Generalized Differentiation of Probability Functions: Parameter Dependent Sets Given by Intersections of Convex Sets and Complements of Convex Sets

In this work we consider probability functions working on parameter dependent sets that are given as an intersection of convex sets and their complements. Such an underlying structure naturally arises when having to handle bilateral inequality systems in various applications, such as energy. We provide conditions under which the probability functions are sub-differentiable as well as a constraint qualification condition under which the probability function turns out to be smooth. Our analysis allows for (nearly) arbitrary distributions of random vectors.