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

Abstract

Let B(n,p) denote a binomial random variable with parameters n and p. Vašek Chvátal conjectured 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. 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.