Notation ${\displaystyle \operatorname {Pois} (\lambda ){\underset {\lambda }{\land ))\pi (\lambda )}$ ${\displaystyle \lambda \in (0,\infty )}$ ${\displaystyle k\in \mathbb {N} _{0))$ ${\displaystyle \int \limits _{0}^{\infty }{\frac {\lambda ^{k)){k!))e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda }$ ${\displaystyle \int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )d\lambda }$ ${\displaystyle \int \limits _{0}^{\infty }(\lambda +(\lambda -\mu _{\pi })^{2})\,\,\pi (\lambda )d\lambda }$ ${\displaystyle {\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-{\frac {3}{2))}\,{\Biggl [}\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{3}\,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]))$ ${\displaystyle M_{\pi }(e^{t}-1)}$, with ${\displaystyle M_{\pi ))$ the MGF of π ${\displaystyle M_{\pi }(e^{it}-1)}$ ${\displaystyle M_{\pi }(z-1)}$

A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that a random variable is Poisson distributed, where the rate parameter itself is considered as a random variable. Hence it is a special case of a compound probability distribution. Mixed Poisson distributions can be found in actuarial mathematics as a general approach for the distribution of the number of claims and is also examined as an epidemiological model.[1] It should not be confused with compound Poisson distribution or compound Poisson process.[2]

## Definition

A random variable X satisfies the mixed Poisson distribution with density π(λ) if it has the probability distribution[3]

${\displaystyle \operatorname {P} (X=k)=\int \limits _{0}^{\infty }{\frac {\lambda ^{k)){k!))e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda }$.

If we denote the probabilities of the Poisson distribution by qλ(k), then

${\displaystyle \operatorname {P} (X=k)=\int \limits _{0}^{\infty }q_{\lambda }(k)\,\,\pi (\lambda )\,\mathrm {d} \lambda }$.

## Properties

In the following let ${\displaystyle \mu _{\pi }=\int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )d\lambda \,}$ be the expected value of the density ${\displaystyle \pi (\lambda )\,}$ and ${\displaystyle \sigma _{\pi }^{2}=\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{2}\,\,\pi (\lambda )d\lambda \,}$ be the variance of the density.

### Expected value

The expected value of the Mixed Poisson Distribution is

${\displaystyle \operatorname {E} (X)=\mu _{\pi ))$.

### Variance

For the variance one gets[3]

${\displaystyle \operatorname {Var} (X)=\mu _{\pi }+\sigma _{\pi }^{2))$.

### Skewness

The skewness can be represented as

${\displaystyle \operatorname {v} (X)={\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-{\frac {3}{2))}\,{\Biggl [}\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{3}\,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]))$.

### Characteristic function

The characteristic function has the form

${\displaystyle \varphi _{X}(s)=M_{\pi }(e^{is}-1)\,}$.

Where ${\displaystyle M_{\pi ))$ is the moment generating function of the density.

### Probability generating function

For the probability generating function, one obtains[3]

${\displaystyle m_{X}(s)=M_{\pi }(s-1)\,}$.

### Moment-generating function

The moment-generating function of the mixed Poisson distribution is

${\displaystyle M_{X}(s)=M_{\pi }(e^{s}-1)\,}$.

## Examples

 .mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em)) Theorem — Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.[3] .mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em;text-align:justify}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em))Proof Let ${\displaystyle \pi (\lambda )={\frac {({\frac {p}{1-p)))^{r)){\Gamma (r)))\lambda ^{r-1}e^{-{\frac {p}{1-p))\lambda ))$ be a density of a ${\displaystyle \mathrm {\Gamma } \left(r,{\frac {p}{1-p))\right)}$ distributed random variable. {\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 }\,\mathrm {d} \lambda \\&={\frac {p^{r}(1-p)^{-r)){\Gamma (r)k!))\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda {\frac {1}{1-p))}\,\mathrm {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 }\,\mathrm {d} \lambda } _{=\Gamma (r+k)}\\&={\frac {\Gamma (r+k)}{\Gamma (r)k!))(1-p)^{k}p^{r}\end{aligned))} Therefore we get ${\displaystyle X\sim \operatorname {NegB} (r,p)}$. Theorem — Compounding a Poisson distribution with rate parameter distributed according to a exponential distribution yields a geometric distribution. Proof Let ${\displaystyle \pi (\lambda )={\frac {1}{\beta ))e^{-{\frac {\lambda }{\beta ))))$ be a density of a ${\displaystyle \mathrm {Exp} \left({\frac {1}{\beta ))\right)}$ distributed random variable. Using integration by parts n times yields: {\displaystyle {\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!))\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {1}{\beta ))e^{-{\frac {\lambda }{\beta ))}\mathrm {d} \lambda \\&={\frac {1}{k!\beta ))\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda \left({\frac {1+\beta }{\beta ))\right)}\,\mathrm {d} \lambda \\&={\frac {1}{k!\beta ))\cdot k!\left({\frac {\beta }{1+\beta ))\right)^{k}\int \limits _{0}^{\infty }e^{-\lambda \left({\frac {1+\beta }{\beta ))\right)}\,\mathrm {d} \lambda \\&=\left({\frac {\beta }{1+\beta ))\right)^{k}\left({\frac {1}{1+\beta ))\right)\end{aligned))} Therefore we get ${\displaystyle X\sim \operatorname {Geo\left({\frac {1}{1+\beta ))\right)} }$.

## Table of mixed Poisson distributions

mixing distribution mixed Poisson distribution[4]
gamma negative binomial
exponential geometric
inverse Gaussian Sichel
Poisson Neyman
generalized inverse Gaussian Poisson-generalized inverse Gaussian
generalized gamma Poisson-generalized gamma
generalized Pareto Poisson-generalized Pareto
inverse-gamma Poisson-inverse gamma
log-normal Poisson-log-normal
Lomax Poisson–Lomax
Pareto Poisson–Pareto
Pearson’s family of distributions Poisson–Pearson family
truncated normal Poisson-truncated normal
uniform Poisson-uniform
shifted gamma Delaporte
beta with specific parameter values Yule

## Literature

• Jan Grandell: Mixed Poisson Processes. Chapman & Hall, London 1997, ISBN 0-412-78700-8 .
• Tom Britton: Stochastic Epidemic Models with Inference. Springer, 2019, doi:10.1007/978-3-030-30900-8

## References

1. ^ Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, New York, NY: Springer New York, vol. 156, pp. 37–49, doi:10.1007/978-1-4613-0111-0_3, ISBN 978-0-387-95135-5, retrieved 2022-07-08
2. ^ Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59–S79. doi:10.1017/S051503610001165X. ISSN 0515-0361.
3. ^ a b c d Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Cambridge. pp. 5–7. doi:10.1017/S051503610001165X.((cite web)): CS1 maint: url-status (link)
4. ^ Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review / Revue Internationale de Statistique. 73 (1): 35–58. ISSN 0306-7734.