Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions

TL;DR

This paper derives sharp, simple bounds for the raw moments of Binomial and Poisson distributions, improving previous bounds by an exponential factor, with tight asymptotics for small moment orders.

Contribution

It introduces new inequalities for raw moments of sub-Poissonian variables, providing tighter bounds than existing results.

Findings

Bounds are tight for small k

Inequalities improve previous bounds exponentially

Applicable to Binomial and Poisson distributions

Abstract

We prove the inequality $E[(X/\mu)^k] \le (\frac{k/\mu}{\log(k/\mu+1)})^k \le \exp(k^2/(2\mu))$ for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean $\mu$. The asymptotics $1+O(k^2/\mu)$ can be shown to be tight for small $k$. This improves over previous uniform bounds for the raw moments of those distributions by a factor exponential in $k$.