On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

Abstract

We study the distributions of the random variables Sn and Vr related to a sequence of dependent Bernoulli variables, where Sn denotes the number of successes in n trials and Vr the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that Sn and Vr can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of Sn and Vr which enable direct computations of the probabilities.