A study on the $F$-distribution motivated by Chvátal's theorem

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 $P(X_{d_1,d_2}\le \kappa E[X_{d_1,d_2}])$. Our motivation comes from Chvátal’s theorem on the binomial distribution.