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

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

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Bibliographic Details
Published in:Statistics & probability letters 2021-12, Vol.179, p.109224, Article 109224
Main Author: Pinelis, Iosif
Format: Article
Language:English
Subjects:
Binomial distribution
Exact bounds
Probability inequalities
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Description
Summary:Let X be a random variable distributed according to the binomial distribution with parameters n and p. It is shown that P(X>EX)⩾1/4 if 1>p⩾c/n, where c≔ln(4/3), the best possible constant factor.
ISSN:0167-7152
1879-2103
DOI:10.1016/j.spl.2021.109224