mixed Poisson distribution

A random variable X satisfies the mixed Poisson distribution with density {{pi}}(λ) if it has the probability distribution{{Cite journal |last=Willmot |first=Gord |date=2014-08-29 |title=Mixed Compound Poisson Distributions |journal=Astin Bulletin |volume=16 |pages=5–7 |doi=10.1017/S051503610001165X|s2cid=17737506 |doi-access=free }}

$$\backslash operatorname\{P\}(X=k)\; =\; \backslash int\_0^\backslash infty\; \backslash frac\{\backslash lambda^k\}\{k!\}e^\{-\backslash lambda\}\; \backslash ,\backslash ,\backslash pi(\backslash lambda)\backslash ,\; d\backslash lambda.$$

If we denote the probabilities of the Poisson distribution by {{math|q_λ(k)}}, then

$$\backslash operatorname\{P\}(X=k)\; =\; \backslash int\_0^\backslash infty\; q\_\backslash lambda(k)\; \backslash ,\backslash ,\backslash pi(\backslash lambda)\backslash ,\; d\backslash lambda.$$

• The variance is always bigger than the expected value. This property is called overdispersion. This is in contrast to the Poisson distribution where mean and variance are the same.
• In practice, almost only densities of gamma distributions, logarithmic normal distributions and inverse Gaussian distributions are used as densities {{pi}}(λ). If we choose the density of the gamma distribution, we get the negative binomial distribution, which explains why this is also called the Poisson gamma distribution.

In the following let $\backslash mu\_\backslash pi=\backslash int\_0^\backslash infty\; \backslash lambda\; \backslash ,\backslash ,\backslash pi(\backslash lambda)\; \backslash ,\; d\backslash lambda\backslash ,$ be the expected value of the density $\backslash pi(\backslash lambda)\backslash ,$ and $\backslash sigma\_\backslash pi^2\; =\; \backslash int\_0^\backslash infty\; (\backslash lambda-\backslash mu\_\backslash pi)^2\; \backslash ,\backslash ,\backslash pi(\backslash lambda)\; \backslash ,\; d\backslash lambda\backslash ,$ be the variance of the density.

The expected value of the mixed Poisson distribution is

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

For the variance one gets

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

The skewness can be represented as

$$\backslash operatorname\{v\}(X)\; =\; \backslash Bigl(\backslash mu\_\backslash pi+\backslash sigma\_\backslash pi^2\backslash Bigr)^\{-3/2\}\; \backslash ,\backslash Biggl[\backslash int\_0^\backslash infty(\backslash lambda-\backslash mu\_\backslash pi)^3\backslash ,\backslash pi(\backslash lambda)\backslash ,d\{\backslash lambda\}+\backslash mu\_\backslash pi\backslash Biggr].$$

The characteristic function has the form

$$\backslash varphi\_X(s)\; =\; M\_\backslash pi(e^\{is\}-1).\backslash ,$$

Where $M\_\backslash pi$ is the moment generating function of the density.

For the probability generating function, one obtains

$$m\_X(s)\; =\; M\_\backslash pi(s-1).\backslash ,$$

The moment-generating function of the mixed Poisson distribution is

$$M\_X(s)\; =\; M\_\backslash pi(e^s-1).\backslash ,$$

{{Math theorem|Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.}}

{{Math proof|Let $\backslash pi(\backslash lambda)=\backslash frac\{(\backslash frac\{p\}\{1-p\})^r\}\{\backslash Gamma(r)\}\; \backslash lambda^\{r-1\}\; e^\{-\backslash frac\{p\}\{1-p\}\backslash lambda\}$ be a density of a $\backslash operatorname\{\backslash Gamma\}\backslash left(r,\backslash frac\{p\}\{1-p\}\backslash right)$ distributed random variable.

$\backslash begin\{align\}$

\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{align}

Therefore we get $X\backslash sim\backslash operatorname\{NegB\}(r,p).$}}

{{Math theorem|Compounding a Poisson distribution with rate parameter distributed according to an exponential distribution yields a geometric distribution.}}

{{Math proof|Let $\backslash pi(\backslash lambda)=\backslash frac1\backslash beta\; e^\{-\backslash frac\; \backslash lambda\backslash beta\}$ be a density of a $\backslash operatorname\{Exp\}\backslash left(\backslash frac1\backslash beta\backslash right)$ distributed random variable. Using integration by parts {{mvar|n}} times yields:

$$\backslash begin\{align\}$$

\operatorname{P}(X=k)&=\frac{1}{k!}\int_0^\infty \lambda^k e^{-\lambda} \frac1\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{align}

Therefore we get $X\backslash sim\backslash operatorname\{Geo\backslash left(\backslash frac\{1\}\{1+\backslash beta\}\backslash right)\}.$}}

{{reflist}}

• {{cite book |first=Jan |last=Grandell |title=Mixed Poisson Processes |publisher=Chapman & Hall |location=London |year=1997 |isbn=0-412-78700-8 }}
• {{cite book |first=Tom |last=Britton |title=Stochastic Epidemic Models with Inference |publisher=Springer |year=2019 |isbn= |doi=10.1007/978-3-030-30900-8 }}

{{Probability distributions}}

Category:Discrete distributions

Category:Types of probability distributions