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Mixed Poisson distribution
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<h2 id="definition">Definition</h2> <p>A <a href="/facts/Random_variable/TwTBXnLT">random variable</a> <i>X</i> satisfies the mixed Poisson distribution with density π(<i>λ</i>) if it has the probability distribution<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p> P ( X = k ) = ∫ 0 ∞ λ k k ! e − λ π ( λ ) d λ . {\displaystyle \operatorname {P} (X=k)=\int _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,d\lambda .} </p><p>If we denote the probabilities of the Poisson distribution by <i>q</i><i>λ</i>(<i>k</i>), then </p><p> P ( X = k ) = ∫ 0 ∞ q λ ( k ) π ( λ ) d λ . {\displaystyle \operatorname {P} (X=k)=\int _{0}^{\infty }q_{\lambda }(k)\,\,\pi (\lambda )\,d\lambda .} </p> <h2 id="properties">Properties</h2> <ul><li>The <a href="/facts/Variance/ULBJKXD1">variance</a> is always bigger than the <a href="/facts/Expected_value/1XV0JKL8">expected value</a>. This property is called <a href="/facts/Overdispersion/NSzgPcjq">overdispersion</a>. This is in contrast to the Poisson distribution where mean and variance are the same.</li> <li>In practice, almost only densities of <a href="/facts/Gamma_distribution/lczcdJmw">gamma distributions</a>, <a href="/facts/Log-normal_distribution/V2RXmmlE">logarithmic normal distributions</a> and <a href="/facts/Inverse_Gaussian_distribution/mUPsu4EJ">inverse Gaussian distributions</a> are used as densities π(<i>λ</i>). If we choose the density of the <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a>, we get the <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a>, which explains why this is also called the Poisson gamma distribution.</li></ul> <p>In the following let μ π = ∫ 0 ∞ λ π ( λ ) d λ {\displaystyle \mu _{\pi }=\int _{0}^{\infty }\lambda \,\,\pi (\lambda )\,d\lambda \,} be the expected value of the density π ( λ ) {\displaystyle \pi (\lambda )\,} and σ π 2 = ∫ 0 ∞ ( λ − μ π ) 2 π ( λ ) d λ {\displaystyle \sigma _{\pi }^{2}=\int _{0}^{\infty }(\lambda -\mu _{\pi })^{2}\,\,\pi (\lambda )\,d\lambda \,} be the variance of the density. </p> <h3>Expected value</h3> <p>The <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of the mixed Poisson distribution is </p><p> E ( X ) = μ π . {\displaystyle \operatorname {E} (X)=\mu _{\pi }.} </p> <h3>Variance</h3> <p>For the <a href="/facts/Variance/ULBJKXD1">variance</a> one gets<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p> Var ( X ) = μ π + σ π 2 . {\displaystyle \operatorname {Var} (X)=\mu _{\pi }+\sigma _{\pi }^{2}.} </p> <h3>Skewness</h3> <p>The <a href="/facts/Skewness/rdS8luFa">skewness</a> can be represented as </p><p> v ( X ) = ( μ π + σ π 2 ) − 3 / 2 [ ∫ 0 ∞ ( λ − μ π ) 3 π ( λ ) d λ + μ π ] . {\displaystyle \operatorname {v} (X)={\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-3/2}\,{\Biggl [}\int _{0}^{\infty }(\lambda -\mu _{\pi })^{3}\,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]}.} </p> <h3>Characteristic function</h3> <p>The <a href="/facts/Characteristic_function/remE9gBw">characteristic function</a> has the form </p><p> φ X ( s ) = M π ( e i s − 1 ) . {\displaystyle \varphi _{X}(s)=M_{\pi }(e^{is}-1).\,} </p><p>Where M π {\displaystyle M_{\pi }} is the <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> of the density. </p> <h3>Probability generating function</h3> <p>For the <a href="/facts/Probability_generating_function/gm7Kxx4g">probability generating function</a>, one obtains<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p> m X ( s ) = M π ( s − 1 ) . {\displaystyle m_{X}(s)=M_{\pi }(s-1).\,} </p> <h3>Moment-generating function</h3> <p>The <a href="/facts/Moment-generating_function/eTmk43QU">moment-generating function</a> of the mixed Poisson distribution is </p><p> M X ( s ) = M π ( e s − 1 ) . {\displaystyle M_{X}(s)=M_{\pi }(e^{s}-1).\,} </p> <h2 id="examples">Examples</h2> <p>Theorem—Compounding a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> with rate parameter distributed according to a <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> yields a <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> Proof <p>Let π ( λ ) = ( p 1 − p ) r Γ ( r ) λ r − 1 e − p 1 − p λ {\displaystyle \pi (\lambda )={\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }} be a density of a Γ ( r , p 1 − p ) {\displaystyle \operatorname {\Gamma } \left(r,{\frac {p}{1-p}}\right)} distributed random variable. </p><p> P ( X = k ) = 1 k ! ∫ 0 ∞ λ k e − λ ( p 1 − p ) r Γ ( r ) λ r − 1 e − p 1 − p λ d λ = p r ( 1 − p ) − r Γ ( r ) k ! ∫ 0 ∞ λ k + r − 1 e − λ 1 1 − p d λ = p r ( 1 − p ) − r Γ ( r ) k ! ( 1 − p ) k + r ∫ 0 ∞ λ k + r − 1 e − λ d λ ⏟ = Γ ( r + k ) = Γ ( r + k ) Γ ( r ) k ! ( 1 − p ) k p r {\displaystyle {\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }\,d\lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda {\frac {1}{1-p}}}\,d\lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}(1-p)^{k+r}\underbrace {\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda }\,d\lambda } _{=\Gamma (r+k)}\\&={\frac {\Gamma (r+k)}{\Gamma (r)k!}}(1-p)^{k}p^{r}\end{aligned}}} </p><p>Therefore we get X ∼ NegB ( r , p ) . {\displaystyle X\sim \operatorname {NegB} (r,p).} </p> <p>Theorem—Compounding a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> with rate parameter distributed according to an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> yields a <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a>. </p> Proof <p>Let π ( λ ) = 1 β e − λ β {\displaystyle \pi (\lambda )={\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}} be a density of a Exp ( 1 β ) {\displaystyle \operatorname {Exp} \left({\frac {1}{\beta }}\right)} distributed random variable. Using <a href="/facts/Integration_by_parts/aV6JgBNd">integration by parts</a> n times yields: P ( X = k ) = 1 k ! ∫ 0 ∞ λ k e − λ 1 β e − λ β d λ = 1 k ! β ∫ 0 ∞ λ k e − λ ( 1 + β β ) d λ = 1 k ! β ⋅ k ! ( β 1 + β ) k ∫ 0 ∞ e − λ ( 1 + β β ) d λ = ( β 1 + β ) k ( 1 1 + β ) {\displaystyle {\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}\,d\lambda \\&={\frac {1}{k!\beta }}\int _{0}^{\infty }\lambda ^{k}e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,d\lambda \\&={\frac {1}{k!\beta }}\cdot k!\left({\frac {\beta }{1+\beta }}\right)^{k}\int _{0}^{\infty }e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,d\lambda \\&=\left({\frac {\beta }{1+\beta }}\right)^{k}\left({\frac {1}{1+\beta }}\right)\end{aligned}}} Therefore we get X ∼ G e o ( 1 1 + β ) . {\displaystyle X\sim \operatorname {Geo\left({\frac {1}{1+\beta }}\right)} .} </p> <h2 id="table-of-mixed-poisson-distributions">Table of mixed Poisson distributions</h2> <table><tbody><tr><th>mixing distribution</th><th>mixed Poisson distribution<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></th></tr><tr><td><a href="/facts/Degenerate_distribution/Tj5b5h2z">Dirac</a></td><td>Poisson</td></tr><tr><td><a href="/facts/Gamma_distribution/lczcdJmw">gamma</a>, <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang</a></td><td><a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial</a></td></tr><tr><td><a href="/facts/Exponential_distribution/yY3EdV0u">exponential</a></td><td><a href="/facts/Geometric_distribution/c8wjfaoV">geometric</a></td></tr><tr><td><a href="/facts/Inverse_Gaussian_distribution/mUPsu4EJ">inverse Gaussian</a></td><td><a href="/facts/Generalized_inverse_Gaussian_distribution/7tdENlkE">Sichel</a></td></tr><tr><td><a href="/facts/Poisson_distribution/CvCzXkHr">Poisson</a></td><td>Neyman</td></tr><tr><td><a href="/facts/Generalized_inverse_Gaussian_distribution/7tdENlkE">generalized inverse Gaussian</a></td><td>Poisson-generalized inverse Gaussian</td></tr><tr><td><a href="/facts/Generalized_gamma_distribution/Nhcj3WeE">generalized gamma</a></td><td>Poisson-generalized gamma</td></tr><tr><td><a href="/facts/Generalized_Pareto_distribution/EnP5hFWh">generalized Pareto</a></td><td>Poisson-generalized Pareto</td></tr><tr><td><a href="/facts/Inverse-gamma_distribution/zpyktNyN">inverse-gamma</a></td><td>Poisson-inverse gamma</td></tr><tr><td><a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal</a></td><td>Poisson-log-normal</td></tr><tr><td><a href="/facts/Lomax_distribution/ch9VY1mn">Lomax</a></td><td>Poisson–Lomax</td></tr><tr><td><a href="/facts/Pareto_distribution/PgMDbLrR">Pareto</a></td><td>Poisson–Pareto</td></tr><tr><td><a href="/facts/Pearson_distribution/L0oULaWK">Pearson’s family of distributions</a></td><td>Poisson–Pearson family</td></tr><tr><td><a href="/facts/Truncated_normal_distribution/q5h0zsuF">truncated normal</a></td><td>Poisson-truncated normal</td></tr><tr><td><a href="/facts/Continuous_uniform_distribution/XbnlVljT">uniform</a></td><td>Poisson-uniform</td></tr><tr><td>shifted gamma</td><td><a href="/facts/Delaporte_distribution/wQYgfmw2">Delaporte</a></td></tr><tr><td>beta with specific parameter values</td><td><a href="/facts/Yule%25E2%2580%2593Simon_distribution/xmXNj82N">Yule</a></td></tr></tbody></table> <h2 id="further-reading">Further reading</h2> <ul><li>Grandell, Jan (1997). <i>Mixed Poisson Processes</i>. London: Chapman & Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-78700-8.</li> <li>Britton, Tom (2019). <i>Stochastic Epidemic Models with Inference</i>. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-030-30900-8">10.1007/978-3-030-30900-8</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, Lecture Notes in Statistics, vol. 156, New York, NY: Springer New York, pp. 37–49, doi:10.1007/978-1-4613-0111-0_3, ISBN 978-0-387-95135-5, retrieved 2022-07-08 <a href="978-0-387-95135-5" target="_blank">978-0-387-95135-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59 – S79. doi:10.1017/S051503610001165X. ISSN 0515-0361. <a href="https://doi.org/10.1017%2FS051503610001165X" target="_blank">https://doi.org/10.1017%2FS051503610001165X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506. <a href="https://doi.org/10.1017%2FS051503610001165X" target="_blank">https://doi.org/10.1017%2FS051503610001165X</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506. <a href="https://doi.org/10.1017%2FS051503610001165X" target="_blank">https://doi.org/10.1017%2FS051503610001165X</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506. <a href="https://doi.org/10.1017%2FS051503610001165X" target="_blank">https://doi.org/10.1017%2FS051503610001165X</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506. <a href="https://doi.org/10.1017%2FS051503610001165X" target="_blank">https://doi.org/10.1017%2FS051503610001165X</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review. 73 (1): 35–58. doi:10.1111/j.1751-5823.2005.tb00250.x. ISSN 0306-7734. JSTOR 25472639. S2CID 53637483. <a href="https://www.jstor.org/stable/25472639" target="_blank">https://www.jstor.org/stable/25472639</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>