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

2021/03/31 by Ahle, Thomas D. · 3 citations
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2103.17027

Abstract

We prove the inequality E[(X/μ)^k] ≤ ((k/μ)/(log(k/μ+1)))^k ≤ exp(k²/(2μ)) for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean μ. The asymptotics 1+O(k²/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.

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