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}.^{[citation needed]}

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]}

There has been applications to insurance claims^{[15]}^{[16]} and x-ray computed tomography.^{[17]}^{[18]}^{[19]}

## 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.- ^
^{a}^{b}^{c}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**(22): 6789–6802. doi:10.1080/03610926.2014.901375. S2CID 125475756.- ^
^{a}^{b}Kemp, C. D. (1967). ""Stuttering – Poisson" distributions".*Journal of the Statistical and Social Enquiry of Ireland*.**21**(5): 151–157. hdl:2262/6987. **^**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".*Journal of the Operational Research Society*.**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. S2CID 119851120.**^**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. Bibcode:1984JHyd...75..357R. 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. Bibcode:1984IJCli...4..609T. doi:10.1002/joc.3370040605.**^**Jørgensen, Bent; Paes De Souza, Marta C. (January 1994). "Fitting Tweedie's compound poisson model to insurance claims data".*Scandinavian Actuarial Journal*.**1994**(1): 69–93. doi:10.1080/03461238.1994.10413930.**^**Smyth, Gordon K.; Jørgensen, Bent (29 August 2014). "Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling".*ASTIN Bulletin*.**32**(1): 143–157. doi:10.2143/AST.32.1.1020.**^**Whiting, Bruce R. (3 May 2002). "Signal statistics in x-ray computed tomography".*Medical Imaging 2002: Physics of Medical Imaging*. International Society for Optics and Photonics.**4682**: 53–60. Bibcode:2002SPIE.4682...53W. doi:10.1117/12.465601. S2CID 116487704.**^**Elbakri, Idris A.; Fessler, Jeffrey A. (16 May 2003). Sonka, Milan; Fitzpatrick, J. Michael (eds.). "Efficient and accurate likelihood for iterative image reconstruction in x-ray computed tomography".*Medical Imaging 2003: Image Processing*. SPIE.**5032**: 1839–1850. Bibcode:2003SPIE.5032.1839E. doi:10.1117/12.480302. S2CID 12215253.**^**Whiting, Bruce R.; Massoumzadeh, Parinaz; Earl, Orville A.; O'Sullivan, Joseph A.; Snyder, Donald L.; Williamson, Jeffrey F. (24 August 2006). "Properties of preprocessed sinogram data in x-ray computed tomography".*Medical Physics*.**33**(9): 3290–3303. Bibcode:2006MedPh..33.3290W. doi:10.1118/1.2230762. PMID 17022224.