Continuous Erlang Mixtures and their Relation to Exponential Mixtures and Poisson Mixtures

This study provides a novel method for obtaining Erlang mixtures from a mixed Poisson process. The study solved the basic differential equations of the Poisson process to obtain the Poisson distribution. The waiting time distribution in a Poisson process is illustrated as an Erlang distribution. The study also presented the Erlang mixture as the first passage time distribution in the mixed Poisson process, which was expressed using both the direct method and the method of moments. Moreover, these two ways of inferring a mathematical identity have been equated. The exponential mixture and Poisson mixture are explained as special cases of the Erlang mixture. A practical example is given, using type II gamma distribution mixtures. Properties of the mixtures, such as raw moments and probability generating function, are analyzed.