The extreme values of two probability functions for the Gamma distribution

TL;DR

This paper investigates the extremal probabilities related to the Gamma distribution, establishing bounds for the probability of the variable being below its mean and within one standard deviation, motivated by conjectures in probability theory.

Contribution

It determines the exact infimum values of two probability functions for the Gamma distribution, providing insights into their extremal behavior.

Findings

Infimum of P{X ≤ E[X]} is 1/2.

Infimum of P{|X - E[X]| ≤ √Var(X)} is approximately 0.6826.

Results are motivated by and relate to existing conjectures in probability theory.

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.