special cases include: shift geometric 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. When some are 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 where and , with . 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 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 For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models. 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