The extreme values of two probability functions for the Gamma distribution

Motivated by Chvátal’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 $P\{\mathrm{\mid }{X}_{\alpha ,\beta }-E[X_{\alpha ,\beta }]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X_{\alpha ,\beta })}\}$. Among other things, we show that ${\inf }_{\alpha ,\beta }P\{X_{\alpha ,\beta }\le E[X_{\alpha ,\beta }]\}=\frac{1}{2}$ and ${\inf }_{\alpha ,\beta }P\{\mathrm{\mid }{X}_{\alpha ,\beta }-E[X_{\alpha ,\beta }]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X_{\alpha ,\beta })}\}=P\{\mathrm{\mid }Z\mathrm{\mid }\le 1\}\approx 0.6826$, where $Z$ is a standard normal random variable.