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
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.
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:
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 (or stuttering-Poisson distribution) . 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. 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. 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.
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 with variance power 1<p<2 (proof via comparison of characteristic function (probability theory)). To be more explicit, if
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
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.
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