Extreme probability distributions of random/fuzzy sets and p-boxes

Abstract: Uncertain information about a system variable described by a random set or an equivalent Dempster-Shafer structure on a finite space of singletons determines an infinite convex set of probability distributions, given by the convex hull of a finite set of extreme distributions. Extreme distributions allow one to evaluate (through the Choquet integral) exact upper/lower bounds of the expectation of monotonic and non-monotonic functions of uncertain variables, for example in reliability evaluation of engineering systems. The paper considers the simple case of a single variable, and details applications to random sets with nested focal elements (consonant random sets or the equivalent fuzzy set) and to p-boxes. A simple direct procedure to derive extreme distributions from a p-box is described through simple numerical examples.