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

2025/03/20 by Guo, Zheng-Yan, Hu, Ze-Chun, Wang, Run-Yu
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.15899

Abstract

Let B(n,p) denote a binomial random variable with parameters n and p. Chvátal's theorem says that for any fixed n≥ 2, as m ranges over \0,1,…,n\, the probability q_m:=P(B(n,m/n)≤ m) is the smallest when m is closest to 2n/3. Let R be the family of random variables of the form X=∑ⁿ_k=1a_kε_k, where n≥ 1, a_k, k=1, …, n, are real numbers with ∑ⁿ_k=1 a_k²=1, and ε_k, k=1, 2, …, are independent Rademacher random variables (i.e., P(ε_k=1)=P(ε_k=-1)=1/2). Tomaszewski's theorem says that inf_{X∈ R}P(|X|≤ 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|≤ √\rm Var (B(n,k/n))) when k ranges over \0,1,…,n\ for any fixed n≥ 1, where \rm Var (⋅) denotes the variance, and prove that it is the smallest when k=1 and n-1.

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