On the Gaussian Mixture Representation of the Laplace Distribution

Under certain conditions, a symmetric unimodal continuous random variable ξ can be represented as a scale mixture of a standard Normal distribution Z, that is, $\xi =\sqrt{W}Z$, where the mixing distribution W is independent of Z. It is well known that if the mixing distribution is inverse Gamma, then ξ has Student’s t distribution. However, it is less well known that if the mixing distribution is Gamma, then ξ has a Laplace distribution. Several existing proofs of the latter result rely on complex calculus or nontrivial change of variables in integrals. We offer two simple and intuitive proofs based on representation and moment generating functions. As a byproduct, our proof by representation makes connections to many existing results in statistics. Supplementary materials for this article are available online.