Using moment generating functions to derive mixture distributions

Mixture models are important in theoretical and applied statistics. Mathematical statistics courses for undergraduate senior or first-year graduate students should expose students to some properties of the mixture model. In current textbooks, the predominant approach to obtain the mixture distribution is based on marginalization of the joint distribution defined by the mixture model. This article proposes the use of moment generating functions (mgf) to obtain the distribution of some mixtures. For mixtures that do not have a mgf, a generalization using characteristic functions is illustrated with an example. The mgf approach tends to be simple and it uses some of the tools already built into the course. Several examples are used to illustrate the proposed methodology.