Probability Models with Discrete and Continuous Parts

In mathematical statistics courses, students learn that the quadratic function is minimized when x is the mean of the random variable X, and that the graphs of this function for any two distributions of X are simply translates of each other. We focus on the problem of minimizing the function defined by in the context of mixtures of probability distributions of the discrete, absolutely continuous, and singular continuous types. This problem is important, for example, in Bayesian statistics, when one attempts to compute the decision function, which minimizes the expected risk with respect to an absolute error loss function. Although the literature considers this problem, it does so only under restrictive conditions on the distribution of the random variable X, by, for example, assuming that the corresponding cumulative distribution function is discrete or absolutely continuous. By using Riemann-Stieltjes integration, we prove a theorem, which solves this minimization problem under completely general conditions on the distribution of X. We also illustrate our result by presenting examples involving mixtures of distributions of the discrete and absolutely continuous types, and for the Cantor distribution, in which case the cumulative distribution function is singular continuous. Finally, we prove a theorem that evaluates the function y(x) when X has the Cantor distribution.