# 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

i.e., *N* is a random variable whose distribution is a Poisson distribution with expected value λ, and that

are identically distributed random variables that are mutually independent and also independent of *N*. Then the probability distribution of the sum of
i.i.d. random variables

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

Then, since E(*N*) = Var(*N*) if *N* is Poisson, these formulae can be reduced to

The probability distribution of *Y* can be determined in terms of characteristic functions:

and hence, using the probability-generating function of the Poisson distribution, we have

An alternative approach is via cumulant generating functions:

Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution *λ* = 1, the cumulants of *Y* are the same as the moments of *X*_{1}.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.[1] And compound Poisson distributions is infinitely divisible by the definition.

## Discrete compound Poisson distribution

When
are non-negative integer-valued i.i.d random variables with
, then this compound Poisson distribution is named **discrete compound Poisson distribution**[2][3][4] (or stuttering-Poisson distribution[5]) . We say that the discrete random variable
satisfying probability generating function characterization

has a discrete compound Poisson(DCP) distribution with parameters , which is denoted by

Moreover, if , we say has a discrete compound Poisson distribution of order . When , DCP becomes Poisson distribution and Hermite distribution, respectively. When , DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively.[6] 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[7] and references therein.

Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v.
is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution.[8] 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 *X*_{1}, ..., *X*_{n} 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[5][9]). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.[3]

When some are non-negative, it is the discrete pseudo compound Poisson distribution.[3] We define that any discrete random variable satisfying probability generating function characterization

has a discrete pseudo compound Poisson distribution with parameters .

## 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[10] with variance power *1<p<2* (proof via comparison of characteristic function (probability theory)). To be more explicit, if

and

i.i.d., then the distribution of

is a reproductive exponential dispersion model with

The mapping of parameters Tweedie parameter to the Poisson and Gamma parameters is the following:

## Compound Poisson processes

A compound Poisson process with rate
and jump size distribution *G* is a continuous-time stochastic process
given by

where the sum is by convention equal to zero as long as *N*(*t*)=0. Here,
is a Poisson process with rate
, and
are independent and identically distributed random variables, with distribution function *G*, which are also independent of
[11]

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.[12]

## 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.[13] Thompson applied the same model to monthly total rainfalls.[14]

## See also

## References

- Lukacs, E. (1970). Characteristic functions. London: Griffin.
- Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, ISBN 978-0-471-27246-5.
- Huiming, Zhang; Yunxiao Liu; Bo Li (2014). "Notes on discrete compound Poisson model with applications to risk theory".
*Insurance: Mathematics and Economics*.**59**: 325–336. doi:10.1016/j.insmatheco.2014.09.012. - Huiming, Zhang; Bo Li (2016). "Characterizations of discrete compound Poisson distributions".
*Communications in Statistics - Theory and Methods*.**45**: 6789–6802. doi:10.1080/03610926.2014.901375. - Kemp, C. D. (1967). ""Stuttering – Poisson" distributions".
*Journal of the Statistical and Social Enquiry of Ireland*.**21**(5): 151–157. - Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering-Poisson distributions. Technometrics, 18(1), 67-73.
- Wimmer, G., Altmann, G. (1996). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical journal, 38(8), 995-1011.
- Feller, W. (1968).
*An Introduction to Probability Theory and its Applications*. Vol. I (3rd ed.). New York: Wiley. - Adelson, R. M. (1966). "Compound Poisson Distributions".
*OR*.**17**(1): 73–75. doi:10.1057/jors.1966.8. - Jørgensen, Bent (1997).
*The theory of dispersion models*. Chapman & Hall. ISBN 978-0412997112. - S. M. Ross (2007).
*Introduction to Probability Models*(ninth ed.). Boston: Academic Press. ISBN 978-0-12-598062-3. - Ata, N.; Özel, G. (2013). "Survival functions for the frailty models based on the discrete compound Poisson process".
*Journal of Statistical Computation and Simulation*.**83**(11): 2105–2116. doi:10.1080/00949655.2012.679943. - Revfeim, K. J. A. (1984). "An initial model of the relationship between rainfall events and daily rainfalls".
*Journal of Hydrology*.**75**(1–4): 357–364. doi:10.1016/0022-1694(84)90059-3. - Thompson, C. S. (1984). "Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model".
*J. Climatology*.**4**(6): 609–619. doi:10.1002/joc.3370040605.