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

TL;DR

This paper investigates the probability that a binomial random variable is at most its expectation, confirming a conjecture for large sample sizes that the minimal probability occurs near a specific parameter value.

Contribution

It proves Chvátal's conjecture regarding the minimal probability for binomial variables at their expectation when the sample size is large.

Findings

The probability is minimized near m ≈ 2n/3 for large n.

The conjecture holds asymptotically as n grows large.

Provides insight into the behavior of binomial distributions at their expectation.

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.