The extreme values of two probability functions for the Gamma distribution

Abstract

Motivated by Chv\'{a}tal's conjecture and Tomaszewaki's conjecture, we investigate the extreme value problem of two probability functions for the Gamma distribution. Let $\alpha,\beta$ be arbitrary positive real numbers and $X_{\alpha,\beta}$ be a Gamma random variable with shape parameter $\alpha$ and scale parameter $\beta$. We study the extreme values of functions $P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\}$ and $P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rm Var}(X_{\alpha,\beta})}\}$. Among other things, we show that $ \inf_{\alpha,\beta}P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\}=\frac{1}{2}$ and $\inf_{\alpha,\beta}P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rm Var}(X_{\alpha,\beta})}\}=P\{|Z|\le 1\}\approx 0.6826$, where $Z$ is a standard normal random variable.