A study on the Poisson, geometric and Pascal distributions motivated by Chvátal’s conjecture. (English) Zbl 1515.60044
Summary: Let \(B (n, p)\) denote a binomial random variable with parameters \(n\) and \(p\). Vašek Chvátal conjectured that for any fixed \(n \geq 2\), as \(m\) ranges over \(\{0, \dots, n\}\), the probability \(q_m := P(B(n, m/n) \leq m)\) is the smallest when \(m\) is closest to \(\frac{2n}{3}\). 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.
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