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Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$.Chv\'{a}tal's theorem says 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 $\frac{2n}{3}$. Motivated by this theorem, in this paper weconsider the infimum value of the probability $P(X\leq \kappa E[X])$, where$\kappa$ is a positive real number, and $X$ is a random variable whosedistribution belongs to some infinitely divisible distributions including theinverse Gaussian, log-normal, Gumbel and logistic distributions.

The infimum values of the probability functions for some infinitely divisible distributions motivated by Chv\'{a}tal's theorem