the above mixture model uniquely, as there are three parameters (π, θ1, θ2) to be determined. Definition Consider a mixture of parametric distributions of the same class. Let be the class of all component distributions. Then the convex hull K of J defines the class of all finite mixture of distributions in J: K is said to be identifiable if all its members are unique, that is, given two members p and in K, being mixtures of k distributions and distributions respectively in J, we have if and only if, first of all, and secondly we can reorder the summations such that and for all i. Parameter estimation and system identification Parametric mixture models are often used when we know the distribution Y and we can sample from X, but we would like to determine the ai and θi values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations. It is common to think of probability mixture modeling as a missing data problem. One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data. When we start, this membership is unknown, or missing. The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions. A variety of approaches to the problem of mixture decomposition have been proposed, many of which focus on maximum likelihood methods such as expectation maximization (EM) or maximum a posteriori estimation (MAP). Generally these methods consider separately the questions of system identification and parameter estimation; methods to determine the number and functional form of components within a mixture are distinguished from methods to estimate the corresponding parameter values. Some notable departures are the graphical methods as outlined in Tarter and Lock and more recently minimum message length (MML) techniques such as Figueiredo and Jain and to some extent the moment matching pattern analysis routines suggested by McWilliam and Loh (2009). Expectation maximization (EM) Expectation maximization (EM) is seemingly the most popular technique used to determine the parameters of a mixture with an a priori given number of components. This is a particular way of implementing maximum likelihood estimation for this problem. EM is of particular appeal for