Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. 1. Introduction. This paper gives the development of recurreftce relations for moments about the origin and mean of binomial, Poisson, and hyper-geometric frequency distributions from the basis of the moment arrays defined by H. E. Soper.2 This procedure has the advantage of expressing the moments in terms of coefficients which are alike for the three distributions and are de-rivable by a single process, thus providing a degree of formal coordination of the distributions. For both kinds of moments, the coefficients satisfy relatively simple recurrence relations, the use of which leads to recurrence relations for the moments, thus unifying the derivation of these relations for the three distributions. The relations derived in this way for the hypergeometric dis-tribution are apparently new. Apparently new recurrence relations for certain auxiliary coefficients in the expression of the moments about the mean of binomial and Poisson distributions are also given. This course of development involves repetition of a number of well-known results which is justified, it is hoped, by the unification obtained.3