Best lower bound on the probability of a binomial exceeding its expectation

Abstract

Let $X$ be a random variable distributed according to the binomial distribution with parameters $n$ and $p$. It is shown that $P(X>EX)\ge 1/4$ if $1>p\ge c/n$, where $c:=\ln (4/3)$, the best possible constant factor.

7 pages; to appear in Statistics and Probability Letters