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

03/31/2021

∙

by   Thomas D. Ahle, et al.

∙

0

∙

share

We prove the inequality E[(X/μ)^k] ≤ (k/μ/log(k/μ+1))^k ≤exp(k^2/(2μ)) for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean μ. The asymptotics 1+O(k^2/μ) 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.