One of the main steps in an uncertainty analysis is the selection of appropriate probability distribution functions for all stochastic variables. In this paper, criteria for such selections are reviewed, the most important among them being any a priori knowledge about the nature of a stochastic variable, and theCentral Limit Theoremof probability theory applied to sums and products of stochastic variables. In applications of these criteria, it is shown that many of the popular selections, such as the uniform distribution for a poorly known variable, require far more knowledge than is actually available. However, the knowledge available is usually sufficient to make use of other, more appropriate distributions. Next, functions of stochastic variables and the selection of probability distributions for their arguments as well as the use of different methods of error propagation through these functions are discussed. From these evaluations, priorities can be assigned to determine which of the stochastic variables in a function need the most care in selecting the type of distribution and its parameters. Finally, a method is proposed to assist in the assignment of an appropriate distribution which is commensurate with the total information on a particular stochastic variable, and is based on the scientific method. Two examples are given to elucidate the method for cases of little or almost no information.