 Compound Poisson Distribution
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Compound Poisson Distribution

In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. In the simplest cases, the result can be either a continuous or a discrete distribution.

## Definition

Suppose that

$N\sim \operatorname {Poisson} (\lambda ),$ i.e., N is a random variable whose distribution is a Poisson distribution with expected value ?, and that

$X_{1},X_{2},X_{3},\dots$ are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of $N$ i.i.d. random variables

$Y=\sum _{n=1}^{N}X_{n}$ is a compound Poisson distribution.

In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, and this joint distribution can be obtained by combining the conditional distribution Y | N with the marginal distribution of N.

## Properties

The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus

$\operatorname {E} (Y)=\operatorname {E} \left[\operatorname {E} (Y\mid N)\right]=\operatorname {E} \left[N\operatorname {E} (X)\right]=\operatorname {E} (N)\operatorname {E} (X),$ {\begin{aligned}\operatorname {Var} (Y)&=E\left[\operatorname {Var} (Y\mid N)\right]+\operatorname {Var} \left[E(Y\mid N)\right]=\operatorname {E} \left[N\operatorname {Var} (X)\right]+\operatorname {Var} \left[N\operatorname {E} (X)\right],\\[6pt]&=\operatorname {E} (N)\operatorname {Var} (X)+\left(\operatorname {E} (X)\right)^{2}\operatorname {Var} (N).\end{aligned}} Then, since E(N) = Var(N) if N is Poisson, these formulae can be reduced to

$\operatorname {E} (Y)=\operatorname {E} (N)\operatorname {E} (X),$ $\operatorname {Var} (Y)=E(N)(\operatorname {Var} (X)+{E(X)}^{2})=E(N){E(X^{2})}.$ The probability distribution of Y can be determined in terms of characteristic functions:

$\varphi _{Y}(t)=\operatorname {E} (e^{itY})=\operatorname {E} _{N}(\left(\operatorname {E} (e^{itX}))^{N}\right)=\operatorname {E} ((\varphi _{X}(t))^{N}),\,$ and hence, using the probability-generating function of the Poisson distribution, we have

$\varphi _{Y}(t)={\textrm {e}}^{\lambda (\varphi _{X}(t)-1)}.\,$ An alternative approach is via cumulant generating functions:

$K_{Y}(t)=\ln E[e^{tY}]=\ln \operatorname {E} [\operatorname {E} [e^{tY}\mid N]]=\ln \operatorname {E} [e^{NK_{X}(t)}]=K_{N}(K_{X}(t)).\,$ Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution ? = 1, the cumulants of Y are the same as the moments of X1.[]

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions. And compound Poisson distributions is infinitely divisible by the definition.

## Discrete compound Poisson distribution

When $X_{1},X_{2},X_{3},\dots$ are non-negative integer-valued i.i.d random variables with $P(X_{1}=k)=\alpha _{k},\ (k=1,2,\ldots )$ , then this compound Poisson distribution is named discrete compound Poisson distribution (or stuttering-Poisson distribution) . We say that the discrete random variable $Y$ satisfying probability generating function characterization

$P_{Y}(z)=\sum \limits _{i=0}^{\infty }P(Y=i)z^{i}=\exp \left(\sum \limits _{k=1}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)$ has a discrete compound Poisson(DCP) distribution with parameters $(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left(\sum _{i=1}^{\infty }\alpha _{i}=1,\alpha _{i}\geq 0,\lambda >0\right)$ , which is denoted by

$X\sim {\text{DCP}}(\lambda {\alpha _{1}},\lambda {\alpha _{r}},\ldots )$ Moreover, if $X\sim {\operatorname {DCP} }(\lambda {\alpha _{1}},\ldots ,\lambda {\alpha _{r}})$ , we say $X$ has a discrete compound Poisson distribution of order $r$ . When $r=1,2$ , DCP becomes Poisson distribution and Hermite distribution, respectively. When $r=3,4$ , DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively. Other special cases include: shiftgeometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria-Delbrück distribution in Luria-Delbrück experiment. For more special case of DCP, see the reviews paper and references therein.

Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v. $X$ is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution. It can be shown that the negative binomial distribution is discrete infinitely divisible, i.e., if X has a negative binomial distribution, then for any positive integer n, there exist discrete i.i.d. random variables X1, ..., Xn whose sum has the same distribution that X has. The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution.

This distribution can model batch arrivals (such as in a bulk queue). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.

When some $\alpha _{k}$ are non-negative, it is the discrete pseudo compound Poisson distribution. We define that any discrete random variable $Y$ satisfying probability generating function characterization

$G_{Y}(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}=\exp \left(\sum \limits _{k=1}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)$ has a discrete pseudo compound Poisson distribution with parameters $(\lambda _{1},\lambda _{2},\ldots )=:(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left({\sum \limits _{k=1}^{\infty }{\alpha _{k}}=1,\sum \limits _{k=1}^{\infty }{\left|{\alpha _{k}}\right|}<\infty ,{\alpha _{k}}\in {\mathbb {R} },\lambda >0}\right)$ .

## Compound Poisson Gamma distribution

If X has a gamma distribution, of which the exponential distribution is a special case, then the conditional distribution of Y | N is again a gamma distribution. The marginal distribution of Y can be shown to be a Tweedie distribution with variance power 1<p<2 (proof via comparison of characteristic function (probability theory)). To be more explicit, if

$N\sim \operatorname {Poisson} (\lambda ),$ and

$X_{i}\sim \operatorname {\Gamma } (\alpha ,\beta )$ i.i.d., then the distribution of

$Y=\sum _{i=1}^{N}X_{i}$ is a reproductive exponential dispersion model $ED(\mu ,\sigma ^{2})$ with

{\begin{aligned}\operatorname {E} [Y]&=\lambda {\frac {\alpha }{\beta }}=:\mu ,\\\operatorname {Var} [Y]&=\lambda {\frac {\alpha (1+\alpha )}{\beta ^{2}}}=:\sigma ^{2}\mu ^{p}.\end{aligned}} The mapping of parameters Tweedie parameter $\mu ,\sigma ^{2},p$ to the Poisson and Gamma parameters $\lambda ,\alpha ,\beta$ is the following:

{\begin{aligned}\lambda &={\frac {\mu ^{2-p}}{(2-p)\sigma ^{2}}},\\\alpha &={\frac {2-p}{p-1}},\\\beta &={\frac {\mu ^{1-p}}{(p-1)\sigma ^{2}}}.\end{aligned}} ## Compound Poisson processes

A compound Poisson process with rate $\lambda >0$ and jump size distribution G is a continuous-time stochastic process $\{\,Y(t):t\geq 0\,\}$ given by

$Y(t)=\sum _{i=1}^{N(t)}D_{i},$ where the sum is by convention equal to zero as long as N(t)=0. Here, $\{\,N(t):t\geq 0\,\}$ is a Poisson process with rate $\lambda$ , and $\{\,D_{i}:i\geq 1\,\}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\{\,N(t):t\geq 0\,\}.\,$ For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.

## Applications

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which has an exponential distribution. Thompson applied the same model to monthly total rainfalls.