On the Chvatal-Janson conjecture

TL;DR

This paper proves a conjecture by Chvatal and Janson regarding the probability that a binomial random variable exceeds its expectation, extending the validity from large n to all n ≥ 2.

Contribution

The paper establishes that the probability bound holds for all n ≥ 2, not just asymptotically for large n, confirming the conjecture in full generality.

Findings

The probability that a binomial variable exceeds its mean is bounded as conjectured.

The result holds for all sample sizes n ≥ 2, not only asymptotically.

The proof extends previous asymptotic results to finite n cases.

Abstract

In a recent paper, Svante Janson has considered a conjecture suggested by Va\v{s}ek Chv\a'atal dealing with the probability that a binomial random variable with parameters $n$ and $m/n$ - where $m$ is an integer - exceeds its expectation $m$. Albeit Janson has provided a proof of this conjecture for large $n$, we show that the result actually holds for each $n\geq 2$.