A study on the negative binomial distribution motivated by Chvátal's theorem

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal’s theorem says that for any fixed $n\ge 2$, as $m$ ranges over $\{0,\dots ,n\}$, the probability $q_m:=P(B(n,m\mathrm{/}n)\le m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this note we consider the infimum value of the probability $P(X\le E[X])$, where $X$ is a negative binomial random variable. As a consequence, we give an affirmative answer to the conjecture posed in [Statistics and Probability Letters, 200 (2023) 109871].