Imprecise random variables, random sets, and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family Xλ of random variables, where the parameter λ ranges in an index set Λ, one may ask for the upper/lower probability that g(Xλ ) belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that g(Xλ ) lies in B. In the second case, one considers the random set generated by all g(Xλ ), λ ∈ Λ, e.g. by transforming Xλ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented.