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class="vector-toc-numb">3</span> <span>Relationship to other distributions</span> </div> </a> <ul id="toc-Relationship_to_other_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generating_random_deviates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generating_random_deviates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generating random deviates</span> </div> </a> <ul id="toc-Generating_random_deviates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 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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:r1247679731">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title"><i>q</i>-exponential distribution</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability density function</div><span typeof="mw:File"><a href="/wiki/File:The_Probability_Density_Function_of_q-exponential_distribution.svg" class="mw-file-description" title="Probability density plots of q-exponential distributions"><img alt="Probability density plots of q-exponential distributions" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/The_Probability_Density_Function_of_q-exponential_distribution.svg/325px-The_Probability_Density_Function_of_q-exponential_distribution.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/The_Probability_Density_Function_of_q-exponential_distribution.svg/488px-The_Probability_Density_Function_of_q-exponential_distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/The_Probability_Density_Function_of_q-exponential_distribution.svg/650px-The_Probability_Density_Function_of_q-exponential_distribution.svg.png 2x" data-file-width="800" data-file-height="600" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q<2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo><</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q<2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caa594082b9ca53836f0fc21aec826a9c0e1263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q<2}"></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 \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"></span> <a href="/wiki/Rate_parameter" class="mw-redirect" title="Rate parameter">rate</a> (<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 x\in [0,\infty ){\text{ for }}q\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty ){\text{ for }}q\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4640aab8831de75db5d83a353dcc69dc6df53de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.52ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty ){\text{ for }}q\geq 1}"></span> <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 x\in \left[0,{\frac {1}{\lambda (1-q)}}\right){\text{ for }}q<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \left[0,{\frac {1}{\lambda (1-q)}}\right){\text{ for }}q<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122bc0113c525046a398184f735f7cb55ba3802c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.043ex; height:6.343ex;" alt="{\displaystyle x\in \left[0,{\frac {1}{\lambda (1-q)}}\right){\text{ for }}q<1}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2-q)\lambda e_{q}^{-\lambda x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>λ</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>x</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2-q)\lambda e_{q}^{-\lambda x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be1bee75ffe534eca20bbc74a96595722849548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.73ex; height:3.176ex;" alt="{\displaystyle (2-q)\lambda e_{q}^{-\lambda x}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-e_{q'}^{-\lambda x/q'}{\text{ where }}q'={\frac {1}{2-q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-e_{q'}^{-\lambda x/q'}{\text{ where }}q'={\frac {1}{2-q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e89dedd846d484a99a58cb16a5e646f25829fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.492ex; height:5.676ex;" alt="{\displaystyle 1-e_{q'}^{-\lambda x/q'}{\text{ where }}q'={\frac {1}{2-q}}}"></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 {\frac {1}{\lambda (3-2q)}}{\text{ for }}q<{\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda (3-2q)}}{\text{ for }}q<{\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d6d81cb5331b09980bdc3d8529b76c29d69469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.349ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\lambda (3-2q)}}{\text{ for }}q<{\frac {3}{2}}}"></span> <br /> Otherwise undefined</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {-q'\ln _{q'}(1/2)}{\lambda }}{\text{ where }}q'={\frac {1}{2-q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> <msub> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mi>λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {-q'\ln _{q'}(1/2)}{\lambda }}{\text{ where }}q'={\frac {1}{2-q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0906091c7893adf4cb0c2cc701753bdc8e18b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.438ex; height:6.343ex;" alt="{\displaystyle {\frac {-q'\ln _{q'}(1/2)}{\lambda }}{\text{ where }}q'={\frac {1}{2-q}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> 0</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q-2}{(2q-3)^{2}(3q-4)\lambda ^{2}}}{\text{ for }}q<{\frac {4}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>q</mi> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mi>q</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q-2}{(2q-3)^{2}(3q-4)\lambda ^{2}}}{\text{ for }}q<{\frac {4}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfeb59a492b6790fd270f2e50d04bb72bc746b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.502ex; height:6.176ex;" alt="{\displaystyle {\frac {q-2}{(2q-3)^{2}(3q-4)\lambda ^{2}}}{\text{ for }}q<{\frac {4}{3}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{5-4q}}{\sqrt {\frac {3q-4}{q-2}}}{\text{ for }}q<{\frac {5}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>5</mn> <mo>−</mo> <mn>4</mn> <mi>q</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>3</mn> <mi>q</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{5-4q}}{\sqrt {\frac {3q-4}{q-2}}}{\text{ for }}q<{\frac {5}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a20441e4578c4547b42d9c4be5284098e370f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.58ex; height:7.509ex;" alt="{\displaystyle {\frac {2}{5-4q}}{\sqrt {\frac {3q-4}{q-2}}}{\text{ for }}q<{\frac {5}{4}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Excess kurtosis</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6{\frac {-4q^{3}+17q^{2}-20q+6}{(q-2)(4q-5)(5q-6)}}{\text{ for }}q<{\frac {6}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>17</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>20</mn> <mi>q</mi> <mo>+</mo> <mn>6</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>q</mi> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>q</mi> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>q</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6{\frac {-4q^{3}+17q^{2}-20q+6}{(q-2)(4q-5)(5q-6)}}{\text{ for }}q<{\frac {6}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd95e98a5863cb85df006ea8f9104bda9e20c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.082ex; height:6.509ex;" alt="{\displaystyle 6{\frac {-4q^{3}+17q^{2}-20q+6}{(q-2)(4q-5)(5q-6)}}{\text{ for }}q<{\frac {6}{5}}}"></span></td></tr></tbody></table> <p>The <b><i>q</i>-exponential distribution</b> is a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> arising from the maximization of the <a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis entropy</a> under appropriate constraints, including constraining the domain to be positive. It is one example of a <a href="/wiki/Tsallis_distribution" title="Tsallis distribution">Tsallis distribution</a>. The <i>q</i>-exponential is a generalization of the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> in the same way that Tsallis entropy is a generalization of standard <a href="/wiki/Entropy_(statistical_thermodynamics)" title="Entropy (statistical thermodynamics)">Boltzmann–Gibbs entropy</a> or <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Shannon entropy</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The exponential distribution is recovered as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\rightarrow 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5979d028913dde69b9ae2571c775951ea2959ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.493ex; height:2.509ex;" alt="{\displaystyle q\rightarrow 1.}"></span> </p><p>Originally proposed by the statisticians <a href="/wiki/George_Box" class="mw-redirect" title="George Box">George Box</a> and <a href="/wiki/David_Cox_(statistician)" title="David Cox (statistician)">David Cox</a> in 1964,<sup id="cite_ref-boxcox_2-0" class="reference"><a href="#cite_note-boxcox-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and known as the reverse <a href="/wiki/Box%E2%80%93Cox_transformation" class="mw-redirect" title="Box–Cox transformation">Box–Cox transformation</a> 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 q=1-\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1-\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d33b8e3fb96fe8d75a199e445fea8f3c74f14c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.173ex; height:2.509ex;" alt="{\displaystyle q=1-\lambda ,}"></span> a particular case of <a href="/wiki/Power_transform" title="Power transform">power transform</a> in statistics. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Characterization">Characterization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=1" title="Edit section: Characterization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Probability_density_function">Probability density function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=2" title="Edit section: Probability density function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>q</i>-exponential distribution has the probability density function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2-q)\lambda e_{q}(-\lambda x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>λ</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>λ</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2-q)\lambda e_{q}(-\lambda x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0b72da3e55f70578dc2cb42e8a912b7c5e1010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.611ex; height:3.009ex;" alt="{\displaystyle (2-q)\lambda e_{q}(-\lambda x)}"></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 e_{q}(x)=[1+(1-q)x]^{1/(1-q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>x</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{q}(x)=[1+(1-q)x]^{1/(1-q)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7212d1a9980d9e5f5903426e902ca335ed8ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.83ex; height:3.509ex;" alt="{\displaystyle e_{q}(x)=[1+(1-q)x]^{1/(1-q)}}"></span></dd></dl> <p>is the <a href="/wiki/Tsallis_statistics#q-exponential" title="Tsallis statistics"><i>q</i>-exponential</a> if <span class="nowrap"><i>q</i> ≠ 1</span>. When <span class="nowrap"><i>q</i> = 1</span>, <i>e</i><sub><i>q</i></sub>(x) is just exp(<i>x</i>). </p> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=3" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a similar procedure to how the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> can be derived (using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive), the <i>q</i>-exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_other_distributions">Relationship to other distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=4" title="Edit section: Relationship to other distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>q</i>-exponential is a special case of the <a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">generalized Pareto distribution</a> 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 \mu =0,\quad \xi ={\frac {q-1}{2-q}},\quad \sigma ={\frac {1}{\lambda (2-q)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =0,\quad \xi ={\frac {q-1}{2-q}},\quad \sigma ={\frac {1}{\lambda (2-q)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d7d5560788811445f20a8a14c2602aa76ad933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.561ex; height:6.176ex;" alt="{\displaystyle \mu =0,\quad \xi ={\frac {q-1}{2-q}},\quad \sigma ={\frac {1}{\lambda (2-q)}}.}"></span></dd></dl> <p>The <i>q</i>-exponential is the generalization of the <a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax distribution</a> (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {2-q}{q-1}},\quad \lambda _{\mathrm {Lomax} }={\frac {1}{\lambda (q-1)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {2-q}{q-1}},\quad \lambda _{\mathrm {Lomax} }={\frac {1}{\lambda (q-1)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38343494cfcde3e09c5d28a60927311d664706cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.165ex; height:6.176ex;" alt="{\displaystyle \alpha ={\frac {2-q}{q-1}},\quad \lambda _{\mathrm {Lomax} }={\frac {1}{\lambda (q-1)}}.}"></span></dd></dl> <p>As the Lomax distribution is a shifted version of the <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>, the <i>q</i>-exponential is a shifted reparameterized generalization of the Pareto. When <span class="nowrap"><i>q</i> > 1</span>, the <i>q</i>-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, 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\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda ){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={\frac {1}{\lambda (q-1)}},\alpha ={\frac {2-q}{q-1}}\right)-x_{m}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">q</mi> </mrow> </mrow> <mtext>-</mtext> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">p</mi> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>Y</mi> <mo>∼</mo> <mrow> <mo>[</mo> <mrow> <mi>Pareto</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda ){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={\frac {1}{\lambda (q-1)}},\alpha ={\frac {2-q}{q-1}}\right)-x_{m}\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632d8ecbba4a7776ac0b5b68461a72f560cfb7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.29ex; height:6.343ex;" alt="{\displaystyle X\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda ){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={\frac {1}{\lambda (q-1)}},\alpha ={\frac {2-q}{q-1}}\right)-x_{m}\right],}"></span></dd></dl> <p>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\sim Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99bee05ade858bf2bceb519ada14600501a9b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.499ex; height:2.176ex;" alt="{\displaystyle X\sim Y.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Generating_random_deviates">Generating random deviates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=5" title="Edit section: Generating random deviates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Random deviates can be drawn using <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>. Given a variable <i>U</i> that is uniformly distributed on the interval (0,1), then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\frac {-q'\ln _{q'}(U)}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> <msub> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">q</mi> </mrow> </mrow> <mtext>-</mtext> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">p</mi> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\frac {-q'\ln _{q'}(U)}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e67db59b1dc3ee2ed1e805b449e4474ccfb59b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.246ex; height:6.009ex;" alt="{\displaystyle X={\frac {-q'\ln _{q'}(U)}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln _{q'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln _{q'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c53aa47a4edd10dad7b35f6a47989375a4c11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.466ex; height:2.843ex;" alt="{\displaystyle \ln _{q'}}"></span> is the <a href="/wiki/Tsallis_statistics#q-logarithm" title="Tsallis statistics"><i>q</i>-logarithm</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 q'={\frac {1}{2-q}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q'={\frac {1}{2-q}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41141c9fe30eba7f21100209a4fa25846c95ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.418ex; height:5.676ex;" alt="{\displaystyle q'={\frac {1}{2-q}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Being a <a href="/wiki/Power_transform" title="Power transform">power transform</a>, it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables. It has been found to be an accurate model for train delays.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.<sup id="cite_ref-sun-16pra2_4-0" class="reference"><a href="#cite_note-sun-16pra2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Constantino_Tsallis" title="Constantino Tsallis">Constantino Tsallis</a></li> <li><a href="/wiki/Tsallis_statistics" title="Tsallis statistics">Tsallis statistics</a></li> <li><a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis entropy</a></li> <li><a href="/wiki/Tsallis_distribution" title="Tsallis distribution">Tsallis distribution</a></li> <li><a href="/wiki/Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)"><i>q</i>-copula</a></li> <li><a href="/wiki/Q-Gaussian" class="mw-redirect" title="Q-Gaussian"><i>q</i>-Gaussian</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=Q-exponential_distribution&action=edit&section=8" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356</span> </li> <li id="cite_note-boxcox-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-boxcox_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBoxCox1964" class="citation journal cs1"><a href="/wiki/George_E._P._Box" title="George E. P. Box">Box, George E. P.</a>; <a href="/wiki/David_Cox_(statistician)" title="David Cox (statistician)">Cox, D. R.</a> (1964). "An analysis of transformations". <i>Journal of the Royal Statistical Society, Series B</i>. <b>26</b> (2): 211–252. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2984418">2984418</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0192611">0192611</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=An+analysis+of+transformations&rft.volume=26&rft.issue=2&rft.pages=211-252&rft.date=1964&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2984418%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D192611%23id-name%3DMR&rft.aulast=Box&rft.aufirst=George+E.+P.&rft.au=Cox%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQ-exponential+distribution" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeith_Briggs_and_Christian_Beck2007" class="citation journal cs1">Keith Briggs and Christian Beck (2007). "Modelling train delays with <i>q</i>-exponential functions". <i>Physica A</i>. <b>378</b> (2): 498–504. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0611097">physics/0611097</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007PhyA..378..498B">2007PhyA..378..498B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physa.2006.11.084">10.1016/j.physa.2006.11.084</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:107475">107475</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A&rft.atitle=Modelling+train+delays+with+q-exponential+functions&rft.volume=378&rft.issue=2&rft.pages=498-504&rft.date=2007&rft_id=info%3Aarxiv%2Fphysics%2F0611097&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A107475%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physa.2006.11.084&rft_id=info%3Abibcode%2F2007PhyA..378..498B&rft.au=Keith+Briggs+and+Christian+Beck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQ-exponential+distribution" class="Z3988"></span></span> </li> <li id="cite_note-sun-16pra2-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-sun-16pra2_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._SunN._A._Sinitsyn2016" class="citation journal cs1">C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution". <i><a href="/wiki/Phys._Rev._A" class="mw-redirect" title="Phys. Rev. A">Phys. Rev. A</a></i>. <b>94</b> (3): 033808. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1606.08430">1606.08430</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhRvA..94c3808S">2016PhRvA..94c3808S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.94.033808">10.1103/PhysRevA.94.033808</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119317114">119317114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+A&rft.atitle=Landau-Zener+extension+of+the+Tavis-Cummings+model%3A+Structure+of+the+solution&rft.volume=94&rft.issue=3&rft.pages=033808&rft.date=2016&rft_id=info%3Aarxiv%2F1606.08430&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119317114%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.94.033808&rft_id=info%3Abibcode%2F2016PhRvA..94c3808S&rft.au=C.+Sun&rft.au=N.+A.+Sinitsyn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQ-exponential+distribution" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Q-exponential_distribution&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Juniper, J. (2007) <a rel="nofollow" class="external text" href="http://e1.newcastle.edu.au/coffee/pubs/wp/2007/07-10.pdf">"The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty"</a>, Centre of Full Employment and Equity, The University of Newcastle, Australia</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=Q-exponential_distribution&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cscs.umich.edu/~crshalizi/notebooks/tsallis.html">Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tsallis" title="Template:Tsallis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tsallis" title="Template talk:Tsallis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tsallis" title="Special:EditPage/Template:Tsallis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tsallis_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Tsallis_statistics" title="Tsallis statistics">Tsallis statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constantino_Tsallis" title="Constantino Tsallis">Constantino Tsallis</a></li> <li><a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis entropy</a></li> <li><a href="/wiki/Tsallis_statistics" title="Tsallis statistics">Tsallis statistics</a></li> <li><a href="/wiki/Tsallis_distribution" title="Tsallis distribution">Tsallis distribution</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution">q-Gaussian distribution</a></li> <li><a class="mw-selflink selflink">q-exponential distribution</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull distribution</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite <br />support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's <i>z</i></a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a 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 class="mw-selflink selflink"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <i>t</i></a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> 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