In this article, we propose methods for expanding generating functions effectively. We develop the formulae for the study of the generating functions systematically and show that the generating functions can be expressed in terms of the Bell polynomials. Several examples associated with waiting time distributions are given in order to illustrate how our theoretical results are employed for the investigation of negative binomial distributions. Finally, we address parameter estimation problems in the shifted negative binomial distributions of order k with two parameters. The methodology will be useful for a systematic study of generating functions to develop a framework of discrete distribution theory.

generating function, Bell polynomials, rational generating function, waiting time, negative binomial distribution, moment estimation, reliability theory, point process.