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

TL;DR

This paper investigates the minimum values of three probability functions related to the Laplace and Student's t distributions, motivated by three conjectures, to understand their behavior across distribution parameters.

Contribution

It provides the first analysis of the infimum probabilities for these distributions, addressing conjectures in probability theory.

Findings

Identifies the infimum values of the three probability functions for the distributions.

Provides insights into the behavior of these probabilities across distribution parameters.

Addresses and potentially resolves three longstanding conjectures.

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.