Poisson Approximation to Power Series Distributions

The purpose of this article is to provide a simple proof for the Poisson approximation to power series distributions.As special cases we obtain the well-known Poisson approximations to the binomial and negative binomial distributions.The proof is based on the continuity theorem for probability generating functions presented in Feller (1968).The result illustrates one justification to use Poisson models as suitable descriptions of phenomena that are the cumulative effect of a large number of improbable events.