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class="vector-toc-numb">4.1</span> <span>Gumbel reparameterization tricks</span> </div> </a> <ul id="toc-Gumbel_reparameterization_tricks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Random_variate_generation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Random_variate_generation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Random variate generation</span> </div> </a> <button aria-controls="toc-Random_variate_generation-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 Random variate generation subsection</span> </button> <ul id="toc-Random_variate_generation-sublist" class="vector-toc-list"> <li id="toc-Probability_paper" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_paper"> <div 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Available in 18 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-18" 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">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_de_Gumbel" title="Distribució de Gumbel – Catalan" lang="ca" hreflang="ca" data-title="Distribució de Gumbel" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gumbel-Verteilung" title="Gumbel-Verteilung – German" lang="de" hreflang="de" data-title="Gumbel-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_Gumbel" title="Distribución de Gumbel – Spanish" lang="es" hreflang="es" data-title="Distribución de Gumbel" 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/Gumbel_banaketa" title="Gumbel banaketa – Basque" lang="eu" hreflang="eu" data-title="Gumbel 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_%DA%AF%D8%A7%D9%85%D8%A8%D9%84" title="توزیع گامبل – Persian" lang="fa" hreflang="fa" data-title="توزیع گامبل" 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_Gumbel" title="Loi de Gumbel – French" lang="fr" hreflang="fr" data-title="Loi de Gumbel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%BC%EB%B2%A8_%EB%B6%84%ED%8F%AC" title="굼벨 분포 – Korean" lang="ko" hreflang="ko" data-title="굼벨 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_di_Gumbel" title="Distribuzione di Gumbel – Italian" lang="it" hreflang="it" data-title="Distribuzione di Gumbel" 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%92%D7%90%D7%9E%D7%91%D7%9C" title="התפלגות גאמבל – Hebrew" lang="he" hreflang="he" data-title="התפלגות גאמבל" 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/Gumbel-eloszl%C3%A1s" title="Gumbel-eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Gumbel-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/Gumbel-verdeling" title="Gumbel-verdeling – Dutch" lang="nl" hreflang="nl" data-title="Gumbel-verdeling" 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%82%AC%E3%83%B3%E3%83%99%E3%83%AB%E5%88%86%E5%B8%83" title="ガンベル分布 – Japanese" lang="ja" hreflang="ja" data-title="ガンベル分布" 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_Fishera-Tippetta" title="Rozkład Fishera-Tippetta – Polish" lang="pl" hreflang="pl" data-title="Rozkład Fishera-Tippetta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_de_Gumbel" title="Distribuição de Gumbel – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição de Gumbel" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gumbel_distribution" title="Gumbel distribution – Simple English" lang="en-simple" hreflang="en-simple" data-title="Gumbel 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/Gumbelova_porazdelitev" title="Gumbelova porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Gumbelova porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%94%98%E5%88%A5%E5%88%86%E4%BD%88" title="甘別分佈 – Cantonese" lang="yue" hreflang="yue" data-title="甘別分佈" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%80%BF%E8%B4%9D%E5%B0%94%E5%88%86%E5%B8%83" title="耿贝尔分布 – Chinese" lang="zh" hreflang="zh" data-title="耿贝尔分布" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Particular case of the generalized extreme value 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">Gumbel</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:Gumbel-Density.svg" class="mw-file-description" title="Probability distribution function"><img alt="Probability distribution function" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gumbel-Density.svg/325px-Gumbel-Density.svg.png" decoding="async" width="325" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gumbel-Density.svg/488px-Gumbel-Density.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gumbel-Density.svg/650px-Gumbel-Density.svg.png 2x" data-file-width="600" data-file-height="480" /></a></span></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:Gumbel-Cumulative.svg" class="mw-file-description" title="Cumulative distribution function"><img alt="Cumulative distribution function" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gumbel-Cumulative.svg/325px-Gumbel-Cumulative.svg.png" decoding="async" width="325" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gumbel-Cumulative.svg/488px-Gumbel-Cumulative.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gumbel-Cumulative.svg/650px-Gumbel-Cumulative.svg.png 2x" data-file-width="600" data-file-height="480" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Notation</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 {\text{Gumbel}}(\mu ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gumbel</mtext> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Gumbel}}(\mu ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3879402b0e24a2bdc9a553b69411ad7a6ba5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.601ex; height:2.843ex;" alt="{\displaystyle {\text{Gumbel}}(\mu ,\beta )}"></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 \mu ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7e1ef161a49a22b500d63307460ad92eeb6a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.049ex; height:2.176ex;" alt="{\displaystyle \mu ,}"></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 \beta >0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84c3bb9095bcad7cf2ca44f9de3b771f0e20842d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.24ex; height:2.509ex;" alt="{\displaystyle \beta >0,}"></span> <a href="/wiki/Scale_parameter" title="Scale parameter">scale</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></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 {1}{\beta }}e^{-(z+e^{-z})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>z</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\beta }}e^{-(z+e^{-z})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09e74a68cf5035d35d98a8a7effacd2b3ffd854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.683ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\beta }}e^{-(z+e^{-z})}}"></span><br /> 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 z={\frac {x-\mu }{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\frac {x-\mu }{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90378d05d7a7444b76933c921a89f3283c9d5478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.594ex; height:5.676ex;" alt="{\displaystyle z={\frac {x-\mu }{\beta }}}"></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 e^{-e^{-(x-\mu )/\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>β</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-e^{-(x-\mu )/\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f60c6422d7f80aee3d67d7243a1e9bd7a006716c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.639ex; height:3.176ex;" alt="{\displaystyle e^{-e^{-(x-\mu )/\beta }}}"></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 \mu -\beta \ln(-\ln(p))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu -\beta \ln(-\ln(p))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7091dc8777919e77468d468d0090dc9d0ac3c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.823ex; height:2.843ex;" alt="{\displaystyle \mu -\beta \ln(-\ln(p))}"></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 \mu +\beta \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>β</mi> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +\beta \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2d6f64ea17939021b198079f95435f738b117c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.836ex; height:2.676ex;" alt="{\displaystyle \mu +\beta \gamma }"></span> <br /> 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a></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 \mu -\beta \ln(\ln 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu -\beta \ln(\ln 2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654a2956b87facdbb0bfa5b104410bc0105745b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.199ex; height:2.843ex;" alt="{\displaystyle \mu -\beta \ln(\ln 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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></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 {\frac {\pi ^{2}}{6}}\beta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{6}}\beta ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9d185d0bf4b13a2122cb53e197b25f79021bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.616ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi ^{2}}{6}}\beta ^{2}}"></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 {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}\approx 1.14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>1.14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}\approx 1.14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47ec5aa625e99c988a833c9e9e4c9ccb61f65f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.946ex; height:6.343ex;" alt="{\displaystyle {\frac {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}\approx 1.14}"></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 {12}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e96f74e3fbb17e9227df448483d03fe57f51fb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {12}{5}}}"></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 \ln(\beta )+\gamma +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(\beta )+\gamma +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee3bde4575e78f2f8ecc41c460222e495953a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.186ex; height:2.843ex;" alt="{\displaystyle \ln(\beta )+\gamma +1}"></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"> <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-\beta t)e^{\mu t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (1-\beta t)e^{\mu t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee3fbb85562616eff82eb8f683fa9b83ca8c2b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.337ex; height:3.009ex;" alt="{\displaystyle \Gamma (1-\beta t)e^{\mu t}}"></span></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 \Gamma (1-i\beta t)e^{i\mu t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>β</mi> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>μ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (1-i\beta t)e^{i\mu t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033d2faa557720fe13217a65a737995f19269394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.707ex; height:3.176ex;" alt="{\displaystyle \Gamma (1-i\beta t)e^{i\mu t}}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>Gumbel distribution</b> (also known as the <b>type-I <a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">generalized extreme value distribution</a></b>) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. </p><p>This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to <a href="/wiki/Extreme_value_theory" title="Extreme value theory">extreme value theory</a>, which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p>The Gumbel distribution is a particular case of the <a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">generalized extreme value distribution</a> (also known as the Fisher–Tippett distribution). It is also known as the <i>log-<a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a></i> and the <i>double exponential distribution</i> (a term that is alternatively sometimes used to refer to the <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a>). It is related to the <a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz distribution</a>: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained. </p><p>In the <a href="/wiki/Latent_variable" class="mw-redirect" title="Latent variable">latent variable</a> formulation of the <a href="/wiki/Multinomial_logit" class="mw-redirect" title="Multinomial logit">multinomial logit</a> model — common in <a href="/wiki/Discrete_choice" title="Discrete choice">discrete choice</a> theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed <a href="/wiki/Random_variable" title="Random variable">random variables</a> has a <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a>. </p><p>The Gumbel distribution is named after <a href="/wiki/Emil_Julius_Gumbel" title="Emil Julius Gumbel">Emil Julius Gumbel</a> (1891–1966), based on his original papers describing the distribution.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<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=Gumbel_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the Gumbel distribution 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;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>β</mi> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3247380127552a36c03de3ee8f080ff8356efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.806ex; height:3.676ex;" alt="{\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Standard_Gumbel_distribution">Standard Gumbel distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gumbel_distribution&action=edit&section=2" title="Edit section: Standard Gumbel distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard Gumbel distribution is the case 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 \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}"></span> 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 \beta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73416922785589e358ae2bb10c7633667b4c24a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =1}"></span> with cumulative distribution function </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)=e^{-e^{(-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> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=e^{-e^{(-x)}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5c233f2ad9e6bb476e138214d7de1e04eb8b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.73ex; height:3.676ex;" alt="{\displaystyle F(x)=e^{-e^{(-x)}}\,}"></span></dd></dl> <p>and probability density function </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)=e^{-(x+e^{-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> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=e^{-(x+e^{-x})}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60d234b7346628446fccb05cbbbf2143a32fb82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.987ex; height:3.509ex;" alt="{\displaystyle f(x)=e^{-(x+e^{-x})}.}"></span></dd></dl> <p>In this case the mode is 0, the median 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 -\ln(\ln(2))\approx 0.3665}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0.3665</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(\ln(2))\approx 0.3665}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee00360bd5cc99ce17ff450b75def34bbca342a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.413ex; height:2.843ex;" alt="{\displaystyle -\ln(\ln(2))\approx 0.3665}"></span>, the mean 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 \gamma \approx 0.5772}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>≈</mo> <mn>0.5772</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \approx 0.5772}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3ed3ff10e446e8e7b345028a3ef4f940d7b146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.82ex; height:2.676ex;" alt="{\displaystyle \gamma \approx 0.5772}"></span> (the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>), and the standard deviation 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 \pi /{\sqrt {6}}\approx 1.2825.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.2825.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /{\sqrt {6}}\approx 1.2825.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7cd79430449d6ce3c861efb59a385cb823054a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.797ex; height:3.009ex;" alt="{\displaystyle \pi /{\sqrt {6}}\approx 1.2825.}"></span> </p><p>The <a href="/wiki/Cumulant" title="Cumulant">cumulants</a>, for <i>n</i> > 1, are 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 \kappa _{n}=(n-1)!\zeta (n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{n}=(n-1)!\zeta (n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9875397785e8b8cfd0fc11f49b8a3c5fc3ad3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.456ex; height:2.843ex;" alt="{\displaystyle \kappa _{n}=(n-1)!\zeta (n).}"></span></dd></dl> <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=Gumbel_distribution&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mode is μ, while the median 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 \mu -\beta \ln \left(\ln 2\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu -\beta \ln \left(\ln 2\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74a2531edaeba1353eed1bd8f42105aa08140cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.846ex; height:2.843ex;" alt="{\displaystyle \mu -\beta \ln \left(\ln 2\right),}"></span> and the mean 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 \operatorname {E} (X)=\mu +\gamma \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>γ</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=\mu +\gamma \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713b2dc96aec595e7dde1a95c7f4e18b168fb7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.307ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (X)=\mu +\gamma \beta }"></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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. </p><p>The standard deviation <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" 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 \sigma }"></span> 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 \beta \pi /{\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \pi /{\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07bd24928d61277f952ff524f21ac3c8b97f24f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.925ex; height:3.009ex;" alt="{\displaystyle \beta \pi /{\sqrt {6}}}"></span> hence <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 =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <mo>≈</mo> <mn>0.78</mn> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0d05140bc25280b5e1b6edcb21bf20e669c493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.562ex; height:3.009ex;" alt="{\displaystyle \beta =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .}"></span> <sup id="cite_ref-Oosterbaan_4-0" class="reference"><a href="#cite_note-Oosterbaan-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>At the mode, 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 x=\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=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20a0e3d1103b2704e577d15b7319bf0870e5d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.176ex;" alt="{\displaystyle x=\mu }"></span>, the value 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 F(x;\mu ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a31ff61b3a1e56f43c43d9ee927a7c4e6b2d8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.681ex; height:2.843ex;" alt="{\displaystyle F(x;\mu ,\beta )}"></span> becomes <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^{-1}\approx 0.37}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≈</mo> <mn>0.37</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-1}\approx 0.37}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f4eb49978a6902f9d3c1413430dc9ebf515b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.649ex; height:2.676ex;" alt="{\displaystyle e^{-1}\approx 0.37}"></span>, irrespective of the value 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 \beta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7eccdb23980e06f136fbea999c8e96e7db1b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.979ex; height:2.509ex;" alt="{\displaystyle \beta .}"></span> </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 G_{1},...,G_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1},...,G_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2712b95a94b10bf8d0515d12aed16acd564760b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.966ex; height:2.509ex;" alt="{\displaystyle G_{1},...,G_{k}}"></span> are iid Gumbel random variables with parameters <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 ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65720dcaf4946dbf2388d61b80d0e133686969fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.577ex; height:2.843ex;" alt="{\displaystyle (\mu ,\beta )}"></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 \max\{G_{1},...,G_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max\{G_{1},...,G_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a98d72336727b27efb8c567854c96b3e232fb7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.617ex; height:2.843ex;" alt="{\displaystyle \max\{G_{1},...,G_{k}\}}"></span> is also a Gumbel random variable with parameters <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 +\beta \ln k,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu +\beta \ln k,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bdfcf3dfc03a8ad3b3ad5646bbf88b8dd16bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.674ex; height:2.843ex;" alt="{\displaystyle (\mu +\beta \ln k,\beta )}"></span>. </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 G_{1},G_{2},...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1},G_{2},...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32da5a222d769d904fcab29dbf1b43e2008710c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.545ex; height:2.509ex;" alt="{\displaystyle G_{1},G_{2},...}"></span> are iid random variables such 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 \max\{G_{1},...,G_{k}\}-\beta \ln k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>−</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max\{G_{1},...,G_{k}\}-\beta \ln k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1e0c6e72d592df3c32f201e04d72993fab56e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.714ex; height:2.843ex;" alt="{\displaystyle \max\{G_{1},...,G_{k}\}-\beta \ln k}"></span> has the same distribution 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 G_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ea4f4668b8334c8a7d3d284b0fd22131ef5f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{1}}"></span> for all natural numbers <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></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 G_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ea4f4668b8334c8a7d3d284b0fd22131ef5f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{1}}"></span> is necessarily Gumbel distributed with scale parameter <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> (actually it suffices to consider just two distinct values of k>1 which are coprime). </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=Gumbel_distribution&action=edit&section=4" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>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 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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has a Gumbel distribution, then the conditional distribution of <i>Y</i> = −<i>X</i> given that <i>Y</i> is positive, or equivalently given that <i>X</i> is negative, has a <a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz distribution</a>. The cdf <i>G</i> of <i>Y</i> is related to <i>F</i>, the cdf of <i>X</i>, by the formula <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(y)=P(Y\leq y)=P(X\geq -y\mid X\leq 0)=(F(0)-F(-y))/F(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>≤</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≥</mo> <mo>−</mo> <mi>y</mi> <mo>∣</mo> <mi>X</mi> <mo>≤</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(y)=P(Y\leq y)=P(X\geq -y\mid X\leq 0)=(F(0)-F(-y))/F(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3384255d806cd8b53fb3a39509147cd6c0a11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.195ex; height:2.843ex;" alt="{\displaystyle G(y)=P(Y\leq y)=P(X\geq -y\mid X\leq 0)=(F(0)-F(-y))/F(0)}"></span> for <i>y</i> > 0. Consequently, the densities are related by <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(y)=f(-y)/F(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(y)=f(-y)/F(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c164368afa3bafaaf745ffb3a4082cbbe2eca17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.106ex; height:2.843ex;" alt="{\displaystyle g(y)=f(-y)/F(0)}"></span>: the <a href="/wiki/Gompertz_function" title="Gompertz function">Gompertz density</a> is proportional to a reflected Gumbel density, restricted to the positive half-line.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>If <i>X</i> is an exponentially distributed variable with mean 1, then −log(<i>X</i>) has a standard Gumbel distribution.</li> <li>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 X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b59c5dec28dfa0c35face6be07580f7efe89d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.398ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}"></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 Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe59997015a5033b1ca9b188d27595f876b5f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.045ex; height:2.843ex;" alt="{\displaystyle Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}"></span> are independent, 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-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>−</mo> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d512d8bc3bf0c212c87dc3443cd58f867797f126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.113ex; height:2.843ex;" alt="{\displaystyle X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}"></span> (see <a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic distribution</a>).</li> <li>Despite this, 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 X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0737ce7b4ae791fc5b3710fe9ebaf394a6668f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.573ex; height:2.843ex;" alt="{\displaystyle X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )}"></span> are independent, 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+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>≁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c375e566cb5d5f30d494133730ca4b9ace06c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.442ex; height:2.843ex;" alt="{\displaystyle X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )}"></span>. This can easily be seen by noting 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 E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mi>γ</mi> <mo>≠</mo> <mn>2</mn> <mi>α</mi> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feac6afafc6214ee7ef06c1ae180cf1b0bd34dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.093ex; height:2.843ex;" alt="{\displaystyle E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}"></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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the Euler-Mascheroni constant). Instead, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.<sup id="cite_ref-Marques_6-0" class="reference"><a href="#cite_note-Marques-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <p>Theory related to the <a href="/wiki/Generalized_multivariate_log-gamma_distribution" title="Generalized multivariate log-gamma distribution">generalized multivariate log-gamma distribution</a> provides a multivariate version of the Gumbel distribution. </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=Gumbel_distribution&action=edit&section=5" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:FitGumbelDistr.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-320px-FitGumbelDistr.tif.png" decoding="async" width="320" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-480px-FitGumbelDistr.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-640px-FitGumbelDistr.tif.png 2x" data-file-width="623" data-file-height="465" /></a><figcaption><a href="/wiki/Distribution_fitting" class="mw-redirect" title="Distribution fitting">Distribution fitting</a> with <a href="/wiki/Confidence_band" class="mw-redirect" title="Confidence band">confidence band</a> of a cumulative Gumbel distribution to maximum one-day October rainfalls.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>Gumbel has shown that the maximum value (or last <a href="/wiki/Order_statistic" title="Order statistic">order statistic</a>) in a sample of <a href="/wiki/Random_variable" title="Random variable">random variables</a> following an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> minus the natural logarithm of the sample size <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> approaches the Gumbel distribution as the sample size increases.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Concretely, 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 \rho (x)=e^{-x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=e^{-x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffdf1356bf833cb4f7a8322ff916a977ca574746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.974ex; height:3.009ex;" alt="{\displaystyle \rho (x)=e^{-x}}"></span> be the probability 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 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> 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 Q(x)=1-e^{-x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=1-e^{-x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a78757d72dbbf7f44b7f23f305213f107fce62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.613ex; height:3.009ex;" alt="{\displaystyle Q(x)=1-e^{-x}}"></span> its cumulative distribution. Then the maximum value out 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 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> realizations 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}"> <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 smaller than <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> if and only if all realizations are smaller than <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. So the cumulative distribution of the maximum value <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 {\tilde {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5c5435030c952a58a756e691ea64f60c1bd240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}}"></span> satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\tilde {x}}-\log(N)\leq X)=P({\tilde {x}}\leq X+\log(N))=[Q(X+\log(N))]^{N}=\left(1-{\frac {e^{-X}}{N}}\right)^{N},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>≤</mo> <mi>X</mi> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>X</mi> </mrow> </msup> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\tilde {x}}-\log(N)\leq X)=P({\tilde {x}}\leq X+\log(N))=[Q(X+\log(N))]^{N}=\left(1-{\frac {e^{-X}}{N}}\right)^{N},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0383cf8c3cef0665dc276748ee575fde74aced8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.482ex; height:6.843ex;" alt="{\displaystyle P({\tilde {x}}-\log(N)\leq X)=P({\tilde {x}}\leq X+\log(N))=[Q(X+\log(N))]^{N}=\left(1-{\frac {e^{-X}}{N}}\right)^{N},}"></span></dd></dl> <p>and, for large <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>, the right-hand-side converges 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 e^{-e^{(-X)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-e^{(-X)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7aff7cdab40610a3da3e1ef325bcc4b017c687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.385ex; height:3.176ex;" alt="{\displaystyle e^{-e^{(-X)}}.}"></span> </p><p>In <a href="/wiki/Hydrology" title="Hydrology">hydrology</a>, therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,<sup id="cite_ref-Oosterbaan_4-1" class="reference"><a href="#cite_note-Oosterbaan-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and also to describe droughts.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Gumbel has also shown that the <a href="/wiki/Estimator" title="Estimator">estimator</a> <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num"><i>r</i></span>⁄<span class="den">(<i>n</i>+1)</span></span> for the probability of an event — where <i>r</i> is the rank number of the observed value in the data series and <i>n</i> is the total number of observations — is an <a href="/wiki/Unbiased_estimator" class="mw-redirect" title="Unbiased estimator">unbiased estimator</a> of the <a href="/wiki/Cumulative_probability" class="mw-redirect" title="Cumulative probability">cumulative probability</a> around the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> of the distribution. Therefore, this estimator is often used as a <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting position</a>. </p><p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, the Gumbel distribution approximates the number of terms in a random <a href="/wiki/Partition_of_an_integer" class="mw-redirect" title="Partition of an integer">partition of an integer</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as well as the trend-adjusted sizes of maximal <a href="/wiki/Prime_gaps" class="mw-redirect" title="Prime gaps">prime gaps</a> and maximal gaps between <a href="/wiki/Prime_constellations" class="mw-redirect" title="Prime constellations">prime constellations</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>It appears in the <a href="/wiki/Coupon_collector%27s_problem" title="Coupon collector's problem">coupon collector's problem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Gumbel_reparameterization_tricks">Gumbel reparameterization tricks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gumbel_distribution&action=edit&section=6" title="Edit section: Gumbel reparameterization tricks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, the Gumbel distribution is sometimes employed to generate samples from the <a href="/wiki/Categorical_distribution" title="Categorical distribution">categorical distribution</a>. This technique is called "Gumbel-max trick" and is a special example of "<a href="/wiki/Reparameterization_trick" title="Reparameterization trick">reparameterization tricks</a>".<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In detail, 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 (\pi _{1},\ldots ,\pi _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi _{1},\ldots ,\pi _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae86da363c79cddc4b31fbc1921cd7db2de6566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.91ex; height:2.843ex;" alt="{\displaystyle (\pi _{1},\ldots ,\pi _{n})}"></span> be nonnegative, and not all zero, and 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 g_{1},\ldots ,g_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1},\ldots ,g_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38895e0a6e1f24d3784c7a87bbe2f8c3075b18ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.009ex;" alt="{\displaystyle g_{1},\ldots ,g_{n}}"></span> be independent samples of Gumbel(0, 1), then by routine integration,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Pr(j=\arg \max _{i}(g_{i}+\log \pi _{i}))={\frac {\pi _{j}}{\sum _{i}\pi _{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Pr(j=\arg \max _{i}(g_{i}+\log \pi _{i}))={\frac {\pi _{j}}{\sum _{i}\pi _{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d13f4f222d8806ce44d7d6e696d5320c635953" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.351ex; height:5.843ex;" alt="{\displaystyle Pr(j=\arg \max _{i}(g_{i}+\log \pi _{i}))={\frac {\pi _{j}}{\sum _{i}\pi _{i}}}}"></span>That 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 \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Categorical</mtext> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb23fdab3f77ee1f9a0e5f138e0c16d4df6fc1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.525ex; height:6.843ex;" alt="{\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}"></span> </p><p>Equivalently, given any <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_{n}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},...,x_{n}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b6573be195ebafda272e9695753d9be43b9fd5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.62ex; height:2.509ex;" alt="{\displaystyle x_{1},...,x_{n}\in \mathbb {R} }"></span>, we can sample from its <a href="/wiki/Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a> by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Pr(j=\arg \max _{i}(g_{i}+x_{i}))={\frac {e^{x_{j}}}{\sum _{i}e^{x_{i}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Pr(j=\arg \max _{i}(g_{i}+x_{i}))={\frac {e^{x_{j}}}{\sum _{i}e^{x_{i}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015c45b3b06c0221006458164fc7b8a2945dab35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.753ex; height:6.009ex;" alt="{\displaystyle Pr(j=\arg \max _{i}(g_{i}+x_{i}))={\frac {e^{x_{j}}}{\sum _{i}e^{x_{i}}}}}"></span>Related equations include:<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <ul><li>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 x\sim \operatorname {Exp} (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>Exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim \operatorname {Exp} (\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc3e83714f25d9a9bb6cb159d453aeb9c5686b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.696ex; height:2.843ex;" alt="{\displaystyle x\sim \operatorname {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 (-\ln x-\gamma )\sim {\text{Gumbel}}(-\gamma +\ln \lambda ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gumbel</mtext> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>γ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>λ</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\ln x-\gamma )\sim {\text{Gumbel}}(-\gamma +\ln \lambda ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6e6ddab6e536a58ad4d7753f8b90c34ea71256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.485ex; height:2.843ex;" alt="{\displaystyle (-\ln x-\gamma )\sim {\text{Gumbel}}(-\gamma +\ln \lambda ,1)}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Categorical</mtext> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb23fdab3f77ee1f9a0e5f138e0c16d4df6fc1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.525ex; height:6.843ex;" alt="{\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Gumbel}}\left(\log \left(\sum _{i}\pi _{i}\right),1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gumbel</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Gumbel}}\left(\log \left(\sum _{i}\pi _{i}\right),1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a553c3e9529acf4c9b54d1dd750f0d2cf6b78d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.275ex; height:7.509ex;" alt="{\displaystyle \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Gumbel}}\left(\log \left(\sum _{i}\pi _{i}\right),1\right)}"></span>. That is, the Gumbel distribution is a max-stable distribution family.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [\max _{i}(g_{i}+\beta x_{i})]=\log \left(\sum _{i}e^{\beta x_{i}}\right)+\gamma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>γ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [\max _{i}(g_{i}+\beta x_{i})]=\log \left(\sum _{i}e^{\beta x_{i}}\right)+\gamma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99c8458ee82797cad86df826c388d9810950131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.256ex; height:7.509ex;" alt="{\displaystyle \mathbb {E} [\max _{i}(g_{i}+\beta x_{i})]=\log \left(\sum _{i}e^{\beta x_{i}}\right)+\gamma .}"></span></li></ul> <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=Gumbel_distribution&action=edit&section=7" title="Edit section: Random variate generation"><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">Further information: <a href="/wiki/Non-uniform_random_variate_generation" title="Non-uniform random variate generation">Non-uniform random variate generation</a></div> <p>Since the quantile function (inverse <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4007e08239424693d368237cd4e460423e8865f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle Q(p)}"></span>, of a Gumbel distribution 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 Q(p)=\mu -\beta \ln(-\ln(p)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(p)=\mu -\beta \ln(-\ln(p)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0b07bdcde781b9614a3031f35a48cc4d607843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.386ex; height:2.843ex;" alt="{\displaystyle Q(p)=\mu -\beta \ln(-\ln(p)),}"></span></dd></dl> <p>the variate <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 Q(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1bd3335b6c2cd12d7dfb5eab5ec68b3cb0fdaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.43ex; height:2.843ex;" alt="{\displaystyle Q(U)}"></span> has a Gumbel distribution with parameters <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> when the random variate <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is drawn from the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> 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 (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Probability_paper">Probability paper</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gumbel_distribution&action=edit&section=8" title="Edit section: Probability paper"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Gumbel_paper.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Gumbel_paper.JPG/320px-Gumbel_paper.JPG" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Gumbel_paper.JPG/480px-Gumbel_paper.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Gumbel_paper.JPG/640px-Gumbel_paper.JPG 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>A piece of graph paper that incorporates the Gumbel distribution.</figcaption></figure> <p>In pre-software times probability paper was used to picture the Gumbel distribution (see illustration). The paper is based on linearization of the cumulative distribution function <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></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 -\ln[-\ln(F)]={\frac {x-\mu }{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln[-\ln(F)]={\frac {x-\mu }{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc12bde733e40e115ccb991d8a9c4ace8dd8743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.619ex; height:5.676ex;" alt="{\displaystyle -\ln[-\ln(F)]={\frac {x-\mu }{\beta }}}"></span></dd></dl> <p>In the paper the horizontal axis is constructed at a double log scale. The vertical axis is linear. By plotting <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> on the horizontal axis of the paper and 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>-variable on the vertical axis, the distribution is represented by a straight line with a slope 1<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle /\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa09701ccda2eb9836f41207b6d15e287f3c18d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.494ex; height:2.843ex;" alt="{\displaystyle /\beta }"></span>. When <a href="/wiki/Distribution_fitting" class="mw-redirect" title="Distribution fitting">distribution fitting</a> software like <a href="/wiki/CumFreq" title="CumFreq">CumFreq</a> became available, the task of plotting the distribution was made easier. </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=Gumbel_distribution&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">Type-2 Gumbel distribution</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value distribution</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Emil_Julius_Gumbel" title="Emil Julius Gumbel">Emil Julius Gumbel</a></li></ul> <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=Gumbel_distribution&action=edit&section=10" title="Edit section: Notes"><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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">This article uses the Gumbel distribution to model the distribution of the maximum value<i>. </i>To model the minimum value, use the negative of the original values.</span> </li> </ol></div></div> <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=Gumbel_distribution&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 40em;"> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></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="CITEREFGumbel1935" class="citation cs2"><a href="/wiki/Emil_Julius_Gumbel" title="Emil Julius Gumbel">Gumbel, E.J.</a> (1935), <a rel="nofollow" class="external text" href="http://archive.numdam.org/article/AIHP_1935__5_2_115_0.pdf">"Les valeurs extrêmes des distributions statistiques"</a> <span class="cs1-format">(PDF)</span>, <i>Annales de l'Institut Henri Poincaré</i>, <b>5</b> (2): 115–158</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+l%27Institut+Henri+Poincar%C3%A9&rft.atitle=Les+valeurs+extr%C3%AAmes+des+distributions+statistiques&rft.volume=5&rft.issue=2&rft.pages=115-158&rft.date=1935&rft.aulast=Gumbel&rft.aufirst=E.J.&rft_id=http%3A%2F%2Farchive.numdam.org%2Farticle%2FAIHP_1935__5_2_115_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGumbel+distribution" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.</span> </li> <li id="cite_note-Oosterbaan-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Oosterbaan_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Oosterbaan_4-1"><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="CITEREFOosterbaan1994" class="citation book cs1">Oosterbaan, R.J. (1994). <a rel="nofollow" class="external text" href="http://www.waterlog.info/pdf/freqtxt.pdf">"Chapter 6 Frequency and Regression Analysis"</a> <span class="cs1-format">(PDF)</span>. In Ritzema, H.P. (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/drainageprincipl0000unse/page/175"><i>Drainage Principles and Applications, Publication 16</i></a>. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. <a rel="nofollow" class="external text" href="https://archive.org/details/drainageprincipl0000unse/page/175">175–224</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-70754-33-9" title="Special:BookSources/90-70754-33-9"><bdi>90-70754-33-9</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+6+Frequency+and+Regression+Analysis&rft.btitle=Drainage+Principles+and+Applications%2C+Publication+16&rft.place=Wageningen%2C+The+Netherlands&rft.pages=175-224&rft.pub=International+Institute+for+Land+Reclamation+and+Improvement+%28ILRI%29&rft.date=1994&rft.isbn=90-70754-33-9&rft.aulast=Oosterbaan&rft.aufirst=R.J.&rft_id=http%3A%2F%2Fwww.waterlog.info%2Fpdf%2Ffreqtxt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGumbel+distribution" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillemseKaas2007" class="citation journal cs1">Willemse, W.J.; Kaas, R. 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PMLR: 371–379. <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/1706.04161">1706.04161</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Conference+on+Machine+Learning&rft.atitle=Lost+Relatives+of+the+Gumbel+Trick&rft.pages=371-379&rft.date=2017-07-17&rft_id=info%3Aarxiv%2F1706.04161&rft.aulast=Balog&rft.aufirst=Matej&rft.au=Tripuraneni%2C+Nilesh&rft.au=Ghahramani%2C+Zoubin&rft.au=Weller%2C+Adrian&rft_id=https%3A%2F%2Fproceedings.mlr.press%2Fv70%2Fbalog17a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGumbel+distribution" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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.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 href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <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 class="mw-selflink selflink">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 distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" 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