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

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: $P(X_{\alpha }\le yE(X_{\alpha }))$, $P(\mathrm{\mid }{X}_{\alpha }-E(X_{\alpha })\mathrm{\mid }\le y\sqrt{Var(X_{\alpha })})$ and $P(\mathrm{\mid }{X}_{\alpha }-E(X_{\alpha })\mathrm{\mid }\ge y\sqrt{Var(X_{\alpha })}),\mathrm{\forall }y>0$, 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.