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vector-toc-level-2"> <a class="vector-toc-link" href="#Convolution"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Convolution</span> </div> </a> <ul id="toc-Convolution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Related distributions</span> </div> </a> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_process" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stochastic_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Stochastic process</span> </div> </a> <ul id="toc-Stochastic_process-sublist" class="vector-toc-list"> </ul> </li> <li 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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">Normal-inverse Gaussian (NIG)</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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> <a href="/wiki/Location_parameter" title="Location parameter">location</a> (<a href="/wiki/Real_number" title="Real number">real</a>)<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> tail heaviness (real)<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> asymmetry parameter (real)<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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> <a href="/wiki/Scale_parameter" title="Scale parameter">scale parameter</a> (real)<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 \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19a4768de11609f9dd9cfde5d34f5cdbcb5a69a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.458ex; height:4.843ex;" alt="{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}"></span></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 (-\infty ;+\infty )\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>;</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (-\infty ;+\infty )\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6240073d5749fc83627bb4f6de7e6a7b8fddbd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:15.057ex; height:2.843ex;" alt="{\displaystyle x\in (-\infty ;+\infty )\!}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>δ</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>γ</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6588f2dcac8b0ac73f3a53cb959f82dafc47becc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.416ex; height:7.509ex;" alt="{\displaystyle {\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}}"></span> <br /><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 K_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d33b5cc4537e660f8ae20089fcab7f25936705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.883ex; height:2.843ex;" alt="{\displaystyle K_{j}}"></span> denotes a modified <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the second kind<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></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +\delta \beta /\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mi>δ</mi> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +\delta \beta /\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a523e33cb96acd6efec1d80e48612fc6fadc61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle \mu +\delta \beta /\gamma }"></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 \delta \alpha ^{2}/\gamma ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \alpha ^{2}/\gamma ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2138e91ee7eeefd76d9076de780c9a71fb859e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.087ex; height:3.176ex;" alt="{\displaystyle \delta \alpha ^{2}/\gamma ^{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 3\beta /(\alpha {\sqrt {\delta \gamma }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>δ</mi> <mi>γ</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\beta /(\alpha {\sqrt {\delta \gamma }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54c1502c4e4c1221aef0768adad550de68db7bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.589ex; height:3.509ex;" alt="{\displaystyle 3\beta /(\alpha {\sqrt {\delta \gamma }})}"></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 3(1+4\beta ^{2}/\alpha ^{2})/(\delta \gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3(1+4\beta ^{2}/\alpha ^{2})/(\delta \gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aca1f732b1f0735338a02e33c21331c0876e9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.515ex; height:3.176ex;" alt="{\displaystyle 3(1+4\beta ^{2}/\alpha ^{2})/(\delta \gamma )}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>z</mi> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3df2756c335eb932be679d2d946a80243b9758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.453ex; height:3.843ex;" alt="{\displaystyle e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>μ</mi> <mi>z</mi> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>+</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459bc37820ea77f068366df8515b0719760307e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.588ex; height:3.843ex;" alt="{\displaystyle e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}}"></span></td></tr></tbody></table> <p>The <b>normal-inverse Gaussian distribution</b> (<b>NIG</b>, also known as the <b>normal-Wald distribution</b>) is a <a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">continuous probability distribution</a> that is defined as the <a href="/wiki/Normal_variance-mean_mixture" title="Normal variance-mean mixture">normal variance-mean mixture</a> where the mixing density is the <a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">inverse Gaussian distribution</a>. The NIG distribution was noted by Blaesild in 1977 as a subclass of the <a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">generalised hyperbolic distribution</a> discovered by <a href="/wiki/Ole_Barndorff-Nielsen" title="Ole Barndorff-Nielsen">Ole Barndorff-Nielsen</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the next year Barndorff-Nielsen published the NIG in another paper.<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 was introduced in the <a href="/wiki/Mathematical_finance" title="Mathematical finance">mathematical finance</a> literature in 1997.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <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=Normal-inverse_Gaussian_distribution&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Moments">Moments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=2" title="Edit section: Moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Linear_transformation">Linear transformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=3" title="Edit section: Linear transformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This class is closed under <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>, since it is a particular case of the <a href="/wiki/Generalized_hyperbolic_distribution" class="mw-redirect" title="Generalized hyperbolic distribution">Generalized hyperbolic distribution</a>, which has the same property. 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 {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>δ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796856515cab0615a82930ec4911cedca5c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.976ex; height:3.009ex;" alt="{\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}"></span></dd></dl> <p>then<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mi>δ</mi> <mo>,</mo> <mi>a</mi> <mi>μ</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db61d77890a431148cbd67acaa09460e057fe1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.248ex; height:6.176ex;" alt="{\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Summation">Summation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=4" title="Edit section: Summation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This class is <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinitely divisible</a>, since it is a particular case of the <a href="/wiki/Generalized_hyperbolic_distribution" class="mw-redirect" title="Generalized hyperbolic distribution">Generalized hyperbolic distribution</a>, which has the same property. </p> <div class="mw-heading mw-heading3"><h3 id="Convolution">Convolution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=5" title="Edit section: Convolution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The class of normal-inverse Gaussian distributions is closed under <a href="/wiki/Convolution_of_probability_distributions" title="Convolution of probability distributions">convolution</a> in the following sense:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"></span> are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a> that are NIG-distributed with the same values of the 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, but possibly different values of the location and scale parameters, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6899621035d3359b9c1c064739b54c7004e220d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{1}}"></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 \delta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797ad61e57939cd394800dfb549e859f144f7d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.087ex; height:2.676ex;" alt="{\displaystyle \delta _{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94005954e390e621d4ecc940bfa3ee399131a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.103ex; height:2.176ex;" alt="{\displaystyle \mu _{2},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28152a70ffd4f5ba1fd1cf6b8f98c02c952a3002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.087ex; height:2.676ex;" alt="{\displaystyle \delta _{2}}"></span>, respectively, 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}+X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}+X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96d604ba472d9c8d3964bfb198de16b68e5f8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.797ex; height:2.509ex;" alt="{\displaystyle X_{1}+X_{2}}"></span> is NIG-distributed with parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> </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/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" 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</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ab677d974cccb0132cac08bd67fc8ac765627e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.979ex; height:2.509ex;" alt="{\displaystyle \beta ,}"></span><span class="mwe-math-element"><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 _{1}+\mu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1}+\mu _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63b3f9c1c00ce3688f617bbecaeb5137de58f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.752ex; height:2.509ex;" alt="{\displaystyle \mu _{1}+\mu _{2}}"></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 \delta _{1}+\delta _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{1}+\delta _{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010b61a92d342bba4a6631ea9c0bdaef9a7ba2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.66ex; height:2.676ex;" alt="{\displaystyle \delta _{1}+\delta _{2}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=6" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</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 N(\mu ,\sigma ^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\mu ,\sigma ^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816ca9ef23cb206393a86e5d871330e361ee9452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.34ex; height:3.176ex;" alt="{\displaystyle N(\mu ,\sigma ^{2}),}"></span> arises as a special case by setting <span class="mwe-math-element"><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,\delta =\sigma ^{2}\alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22fab95b6a05f6d27b8755306235a4c48bb2063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.293ex; height:3.009ex;" alt="{\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,}"></span> and letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0337f6405bed43834d138cd765f090040bc81870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.425ex; height:1.843ex;" alt="{\displaystyle \alpha \rightarrow \infty }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Stochastic_process">Stochastic process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=7" title="Edit section: Stochastic process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (<a href="/wiki/Wiener_process" title="Wiener process">Wiener process</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 W^{(\gamma )}(t)=W(t)+\gamma t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2eb85bd8a5e45ca8b359e2c3cf7dbaa2d806d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.686ex; height:3.343ex;" alt="{\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t}"></span>, we can define the inverse Gaussian process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734e62b6f9e2e61a9aa2ba1855dabbe0e075ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.378ex; height:3.343ex;" alt="{\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.}"></span> Then given a second independent drifting Brownian motion, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>W</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a2048d49269d5ea4e6100a5030940e35c6b337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.805ex; height:3.343ex;" alt="{\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}"></span>, the normal-inverse Gaussian process is the time-changed process <span class="mwe-math-element"><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_{t}=W^{(\beta )}(A_{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}=W^{(\beta )}(A_{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89d30001319ed5795180d9bab54db3e51d1f4be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.188ex; height:3.343ex;" alt="{\displaystyle X_{t}=W^{(\beta )}(A_{t})}"></span>. The process <span class="mwe-math-element"><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(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"></span> at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=1}"></span> has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of <a href="/wiki/L%C3%A9vy_processes" class="mw-redirect" title="Lévy processes">Lévy processes</a>. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="As_a_variance-mean_mixture">As a variance-mean mixture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal-inverse_Gaussian_distribution&action=edit&section=8" title="Edit section: As a variance-mean mixture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {IG}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {IG}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b9d60c57be31c7f185e2fc91c3a5c66b2273bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-left: -0.069ex; width:2.953ex; height:2.343ex;" alt="{\displaystyle {\mathcal {IG}}}"></span> denote the <a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">inverse Gaussian 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 {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span> denote the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b004dc22f5c6e6deaecf340def8e1a77ce57a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.225ex; height:2.843ex;" alt="{\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19a4768de11609f9dd9cfde5d34f5cdbcb5a69a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.458ex; height:4.843ex;" alt="{\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}"></span>; and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>β</mi> <mi>z</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c58b53c68d099c88cb1bfee4a4bb6669600ff22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.297ex; height:3.009ex;" alt="{\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}"></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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> follows the NIG distribution, with parameters, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\delta ,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>δ</mi> <mo>,</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\delta ,\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27233f8a366c6fa20fcaff6c6463d1e12839b631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta ,\delta ,\mu }"></span>. This can be used to generate NIG variates by <a href="/w/index.php?title=Ancestral_sampling&action=edit&redlink=1" class="new" title="Ancestral sampling (page does not exist)">ancestral sampling</a>. It can also be used to derive an <a href="/wiki/EM_algorithm" class="mw-redirect" title="EM algorithm">EM algorithm</a> for <a href="/wiki/Maximum-likelihood" class="mw-redirect" title="Maximum-likelihood">maximum-likelihood</a> estimation of the NIG parameters.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <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=Normal-inverse_Gaussian_distribution&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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">Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 <i>Note: in the literature this function is also referred to as Modified Bessel function of the third kind</i></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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="CITEREFBarndorff-Nielsen1977" class="citation journal cs1">Barndorff-Nielsen, Ole (1977). "Exponentially decreasing distributions for the logarithm of particle size". <i>Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences</i>. <b>353</b> (1674). The Royal Society: 401–409. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1977.0041">10.1098/rspa.1977.0041</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/79167">79167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+Series+A%2C+Mathematical+and+Physical+Sciences&rft.atitle=Exponentially+decreasing+distributions+for+the+logarithm+of+particle+size&rft.volume=353&rft.issue=1674&rft.pages=401-409&rft.date=1977&rft_id=info%3Adoi%2F10.1098%2Frspa.1977.0041&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F79167%23id-name%3DJSTOR&rft.aulast=Barndorff-Nielsen&rft.aufirst=Ole&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal-inverse+Gaussian+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">O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">O. Barndorff-Nielsen, Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 1997</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">S.T Rachev, Handbook of Heavy Tailed Distributions in Finance, Volume 1: Handbooks in Finance, Book 1, North Holland 2003</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaolella2007" class="citation book cs1">Paolella, Marc S (2007). <i>Intermediate Probability: A computational Approach</i>. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intermediate+Probability%3A+A+computational+Approach&rft.pub=John+Wiley+%26+Sons&rft.date=2007&rft.aulast=Paolella&rft.aufirst=Marc+S&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal-inverse+Gaussian+distribution" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarlis2002" class="citation journal cs1">Karlis, Dimitris (2002). "An EM Type Algorithm for ML estimation for the Normal–Inverse Gaussian Distribution". <i>Statistics and Probability Letters</i>. <b>57</b>: 43–52.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+and+Probability+Letters&rft.atitle=An+EM+Type+Algorithm+for+ML+estimation+for+the+Normal%E2%80%93Inverse+Gaussian+Distribution&rft.volume=57&rft.pages=43-52&rft.date=2002&rft.aulast=Karlis&rft.aufirst=Dimitris&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal-inverse+Gaussian+distribution" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Probability_distributions_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite <br />support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a 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 class="mw-selflink selflink">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <i>t</i></a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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