Numerical Evaluation of Compound Poisson Distributions

Abstract

A useful interpretation is applied to probability generating functions for discrete random variables in order to provide an algorithm for calculating probability mass functions for compound Poisson distributions. This interpretation involves a theorem for geometric transforms which is developed to provide a method for calculating the inverse transform of exp (az ^k). The result of this theorem is then applied to formulate an algorithm for numerically inverting the geometric transform of a compound Poisson process. The evaluation of these probabilities may be carried out to a prespecified point with an accuracy dependent only on the accuracy of the input distributions. The procedure is illustrated by the numerical calculation of the distribution of daily demand probabilities required in the analysis of an inventory system.