A study on the Poisson, geometric and Pascal distributions motivated by Chv\'{a}tal's conjecture

TL;DR

This paper investigates the minimum probability that a Poisson, geometric, or Pascal distributed random variable does not exceed its expectation, inspired by Chvátal's conjecture on binomial probabilities.

Contribution

It extends the analysis of probability bounds to Poisson, geometric, and Pascal distributions motivated by a conjecture on binomial probabilities.

Findings

Identifies the minimum probability for Poisson distribution.

Determines the minimum probability for geometric distribution.

Analyzes the Pascal distribution in the context of the conjecture.

Abstract

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Vas\v{e}k Chv\'{a}tal conjectured that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, 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.