A study on the Poisson, geometric and Pascal distributions motivated by Chvátal's conjecture

2022/10/29 by Fu-Bo Li, Kun Xu, Li, Fu-Bo +3
Computer Science · Mathematics · #Analytic Number Theory Research #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2210.16515

openalex publication_date 2022/10/29 · openalex created_date 2022/11/06 · openalex updated_date 2026/07/28

Abstract

Let B(n,p) denote a binomial random variable with parameters n and p. Vasěk Chvátal conjectured 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). This conjecture has been solved recently. Motivated by this conjecture, in this paper, we consider the corresponding minimum value problem on the probability that a random variable is not more than its expectation, when its distribution is the Poisson distribution, the geometric distribution or the Pascal distribution.

Related

• A study on the Weibull and Pareto distributions motivated by Chvátal's theorem
• A study on the negative binomial distribution motivated by Chvátal's theorem
• On the probability that a binomial variable is at most its expectation
• A note on the binomial distribution motivated by Chvátal's theorem and Tomasewski's theorem
• The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem