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 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 = Var 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 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 are non-negative integer-valued i.i.d random variables with, then this compound Poisson distribution is named discrete compound Poisson distribution . We say that the discrete random variable satisfying probability generating function characterization has a discrete compound Poisson 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. The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount. When some are non-negative, it is the discrete pseudo compound Poisson distribution. We define that any discrete random variable satisfying probability generating function characterization has a discrete pseudo compound Poisson distribution with parameters.
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. 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:
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. There has been applications to insurance claims and x-ray computed tomography.
