On the negative binomial distribution and its generalizations

It is shown that the negative binomial distribution NB(r,p)NB(r,p) may arise out of an identical but dependent geometric sequence. Using a general characterization result for NB(r,p)NB(r,p), based on a non-negative integer (Z+)(Z+)-valued sequence, we show that NB(2,p)NB(2,p) may arise as the distribution of the sum of Z+Z+-valued random variables which are neither geometric nor independent. We show also that NB(r,p)NB(r,p) arises, as the distribution of the number of trials for the rrth 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.