Representation for the Gauss-Laplace Transmutation

TL;DR

This paper explores the representation of symmetric unimodal distributions as scale mixtures of normal distributions, highlighting the specific case where a Gamma mixing distribution yields a Laplace distribution, and provides simple proofs of this fact.

Contribution

It offers two straightforward and intuitive proofs for the representation of Laplace distribution as a scale mixture of normals with Gamma mixing, simplifying previous complex derivations.

Findings

Gamma mixing leads to Laplace distribution from normal

Two simple proofs provided for the representation

Clarifies the relationship between Gamma, Normal, and Laplace distributions

Abstract

Under certain conditions, a symmetric unimodal continuous random variable $\xi$ can be represented as a scale mixture of the standard Normal distribution $Z$, i.e., $\xi = \sqrt{W} Z$, where the mixing distribution $W$ is independent of $Z.$ It is well known that if the mixing distribution is inverse Gamma, then $\xi$ is student's $t$ distribution. However, it is less well known that if the mixing distribution is Gamma, then $\xi$ is a Laplace distribution. Several existing proofs of the latter result rely on complex calculus and change of variables in integrals. We offer two simple and intuitive proofs based on representation and moment generating functions.