The infimum values of two probability functions for the Gamma distribution

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2024年1月17日

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Ping Sun,
Ze-Chun Hu,
Wei Sun

摘要

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 $(α, β) \mapsto P {X_{α, β} \leq κ E [X_{α, β}]}$ $(\alpha ,\beta )\mapsto P\{X_{\alpha ,\beta}\le \kappa E[X_{\alpha ,\beta}] \}$ for any fixed $κ > 0$ $\kappa >0$ and the function $(α, β) \mapsto P {| X_{α, β} - E [X_{α, β}] | \leq \sqrt{Var (X_{α, β})}}$ $(\alpha ,\beta )\mapsto P\{|X_{\alpha ,\beta}-E[X_{\alpha ,\beta}]| \le \sqrt{\operatorname{Var}(X_{\alpha ,\beta})}\}$ . We show that ${inf}_{α, β} P {X_{α, β} \leq E [X_{α, β}]} = \frac{1}{2}$ $\inf_{\alpha ,\beta}P\{X_{\alpha ,\beta}\le E[X_{\alpha ,\beta}]\}= \frac{1}{2}$ and ${inf}_{α, β} P {| X_{α, β} - E [X_{α, β}] | \leq \sqrt{Var (X_{α, β})}} = P {| Z | \leq 1} \approx 0.6827$ $\inf_{\alpha ,\beta}P\{|X_{\alpha ,\beta}-E[X_{\alpha ,\beta}]|\le \sqrt{\operatorname{Var}(X_{\alpha ,\beta})}\}=P\{|Z|\le 1\}\approx 0.6827$ , where Z is a standard normal random variable.