On the probability that a binomial variable is at most its expectation

On the probability that a binomial variable is at most its expectation

Consider the probability that a binomial random variable Bi(n,m∕n) with integer expectation m is at most its expectation. Chvátal 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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Veröffentlicht in:Statistics & probability letters 2021-04, Vol.171, p.109020, Article 109020
1. Verfasser: Janson, Svante
Format: Artikel
Sprache:eng
Schlagworte:
Binomial probabilities
Edgeworth expansions
Median
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Zusammenfassung:Consider the probability that a binomial random variable Bi(n,m∕n) with integer expectation m is at most its expectation. Chvátal 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.
ISSN:0167-7152
1879-2103
1879-2103
DOI:10.1016/j.spl.2020.109020