Representation for the Gauss-Laplace Transmutation

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.