A study on the $F$-distribution motivated by Chv\'{a}tal's theorem

TL;DR

This paper investigates the probability bounds of an $F$-distribution relative to its expectation, inspired by Chvátal's theorem on the binomial distribution, aiming to understand extremal probability behaviors.

Contribution

It introduces a new analysis of the $F$-distribution's probability bounds based on its expectation, extending Chvátal's theorem insights to this distribution.

Findings

Derived bounds for $P(X_{d_1, d_2} \\leq \\kappa E[X_{d_1, d_2}]$

Identified the infimum probability value for various parameters

Connected $F$-distribution properties to binomial distribution theorems

Abstract

Let $X_{d_1, d_2}$ be an $F$-random variable with parameters $d_1$ and $d_2,$ and expectation $E[X_{d_1, d_2}]$. In this paper, for any $\kappa>0,$ we investigate the infimum value of the probability $P(X_{d_1, d_2}\leq \kappa E[X_{d_1, d_2}])$. Our motivation comes from Chv\'{a}tal's theorem on the binomial distribution.