The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series

Abstract

For each of the four classical discrete laws --- binomial, Poisson, negative binomial and hypergeometric --- the mean absolute deviation about the mean collapses to a single point mass. We give a common telescoping proof of these identities and interpret the resulting closed forms by size biasing. We then derive complete asymptotic expansions for the Poisson ($\lambda\to\infty$), negative binomial ($r\to\infty$, $p$ fixed) and hypergeometric ($N\to\infty$, margins in fixed proportion) cases, extending the binomial expansion from the companion papers. The coefficients are given in closed Bernoulli-polynomial form and carry the lattice displacement of the mean exactly. At integer means the expansions reduce to sign-alternating odd series, and a single Binet-kernel argument shows that these series envelop the logarithm of the normalised mean absolute deviation: successive partial sums bracket it.