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

Let ${X_α}$ be a family of random variables satisfying some distribution with a parameter $α$, $E(X_α)$ be the expectation, and $Var(X_α)$ be the variance. In this paper, we study the infimum values of three probability functions: $P(X_α\leq y E(X_α))$, $P\left(|X_α-E(X_α)|\leq y\sqrt{Var(X_α)}\right)$ and $P\left(|X_α-E(X_α)|\geq y\sqrt{Var(X_α)}\right), \forall y>0$, with respect to the parameter $α$ 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.

💡 Research Summary

The paper investigates the infimum (greatest lower bound) of three probability functions defined on a family of random variables ({X_\alpha}) that depend on a parameter (\alpha). For any positive real number (y), the authors consider

1. (C(y)=\inf_{\alpha},P\bigl(X_\alpha\le y,E