A note on the binomial distribution motivated by Chvátal's theorem and Tomasewski's theorem

Abstract

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 $\{0,1,\dots ,n\}$, the probability $q_m:=P(B(n,m/n)\le m)$ is the smallest when $m$ is closest to $2n/3$. Let $\mathcal{R}$ be the family of random variables of the form $X={\sum }_{k=1}^n{a}_k{\epsilon }_k$, where $n\ge 1$, $a_k,k=1,\dots ,n,$ are real numbers with ${\sum }_{k=1}^n{a}_k^2=1$, and ${\epsilon }_k$, $k=1,2,\dots$, are independent Rademacher random variables (i.e., $P({\epsilon }_k=1)=P({\epsilon }_k=-1)=1/2$). Tomaszewski's theorem says that ${\inf }_{X\in \mathcal{R}}P(|X|\le 1)=1/2$. Motivated by Chvátal's Theorem and Tomasewski's Theorem, in this note, we study the minimum value of the probability $f_n(k):=P(|B(n,k/n)-k|\le \sqrt{\mathrm{Var}(B(n,k/n))})$ when $k$ ranges over $\{0,1,\dots ,n\}$ for any fixed $n\ge 1$, where $\mathrm{Var}(\cdot )$ denotes the variance, and prove that it is the smallest when $k=1$ and $n-1$.

11 pages