On the Distribution of Sum of Independent Positive Binomial Variables

Let X 1, X 2, …, X n be n independent and identically distributed random variables having the positive binomial probability function 1 where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X 1 + X 2 + … +X n . The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.