On a Contagious Distribution

Abstract

The purpose of this paper is to discuss the probability distribution that arises when the probability of success at any trial depends linearly upon the number of previous successes. Such a scheme has obvious uses in both biological and economic fields. It will be shown that by assuming a simple linear relationship between the number of previous successes and the probability of success in the next trial, we can derive a distribution that is reasonably easy to handle, provides as good a fit as more usual distributions, and has parameters which are capable of easy physical interpretation. Moreover, for appropriate values of the parameters the negative binomial and the Gram-Charlier systems can be shown to be close approximations.