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

2023/11/12 by Zheng-Yan Guo, Guo, Zheng-Yan, Ze-Yu Tao +3
Decision Sciences · Mathematics · #FOS: Mathematics #Fuzzy Systems and Optimization #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications

paper · pdf · doi:10.48550/arxiv.2311.06877

openalex publication_date 2023/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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,…,n\, the probability q_m:=P(B(n,m/n)≤ m) is the smallest when m is closest to (2n)/(3). Motivated by this theorem, in this note we consider the infimum value of the probability P(X≤ 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].

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