When information about a distribution consists of statistical moments only, a self-consistent approach to deriving a subjective probability density function (pdf) is Maximum Entropy. Nonetheless, the available information may have uncertainty, and statistical moments maybe known only to lie in a certain domain. If Maximum Entropy is used to find the distribution with the largest entropy whose statistical moments lie within the domain, the information at only a single point in the domain would be used and other information would be discarded. In this paper, the bounded information on statistical moments is used to construct a family of Maximum Entropy distributions, leading to an uncertain probability function. This uncertainty description enables the investigation of how the uncertainty in the probabilistic assignment affects the predicted performance of an engineering system with respect to safety, quality and design constraints. It is shown that the pdf which maximizes
(or equivalently minimizes) an engineering metric is potentially different from the pdf which maximizes the entropy. The feasibility of the proposed uncertainty model is shown through its app lication to: (i) fatigue failure analysis of a structural joint; (ii) evaluation of the probability that a response variable of an engineering system exceeds a critical level, and (iii) random vibration.