Exploring the Higher-Order Moment Behavior of Certain Probability Distributions by Combinatorial and Asymptotic Analysis

For the binomial, Poisson and normal distributions, the n-th central-raw ratios E(ξ−Eξ)n/Eξn are known to be infinitesimal with distinct patterns. In this paper, the higher-order moments of the negative binomial distribution NB(r,p) and the Gamma distribution Ga(α,β) are derived using combinatorial and asymptotic methods. Consequently, the asymptotic central-raw ratios of these distributions are shown to be non-zero constants: qqpr (where q=1−p) for NB(r,p) and e−α for Ga(α,β) as n→+∞.