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.
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 conditioned on the number of these variables ():
is a well-defined distribution. In the case N = 0, then the value of Y is 0, so that then Y | N=0 has a degenerate distribution.
The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, where this joint distribution is obtained by combining the conditional distribution Y | N with the marginal distribution of N.
Then, since E(N)=Var(N) if N is Poisson, and dropping the unnecessary subscripts, 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,
An alternative approach is via cumulant generating functions:
When are non-negative discrete random variables, then this compound Poisson distribution is named stuttering Poisson distribution (non-negative discrete compound Poisson distribution) which can model batch arrivals (such as in a bulk queue). If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the "compound Poisson distribution" outlined above is essentially the same as the more general class of compound probability distributions. However, the properties outlined above do depend on its formulation as the sum of a Poisson-distributed number of random variables. The distribution of Y in the case of the compound Poisson distribution with exponentially-distributed summands can be written in an form.
Compound Poisson processes
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 
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.
The discrete compound Poisson distribution is also widely used in actuarial science  for modelling the distribution of the total claim amount of the sum of a Poisson random number of independent and identically distributed claim amounts X1, X2, ..., XN.
||This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (November 2010)|
- Adelson, R. M. (1966). "Compound Poisson Distributions". OR (Operational Research Society) 17 (1): 73–75. JSTOR 3007241.
- Kemp, C. D. (1967). ""Stuttering - Poisson" distributions". Journal of the Statistical and Social Enquiry of Ireland 21 (5): 151–157.
- Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357-364.
- Thompson, C.S. (1984) Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model. J. Climatology, 4, 609 – 619.
- Huiming, Zhang; Lili Chu, Yu Diao (2012). "Some Properties of the Generalized Stuttering Poisson Distribution and its Applications". Studies in Mathematical Sciences 5 (1): 11–26. doi:10.3968/j.sms.1923845220120501.Z0697.