Envelopes around cumulative distribution functions from interval parameters of standard continuous distributions

Abstract

A cumulative distribution function (CDF) states the probability that a sample of a random variable will be no greater than a value x, where x is a real value. Closed form expressions for important CDFs have parameters, such as mean and variance. If these parameters are not point values but rather intervals, sharp or fuzzy, then a single CDF is not specified. Instead, a family of CDFs is specified. Sharp intervals lead to sharp boundaries ("envelopes") around the family, while fuzzy intervals lead to fuzzy boundaries. Algorithms exist that compute the family of CDFs possible for some function g(v) where v is a vector of distributions or bounded families of distribution. We investigate the bounds on families of CDFs implied by interval values for their parameters. These bounds can then be used as inputs to algorithms that manipulate distributions and bounded spaces defining families of distributions (sometimes called probability boxes or p-boxes). For example, problems defining inputs this way may be found in. In this paper, we present the bounds for the families of a few common CDFs when parameters to those CDFs are intervals.