Factorial Distributions

Abstract

This note deals with a two-parameter family of discrete distributions which has interesting moment properties. We are indebted to Arthur Schleifer, Jr. for having informed us that the distributions derived below by formal methods are integer parameter cases of what are called Beta-Pascal in Raiffa and Schlaifer (1961), p. 238. We were led to the present family by searching for discrete distributions to be used in numerical work with inventory problems. In particular, we were interested in enlarging our discrete repertoire (binomial, Poisson, geometric, negative binomial, etc.) in two directions. First, we sought convenient closed form expressions for various associated probabilities and expectations. This would of course simplify computation of penalty functions and optimal inventory levels. Second, we sought representation of rather extreme behavior so as to subject our theoretical formulations to stresses such as large dispersions about average future demand. It turns out that the present family meets both criteria. Not only is it highly tractable but corresponding to each value $r = 0, 1, 2, \cdots$ there is a member distribution possessing its $r$th moment but whose $(r + 1)$st moment fails to exist. Our approach is entirely analogous to the following formal "development" of geometric distributions. We start with the observation that $\sum^\infty_{i = 1} q^i = 1/(1 - q)$ is a convergent series of positive terms for $0 < q < 1$. Then we normalize to unit sum and consider the series to represent a probability distribution: $\sum^\infty_{i = 1} pq^i = 1$ where $p = 1 - q$. If this is done, we obtain the one-parameter geometric distribution whose mode is zero, whose mean equals $q/p$ and whose variance-to-mean ratio equals ($1 + mean$). Our present effort similarly begins with a convergent series (of factorial power functions) and we find a two-parameter family where, interestingly enough, not only is the mode zero but the variance-to-mean ratio, when it exists, approaches the quantity $(1 + \text{mean})$.