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: $P(X_{\alpha}\leq y E(X_{\alpha}))$, $P\left(|X_{\alpha}-E(X_{\alpha})|\leq y\sqrt{Var(X_{\alpha})}\right)$ and $P\left(|X_{\alpha}-E(X_{\alpha})|\geq y\sqrt{Var(X_{\alpha})}\right), \forall y>0$, with respect to the parameter $\alpha$ for the Laplace distribution and the student's $t$ distribution. Our motivation comes from three former conjectures: Chv\'{a}tal's conjecture, Tomaszewski's conjecture and Hitczenko-Kwapie\'{n}'s conjecture.