Bernoulli Sums: The only random variables that count

TL;DR

This paper introduces a new multinomial theorem for commutative idempotents, leading to novel formulas for moments and generating functions of sums of Bernoulli variables, applicable to many classic distributions and problems.

Contribution

It develops a new theoretical framework for calculating moments of sums of Bernoulli variables, unifying and extending results across various distributions.

Findings

Derived new formulas for moments of Bernoulli sums.

Applied results to classic distributions like binomial, Poisson, and hypergeometric.

Provided expressions for moments in terms of tail probabilities.

Abstract

A novel multinomial theorem for commutative idempotents is shown to lead to new results about the moments, central moments, factorial moments, and their generating functions for any random variable $X = \sum_{i} Y_i $ expressible as a sum of Bernoulli indicator random variables $Y_i$. The resulting expressions are functions of the expectation of products of the Bernoulli indicator random variables. These results are used to derive novel expressions for the various moments in a large number of familiar examples and classic problems including: the binomial, hypergeometric, Poisson limit of the binomial, Poisson, Conway-Maxwell-Poisson binomial, Ideal Soliton, and Benford distributions as well as the empty urns and the matching problems. Other general results include expressions for the moments of an arbitrary count random variable in terms of its upper tail probabilities.