Estimating and Validating the Cumulative Distribution of a Function of Random Variables: Toward the Development of Distribution Arithmetic

A method for estimating and validating the cumulative distribution of a function of random variables (independent or dependent) is presented and examined. The method creates a sequence of bounds that will converge to the distribution function in the limit for functions of independent random variables or of random variables of known dependencies. Moreover, an approximation is constructed from and contained in these bounds. Preliminary numerical experiments indicate that this approximation is close to the actual distribution after a few iterations. Several examples are given to illustrate the method.