The infimum values of three probability functions for the Laplace distribution and the student's $t$ distribution

Abstract

Let $\{X_{\alpha }\}$ be a family of random variables satisfying some distribution with a parameter $\alpha$, $E(X_{\alpha })$ be the expectation, and $Var(X_{\alpha })$ be the variance. In this paper, we study the infimum values of three probability functions: $P(X_{\alpha }\le yE(X_{\alpha }))$, $P\left(|X_{\alpha }-E(X_{\alpha })|\le y\sqrt{Var(X_{\alpha })}\right)$ and $P\left(|X_{\alpha }-E(X_{\alpha })|\ge y\sqrt{Var(X_{\alpha })}\right),\forall y>0$, with respect to the parameter $\alpha$ for the Laplace distribution and the student's $t$ distribution. Our motivation comes from three former conjectures: Chvátal's conjecture, Tomaszewski's conjecture and Hitczenko-Kwapień's conjecture.

arXiv comment28 pages