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Beta negative binomial distribution - Wikipedia
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derivation subsection</span> </button> <ul id="toc-Definition_and_derivation-sublist" class="vector-toc-list"> <li id="toc-PMF_expressed_with_Gamma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PMF_expressed_with_Gamma"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>PMF expressed with Gamma</span> </div> </a> <ul id="toc-PMF_expressed_with_Gamma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-PMF_expressed_with_the_rising_Pochammer_symbol" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PMF_expressed_with_the_rising_Pochammer_symbol"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>PMF expressed with the rising Pochammer symbol</span> </div> </a> <ul id="toc-PMF_expressed_with_the_rising_Pochammer_symbol-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Factorial_Moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Factorial_Moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Factorial Moments</span> </div> </a> <ul id="toc-Factorial_Moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-identifiable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-identifiable"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Non-identifiable</span> </div> </a> <ul id="toc-Non-identifiable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_other_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_other_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Relation to other distributions</span> </div> </a> <ul id="toc-Relation_to_other_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heavy_tailed" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heavy_tailed"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Heavy tailed</span> </div> </a> <ul id="toc-Heavy_tailed-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Beta_geometric_distribution" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beta_geometric_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Beta geometric distribution</span> </div> </a> <ul id="toc-Beta_geometric_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beta_negative_binomial_as_a_Pólya_urn_model" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beta_negative_binomial_as_a_Pólya_urn_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Beta negative binomial as a Pólya urn model</span> </div> </a> <ul id="toc-Beta_negative_binomial_as_a_Pólya_urn_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1259743904">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1);padding:0.3em 0.4em}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Beta Negative Binomial</caption><tbody><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (<a href="/wiki/Real_number" title="Real number">real</a>)<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> <mo>></mo> <mn>0</mn> </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/4a87dc52878418173659e6d0ff8e77ab2897eac9" 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 >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a> (<a href="/wiki/Real_number" title="Real number">real</a>) <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"></span> — number of successes until the experiment is stopped (<a href="/wiki/Integer" title="Integer">integer</a> but can be extended to <a href="/wiki/Real_number" title="Real number">real</a>)</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 k\in \{0,1,2,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \{0,1,2,\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb35fee530f159c49680d3b6ee3cf1c5cb16e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.689ex; height:2.843ex;" alt="{\displaystyle k\in \{0,1,2,\ldots \}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><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 {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3fadbd82549bf3c1d92a441020fa1641f9a4fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.567ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>β<!-- β --></mi> </mrow> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mi>α<!-- α --></mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mtext> </mtext> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dacc94c612225c874abf34b5aa11fab3869ca1cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.094ex; height:7.509ex;" alt="{\displaystyle {\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>β<!-- β --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mi>α<!-- α --></mi> <mo>></mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mtext> </mtext> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07bbf03913728c8b0434dd21078dcf4650097019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.729ex; height:7.843ex;" alt="{\displaystyle {\begin{cases}{\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\frac {(2r+\alpha -1)(2\beta +\alpha -1)}{(\alpha -3){\sqrt {\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>r</mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>β<!-- β --></mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>r</mi> <mi>β<!-- β --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mtext> </mtext> <mi>α<!-- α --></mi> <mo>></mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mtext> </mtext> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\frac {(2r+\alpha -1)(2\beta +\alpha -1)}{(\alpha -3){\sqrt {\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f307ea18239e6b098bcae43bf9db0bd2accb468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:35.05ex; height:9.843ex;" alt="{\displaystyle {\begin{cases}{\frac {(2r+\alpha -1)(2\beta +\alpha -1)}{(\alpha -3){\sqrt {\frac {r\beta (r+\alpha -1)(\beta +\alpha -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> does not exist</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;e^{it}){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo>;</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>r</mi> <mo>;</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;e^{it}){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59b7131b638563a4394eda611e9e6ecdb83d9ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-right: -0.108ex; width:33.201ex; height:7.009ex;" alt="{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;e^{it}){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}\!}"></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 (x)^{(r)}={\frac {\Gamma (x+r)}{\Gamma (x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)^{(r)}={\frac {\Gamma (x+r)}{\Gamma (x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9b4eb4d287aae9c4ff3b4658a2fca0ef14a719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.808ex; height:6.509ex;" alt="{\displaystyle (x)^{(r)}={\frac {\Gamma (x+r)}{\Gamma (x)}}}"></span> is the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7506f01afc6742d7168ab0cb201337ca2a216ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.603ex; height:2.509ex;" alt="{\displaystyle {}_{2}F_{1}}"></span> is the <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>.</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability-generating_function" title="Probability-generating function">PGF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;z){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo>;</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>r</mi> <mo>;</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;z){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1a0dcf561cf3ef33a947ad2b40ee61d7a7caa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.091ex; height:7.009ex;" alt="{\displaystyle {}_{2}F_{1}(\beta ,r;\alpha +\beta +r;z){\frac {(\alpha )^{(r)}}{(\alpha +\beta )^{(r)}}}}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, a <b>beta negative binomial distribution</b> is the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of a <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete</a> <a href="/wiki/Random_variable" title="Random variable">random variable</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 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> equal to the number of failures needed to get <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> successes in a sequence of <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a> <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a>. The probability <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> of success on each trial stays constant within any given experiment but varies across different experiments following a <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a>. Thus the distribution is a <a href="/wiki/Compound_probability_distribution" title="Compound probability distribution">compound probability distribution</a>. </p><p>This distribution has also been called both the <b>inverse Markov-Pólya distribution</b> and the <b>generalized Waring distribution</b><sup id="cite_ref-Johnson_1-0" class="reference"><a href="#cite_note-Johnson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or simply abbreviated as the <b>BNB</b> distribution. A shifted form of the distribution has been called the <b>beta-Pascal distribution</b>.<sup id="cite_ref-Johnson_1-1" class="reference"><a href="#cite_note-Johnson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>If parameters of the beta distribution are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> 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<!-- 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>, and if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mid p\sim \mathrm {NB} (r,p),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∣<!-- ∣ --></mo> <mi>p</mi> <mo>∼<!-- ∼ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mid p\sim \mathrm {NB} (r,p),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d656c4c0c375fa17f129fa608749a7f59238f53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.282ex; height:2.843ex;" alt="{\displaystyle X\mid p\sim \mathrm {NB} (r,p),}"></span></dd></dl> <p>where </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\sim {\textrm {B}}(\alpha ,\beta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∼<!-- ∼ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f1e575c7f711a315ff0145ba5dbdf5185c3ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.313ex; height:2.843ex;" alt="{\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}"></span></dd></dl> <p>then the marginal 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/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> (i.e. the <a href="/wiki/Posterior_predictive_distribution" title="Posterior predictive distribution">posterior predictive distribution</a>) is a beta negative binomial distribution: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\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">B</mi> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737361795a433af8e93ffb5adc06082ce43ef739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.505ex; height:2.843ex;" alt="{\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}"></span></dd></dl> <p>In the above, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {NB} (r,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {NB} (r,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf0057448fa226a315bd8b0a939a1f52d81e294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.45ex; height:2.843ex;" alt="{\displaystyle \mathrm {NB} (r,p)}"></span> is the <a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial distribution</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textrm {B}}(\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {B}}(\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bd89404dacda8eb903827d869204e57a83225d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.308ex; height:2.843ex;" alt="{\displaystyle {\textrm {B}}(\alpha ,\beta )}"></span> is the <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_derivation">Definition and derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=1" title="Edit section: Definition and derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Denoting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecccf782d34b8d135fbcbad905379f008c1fffba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.404ex; height:3.176ex;" alt="{\displaystyle f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )}"></span> the densities of the negative binomial and beta distributions respectively, we obtain the PMF <span class="mwe-math-element"><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(k|\alpha ,\beta ,r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0009a1cf2b0bf818049205798bbf3d0086f087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.882ex; height:2.843ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)}"></span> of the BNB distribution by marginalization: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(k|\alpha ,\beta ,r)\;=&\;\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q\\=&\;\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q\\=&\;{\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>=</mo> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β<!-- β --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(k|\alpha ,\beta ,r)\;=&\;\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q\\=&\;\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q\\=&\;{\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203954530ceac152d4a4b380292f7a9eedf8e1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:61.715ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}f(k|\alpha ,\beta ,r)\;=&\;\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q\\=&\;\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q\\=&\;{\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q\end{aligned}}}"></span></dd></dl> <p>Noting that the integral evaluates to: </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 \int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d56114140e561cbf858b9c40700ca9f301ad67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.262ex; height:6.509ex;" alt="{\displaystyle \int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}}"></span></dd></dl> <p>we can arrive at the following formulas by relatively simple manipulations. </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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is an integer, then the PMF can be written in terms of the <a href="/wiki/Beta_function" title="Beta function">beta function</a>,: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89bb513505053b1413448182e5973f7f05cea470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.59ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"></span>.</dd></dl> <p>More generally, the PMF can be written </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(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c64110c4d9e78d4c4610ceaab50639dec57ae81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.264ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8540b85975f69baa655128f7f27e9054b402b7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.548ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="PMF_expressed_with_Gamma">PMF expressed with Gamma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=2" title="Edit section: PMF expressed with Gamma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the properties of the <a href="/wiki/Beta_function" title="Beta function">Beta function</a>, the PMF with integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> can be rewritten as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ae18813ace5eb6f45a56051c06d9baeddb3b4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.548ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}"></span>.</dd></dl> <p>More generally, the PMF can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0e04fafb962b1773f260389a8bc7185caa2942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.222ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="PMF_expressed_with_the_rising_Pochammer_symbol">PMF expressed with the rising Pochammer symbol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=3" title="Edit section: PMF expressed with the rising Pochammer symbol"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The PMF is often also presented in terms of the <a href="/wiki/Pochammer_symbol" class="mw-redirect" title="Pochammer symbol">Pochammer symbol</a> for integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></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 f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r+k)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>β<!-- β --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r+k)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c7046239233250493272b7a3305d88c679c0d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.532ex; height:6.843ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r+k)}}}}"></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=Beta_negative_binomial_distribution&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Factorial_Moments">Factorial Moments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=5" title="Edit section: Factorial Moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <span class="texhtml"><i>k</i></span>-th <a href="/wiki/Factorial_moment" title="Factorial moment">factorial moment</a> of a beta negative binomial random variable <span class="texhtml"><i>X</i></span> is defined for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e38adc86ddeaba8e4443c5c2332997f0b3d779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.797ex; height:2.176ex;" alt="{\displaystyle k<\alpha }"></span> and in this case is equal to </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} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d2da3ee23b85c596422287cf0c382298b12318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.85ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-identifiable">Non-identifiable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=6" title="Edit section: Non-identifiable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The beta negative binomial is <a href="/wiki/Identifiability" title="Identifiability">non-identifiable</a> which can be seen easily by simply swapping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></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<!-- 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> in the above density or <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> and noting that it is unchanged. Thus <a href="/wiki/Estimation_theory" title="Estimation theory">estimation</a> demands that a <a href="/wiki/Constraint_(mathematics)" title="Constraint (mathematics)">constraint</a> be placed on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- 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> or both. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_other_distributions">Relation to other distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=7" title="Edit section: Relation to other distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The beta negative binomial distribution contains the beta geometric distribution as a special case when either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span> or <span class="mwe-math-element"><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<!-- 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>. It can therefore approximate the <a href="/wiki/Geometric_distribution" title="Geometric distribution">geometric distribution</a> arbitrarily well. It also approximates the negative binomial distribution arbitrary well 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. It can therefore approximate the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> arbitrarily well 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <span class="mwe-math-element"><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<!-- 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> 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Heavy_tailed">Heavy tailed</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=8" title="Edit section: Heavy tailed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a> to the beta function, it can be easily shown that 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 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> </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(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∼<!-- ∼ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>+</mo> <mi>α<!-- α --></mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63c2fb02874ecefc27cc03630014c87f504cdba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.769ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}}"></span></dd></dl> <p>which implies that the beta negative binomial distribution is <a href="/wiki/Heavy_tail_distribution" class="mw-redirect" title="Heavy tail distribution">heavy tailed</a> and that <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> less than or equal 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> do not exist. </p> <div class="mw-heading mw-heading2"><h2 id="Beta_geometric_distribution">Beta geometric distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=9" title="Edit section: Beta geometric distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span>. In this case the pmf simplifies to </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(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>β<!-- β --></mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb32314971bead8adc4f9fe099ee7b7ded3df75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.097ex; height:6.509ex;" alt="{\displaystyle f(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}"></span>.</dd></dl> <p>This distribution is used in some <a href="/wiki/Buy_Till_you_Die" title="Buy Till you Die">Buy Till you Die</a> (BTYD) models. </p><p>Further, when <span class="mwe-math-element"><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<!-- 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> the beta geometric reduces to the <a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon distribution</a>. However, it is more common to define the Yule-Simon distribution in terms of a shifted version of the beta geometric. In particular, 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 BG(\alpha ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼<!-- ∼ --></mo> <mi>B</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim BG(\alpha ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b027fb412699cdc98ce7caafac295fca2526c642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.163ex; height:2.843ex;" alt="{\displaystyle X\sim BG(\alpha ,1)}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+1\sim YS(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mn>1</mn> <mo>∼<!-- ∼ --></mo> <mi>Y</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+1\sim YS(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/017f06f3332863514d04b85d65376944176f4bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.651ex; height:2.843ex;" alt="{\displaystyle X+1\sim YS(\alpha )}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Beta_negative_binomial_as_a_Pólya_urn_model"><span id="Beta_negative_binomial_as_a_P.C3.B3lya_urn_model"></span>Beta negative binomial as a Pólya urn model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=10" title="Edit section: Beta negative binomial as a Pólya urn model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case when the 3 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 r,\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4e6978f669e426aa95a3f8953a6426f0aa6ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.57ex; height:2.009ex;" alt="{\displaystyle r,\alpha }"></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<!-- 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> are positive integers, the Beta negative binomial can also be motivated by an <a href="/wiki/Urn_model" class="mw-redirect" title="Urn model">urn model</a> - or more specifically a basic <a href="/wiki/P%C3%B3lya_urn_model" title="Pólya urn model">Pólya urn model</a>. Consider an urn initially containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> red balls (the stopping color) 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<!-- 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> blue balls. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> red colored balls are drawn. The random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> of observed draws of blue balls are distributed according to 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 \mathrm {BNB} (r,\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {BNB} (r,\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e0b72eaf0b6e9a4ea5d58ebaa3bdcacafeb753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.78ex; height:2.843ex;" alt="{\displaystyle \mathrm {BNB} (r,\alpha ,\beta )}"></span>. Note, at the end of the experiment, the urn always contains the fixed number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa44b219d3f3a209d59b00ec373b826bd3f046f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.377ex; height:2.176ex;" alt="{\displaystyle r+\alpha }"></span> of red balls while containing the random number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b70001ed83cfcb9d7d71c0c07f9e2ffcb6eea9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.152ex; height:2.509ex;" alt="{\displaystyle X+\beta }"></span> blue balls. </p><p>By the non-identifiability property, <span class="mwe-math-element"><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> can be equivalently generated with the urn initially containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> red balls (the stopping color) 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> blue balls and stopping when <span class="mwe-math-element"><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<!-- 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> red balls are observed. </p><p><br /> </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=Beta_negative_binomial_distribution&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial distribution</a></li> <li><a href="/wiki/Dirichlet_negative_multinomial_distribution" title="Dirichlet negative multinomial distribution">Dirichlet negative multinomial distribution</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=Beta_negative_binomial_distribution&action=edit&section=12" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Johnson-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Johnson_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Johnson_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Johnson et al. (1993)</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=Beta_negative_binomial_distribution&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) <i>Univariate Discrete Distributions</i>, 2nd edition, Wiley <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54897-9" title="Special:BookSources/0-471-54897-9">0-471-54897-9</a> (Section 6.2.3)</li> <li>Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions<i>, </i><a href="/wiki/Journal_of_the_Royal_Statistical_Society" title="Journal of the Royal Statistical Society">Journal of the Royal Statistical Society</a><i>, Series B, 18, 202–211 </i></li> <li>Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", <i>Journal of Statistical Planning and Inference</i>, 141 (3), 1153-1160 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jspi.2010.09.020">10.1016/j.jspi.2010.09.020</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Beta_negative_binomial_distribution&action=edit&section=14" title="Edit section: External 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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 class="mw-selflink selflink">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 href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral <i>t</i></a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_variate_beta_distribution" title="Matrix variate beta distribution">Matrix beta</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <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="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a 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