Some Elementary Results on Poisson Approximation in a Sequence of Bernoulli Trials

In a finite series of independent success-failure trials, the total number of successes has a binomial probability distribution. It is a classical result that this probability distribution is subject to approximation by a Poisson distribution if the number of trials is sufficiently large and the probability of success in a trial is sufficiently small. Recent interest has focused upon the cases of trials which are possibly dependent, or which have possibly differing success probabilities, or which have more than two possible outcomes. Further, in connection with appropriate metrics for measuring the disparity between two probability distributions, the problem of bounds on the error of approximation has received attention. The present article provides a unified treatment of some of the more elementary of these recent developments. The aim is to offer a springboard for heightened utilization of the Poisson approximation in probability analysis, both theoretical and applied, a source for up-to-date teaching on the subject, and an introduction to a field of research that continues to be interesting and fruitful. The paper also briefly discusses some recent varieties of application of the Poisson approximation, in connection with reliability theory, stochastic processes, urn models, covering problems, and chain letter schemes.