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On the probability that a binomial variable is at most its expectation
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abstract
Consider the probability that a binomial random variable Bi$(n,m/n)$ with integer expectation $m$ is at most its expectation. Chv\'atal conjectured that for any given $n$, this probability is smallest when $m$ is the integer closest to $2n/3$. We show that this holds when $n$ is large.
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Cited by 1 Pith paper
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Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion
Binomial masses a fixed distance from the upper mode have an all-order expansion whose elementary tail is exactly the size-bias factor, leaving pure Appell coefficients.
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