The infimum values of two probability functions for the Gamma distribution

Abstract Let α, β be positive real numbers and let X α , β $X_{\alpha ,\beta}$ be a Gamma random variable with shape parameter α and scale parameter β. We study infimum values of the function ( α , β ) ↦ P { X α , β ≤ κ E [ X α , β ] } $(\alpha ,\beta )\mapsto P\{X_{\alpha ,\beta}\le \kappa E[X_{\alpha ,\beta}] \}$ for any fixed κ > 0 $\kappa >0$ and the function ( α , β ) ↦ P { | X α , β − E [ X α , β ] | ≤ Var ( X α , β ) } $(\alpha ,\beta )\mapsto P\{|X_{\alpha ,\beta}-E[X_{\alpha ,\beta}]| \le \sqrt{\operatorname{Var}(X_{\alpha ,\beta})}\}$ . We show that inf α , β P { X α , β ≤ E [ X α , β ] } = 1 2 $\inf_{\alpha ,\beta}P\{X_{\alpha ,\beta}\le E[X_{\alpha ,\beta}]\}= \frac{1}{2}$ and inf α , β P { | X α , β − E [ X α , β ] | ≤ Var ( X α , β ) } = P { | Z | ≤ 1 } ≈ 0.6827 $\inf_{\alpha ,\beta}P\{|X_{\alpha ,\beta}-E[X_{\alpha ,\beta}]|\le \sqrt{\operatorname{Var}(X_{\alpha ,\beta})}\}=P\{|Z|\le 1\}\approx 0.6827$ , where Z is a standard normal random variable.