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Best lower bound on the probability of a binomial exceeding its expectation
Iosif Pinelis · 2021-08-11 · via math.PR updates on arXiv.org

Let $X$ be a random variable distributed according to the binomial distribution with parameters $n$ and $p$. It is shown that $P(X>EX)\ge1/4$ if $1>p\ge c/n$, where $c:=\ln(4/3)$, the best possible constant factor.

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