Useful moment and CDF formulations for the COM–Poisson distribution

The Poisson distribution is as important for discrete events as the normal distribution is to large sample data. In this note, we discuss a generalized Poisson distribution recently introduced in the statistics literature. We derive—for the first time—exact and explicit expressions for its moments and the cumulative distribution function for the case of over-dispersion. Computational issues are discussed to show the real value of these expressions.