On the negative binomial distribution and its generalizations

Abstract

It is shown that the negative binomial distribution NB(r,p) may arise out of an identical but dependent geometric sequence. Using a general characterization result for NB(r,p), based on a non-negative integer (Z(+))-valued sequence, we show that NB(2,p) may arise as the distribution of the sum of Z(+)-valued random variables which are neither geometric nor independent. We show also that NB(r,p) arises, as the distribution of the number of trials for the rth success, based on a sequence of dependent Bernoulli variables. The generalized negative binomial distributions arising out of certain dependent Bernoulli sequences are also investigated. In particular, certain erroneous results in the literature are corrected. (c) 2006