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.
Categories: math.PR