id-name%3DPMC&rft_id=info%3Apmid%2F16578724&rft_id=info%3Adoi%2F10.1073%2Fpnas.72.5.1654&rft_id=info%3Abibcode%2F1975PNAS...72.1654I&rft.au=Yuji+Ijiri&rft.au=Simon%2C+Herbert+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC432601&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarchol-BalterDowney1997" class="citation journal cs1"><a href="/wiki/Mor_Harchol-Balter" title="Mor Harchol-Balter">Harchol-Balter, Mor</a>; Downey, Allen (August 1997). <a rel="nofollow" class="external text" href="https://users.soe.ucsc.edu/~scott/courses/Fall11/221/Papers/Sync/harcholbalter-tocs97.pdf">"Exploiting Process Lifetime Distributions for Dynamic Load Balancing"</a> <span class="cs1-format">(PDF)</span>. <i>ACM Transactions on Computer Systems</i>. <b>15</b> (3): 253–258. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F263326.263344">10.1145/263326.263344</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:52861447">52861447</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Computer+Systems&rft.atitle=Exploiting+Process+Lifetime+Distributions+for+Dynamic+Load+Balancing&rft.volume=15&rft.issue=3&rft.pages=253-258&rft.date=1997-08&rft_id=info%3Adoi%2F10.1145%2F263326.263344&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52861447%23id-name%3DS2CID&rft.aulast=Harchol-Balter&rft.aufirst=Mor&rft.au=Downey%2C+Allen&rft_id=https%3A%2F%2Fusers.soe.ucsc.edu%2F~scott%2Fcourses%2FFall11%2F221%2FPapers%2FSync%2Fharcholbalter-tocs97.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Kleiber and Kotz (2003): p. 94.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeal1980" class="citation journal cs1">Seal, H. (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0515036100006620">"Survival probabilities based on Pareto claim distributions"</a>. <i>ASTIN Bulletin</i>. <b>11</b>: 61–71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0515036100006620">10.1017/S0515036100006620</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ASTIN+Bulletin&rft.atitle=Survival+probabilities+based+on+Pareto+claim+distributions&rft.volume=11&rft.pages=61-71&rft.date=1980&rft_id=info%3Adoi%2F10.1017%2FS0515036100006620&rft.aulast=Seal&rft.aufirst=H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252FS0515036100006620&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">CumFreq, software for cumulative frequency analysis and probability distribution fitting <a rel="nofollow" class="external autonumber" href="https://www.waterlog.info/cumfreq.htm">[1]</a></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy2010" class="citation journal cs1">Hardy, Michael (2010). "Pareto's Law". <i><a href="/wiki/Mathematical_Intelligencer" class="mw-redirect" title="Mathematical Intelligencer">Mathematical Intelligencer</a></i>. <b>32</b> (3): 38–43. <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%2Fs00283-010-9159-2">10.1007/s00283-010-9159-2</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:121797873">121797873</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Intelligencer&rft.atitle=Pareto%27s+Law&rft.volume=32&rft.issue=3&rft.pages=38-43&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2Fs00283-010-9159-2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121797873%23id-name%3DS2CID&rft.aulast=Hardy&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report">"62 people own the same as half the world, reveals Oxfam Davos report"</a>. Oxfam. Jan 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=62+people+own+the+same+as+half+the+world%2C+reveals+Oxfam+Davos+report&rft.pub=Oxfam&rft.date=2016-01&rft_id=https%3A%2F%2Fwww.oxfam.org%2Fen%2Fpressroom%2Fpressreleases%2F2016-01-18%2F62-people-own-same-half-world-reveals-oxfam-davos-report&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83">"Global Wealth Report 2013"</a>. Credit Suisse. Oct 2013. p. 22. Archived from <a rel="nofollow" class="external text" href="https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83">the original</a> on 2015-02-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-01-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Global+Wealth+Report+2013&rft.pages=22&rft.pub=Credit+Suisse&rft.date=2013-10&rft_id=https%3A%2F%2Fpublications.credit-suisse.com%2Ftasks%2Frender%2Ffile%2F%3FfileID%3DBCDB1364-A105-0560-1332EC9100FF5C83&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTanizaki2004" class="citation book cs1">Tanizaki, Hisashi (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pOGAUcn13fMC"><i>Computational Methods in Statistics and Econometrics</i></a>. CRC Press. p. 133. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780824750886" title="Special:BookSources/9780824750886"><bdi>9780824750886</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Methods+in+Statistics+and+Econometrics&rft.pages=133&rft.pub=CRC+Press&rft.date=2004&rft.isbn=9780824750886&rft.aulast=Tanizaki&rft.aufirst=Hisashi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpOGAUcn13fMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=34" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM._O._Lorenz1905" class="citation journal cs1">M. O. Lorenz (1905). "Methods of measuring the concentration of wealth". <i><a href="/wiki/Publications_of_the_American_Statistical_Association" class="mw-redirect" title="Publications of the American Statistical Association">Publications of the American Statistical Association</a></i>. <b>9</b> (70): 209–19. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1905PAmSA...9..209L">1905PAmSA...9..209L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2276207">10.2307/2276207</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/2276207">2276207</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:154048722">154048722</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+of+the+American+Statistical+Association&rft.atitle=Methods+of+measuring+the+concentration+of+wealth&rft.volume=9&rft.issue=70&rft.pages=209-19&rft.date=1905&rft_id=info%3Adoi%2F10.2307%2F2276207&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154048722%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2276207%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1905PAmSA...9..209L&rft.au=M.+O.+Lorenz&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPareto1965" class="citation book cs1">Pareto, Vilfredo (1965). Librairie Droz (ed.). <i>Ecrits sur la courbe de la répartition de la richesse</i>. Œuvres complètes : T. III. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9782600040211" title="Special:BookSources/9782600040211"><bdi>9782600040211</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ecrits+sur+la+courbe+de+la+r%C3%A9partition+de+la+richesse&rft.series=%C5%92uvres+compl%C3%A8tes+%3A+T.+III&rft.pages=48&rft.date=1965&rft.isbn=9782600040211&rft.aulast=Pareto&rft.aufirst=Vilfredo&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPareto1895" class="citation journal cs1">Pareto, Vilfredo (1895). "La legge della domanda". <i>Giornale degli Economisti</i>. <b>10</b>: 59–68.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Giornale+degli+Economisti&rft.atitle=La+legge+della+domanda&rft.volume=10&rft.pages=59-68&rft.date=1895&rft.aulast=Pareto&rft.aufirst=Vilfredo&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPareto1896" class="citation journal cs1">Pareto, Vilfredo (1896). "Cours d'économie politique". <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F000271629700900314">10.1177/000271629700900314</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:143528002">143528002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Cours+d%27%C3%A9conomie+politique&rft.date=1896&rft_id=info%3Adoi%2F10.1177%2F000271629700900314&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143528002%23id-name%3DS2CID&rft.aulast=Pareto&rft.aufirst=Vilfredo&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pareto_distribution&action=edit&section=35" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Pareto_distribution">"Pareto distribution"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pareto+distribution&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPareto_distribution&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Pareto_distribution"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ParetoDistribution.html">"Pareto distribution"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Pareto+distribution&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FParetoDistribution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAabergé2005" class="citation cs1">Aabergé, Rolf (May 2005). "Gini's Nuclear Family". <a rel="nofollow" class="external text" href="http://www3.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf"><i>International Conference to Honor Two Eminent Social Scientists</i></a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Gini%27s+Nuclear+Family&rft.btitle=International+Conference+to+Honor+Two+Eminent+Social+Scientists&rft.date=2005-05&rft.aulast=Aaberg%C3%A9&rft.aufirst=Rolf&rft_id=http%3A%2F%2Fwww3.unisi.it%2Feventi%2FGiniLorenz05%2F25%2520may%2520paper%2FPAPER_Aaberge.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrovellaBestavros1997" class="citation conference cs1"><a href="/wiki/Mark_Crovella" title="Mark Crovella">Crovella, Mark E.</a>; Bestavros, Azer (December 1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304190612/http://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf"><i>Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes</i></a> <span class="cs1-format">(PDF)</span>. IEEE/ACM Transactions on Networking. Vol. 5. pp. 835–846. Archived from <a rel="nofollow" class="external text" href="https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-02-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Self-Similarity+in+World+Wide+Web+Traffic%3A+Evidence+and+Possible+Causes&rft.pages=835-846&rft.date=1997-12&rft.aulast=Crovella&rft.aufirst=Mark+E.&rft.au=Bestavros%2C+Azer&rft_id=https%3A%2F%2Fwww.cs.bu.edu%2F~crovella%2Fpaper-archive%2Fself-sim%2Fjournal-version.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APareto+distribution" class="Z3988"></span></li></ul> <ul><li><a rel="nofollow" class="external text" href="http://www.csee.usf.edu/~kchriste/tools/syntraf1.c">syntraf1.c</a> is a <a href="/wiki/C_program" class="mw-redirect" title="C program">C program</a> to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" 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 <br />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 <br />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 <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />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 <br />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 <br />semi-infinite <br />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"><i>F</i></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 <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a class="mw-selflink selflink">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 <br />on the whole <br />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 <i>z</i></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 <i>q</i></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 <i>S<sub>U</sub></i></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 <i>t</i></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 <i>t</i></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 <br />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 <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-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 <br />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-<br />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 <br />(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><span class="nobold"><i>Discrete: </i></span></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><span class="nobold"><i>Continuous: </i></span></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 <i>t</i></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><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</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 <i>t</i></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><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></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><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></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> <br />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><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></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 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Pareto distribution - Wikipedia