The Poisson Distribution as a Limit for Dependent Binomial Events with Unequal Probabilities

Many operations-research type problems are concerned with a number of events each of which has two possible outcomes (denoted by, say, “success” and “failure”). Very frequently for these problems the quantity of interest can be stated in terms of the probabilities of specified numbers of “successes” for the group of binomial events considered. The evaluation of these probabilities can be greatly simplified if it is known that they can be approximated by a Poisson distribution. For the special case where each event has the same “success” probability and the events are statistically independent, the binomial probability distribution is applicable. It is well known that the Poisson distribution approximates the binomial distribution if the number of events is large and the probability of “success” is small (see, e.g., Hoel, P. G. 1947. Introduction to Mathematical Statistics . Wiley, 50.). In Koopman [Koopman, B. O. 1950. Necessary and sufficient conditions for Poisson's distribution. Proc. Amer. Math. Soc. 1 813--823.] this result was extended to the case of independent binomial events with possibly different probabilities for “success.” This paper presents a further extension in which an event is not required to be statistically independent of all the other events. These results indicate that the Poisson distribution often is applicable to operations-research problems dealing with large numbers of binomial events that have small “success” probabilities. Operations Research , ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.