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.