Mean and variance bounds and propagation for ill-specified random variables

Foundations, models, and algorithms are provided for identifying optimal mean and variance bounds of an ill-specified random variable. A random variable is ill-specified when at least one of its possible realizations and/or its respective probability mass is not restricted to a point but rather belongs to a set or an interval. We show that a nonexhaustive sensitivity-analysis approach does not always identify the optimal bounds. Also, a procedure for determining the mean and variance bounds of an arithmetic function of ill-specified random variables is presented. Estimates of pairwise correlation among the random variables can be incorporated into the function. The procedure is illustrated in the context of a case study in which exposure to contaminants through the inhalation pathway is modeled.