In this paper, we consider generalised distributions in the context of modelling dispersion but with focus on probability generating function (pgf) which is an important tool in studying statistical properties of a discrete distribution. The aim of this paper is twofold, one is to demonstrate the power of generalising in determination of pgf and two is to show that relationship between power series can naturally lead to pgf of a generalised distribution. Generalised Poisson distributions such as negative binomial, Pólya-Aeppli and Neyman type A are used to model overdispersed (clustered) populations and they all have Poisson as a limiting distribution as contagion breaks down to randomness. In particular, the Pólya-Aeppli distribution served as a typical example in underpinning the inherent power of generalising in determining the pgf. Based on the power series distributions, it is affirmed that negative binomial distribution is a generalised Poisson distribution by utilising the relationship between exponential, logarithmic and negative binomial series and thus readily provides an alternative proof to Quenouille [7] version.

generalized distribution, overdispersion, probability generating function, power of generalizing, power series connection.