Some inequalities on Binomial and Poisson probabilities

Abstract

Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a sufficient condition for the tail probability $\mathbb{P}(S\ge \mathbb{E}(S))$ to be larger than $\mathbb{P}(S+X\ge \mathbb{E}(S+X))$. We also apply this result to sums of independent binomial and Poisson random variables.

Compared to the previous version, the results have been extended to cover the case of random variables having integer mean

References in corpus (1)

• Isaac Newton as a Probabilist