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    A study on the negative binomial distribution motivated by Chv\'atal's theorem

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    Let B(n,p)B(n,p) denote a binomial random variable with parameters nn and pp. Chv\'{a}tal's theorem says that for any fixed n≥2n\geq 2, as mm ranges over {0,…,n}\{0,\ldots,n\}, the probability qm:=P(B(n,m/n)≤m)q_m:=P(B(n,m/n)\leq m) is the smallest when mm is closest to 2n3\frac{2n}{3}. Motivated by this theorem, in this note we consider the infimum value of the probability P(X≤E[X])P(X\leq E[X]), where XX 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].Comment: 10 page

    DNA methylation and carcinogenesis in digestive

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