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

Zheng-Yan Guo, Ze-Yu Tao and Ze-Chun Hu

Statistics & Probability Letters, 2024, vol. 207, issue C

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 qm≔P(B(n,m/n)≤m) is the smallest when m is closest to 2n3. 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 Li et al. (2023).

Keywords: Negative binomial distribution; Chvátal’s theorem; Beta function; Gamma function (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spl.2024.110037

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