Reliability for lifetime distributions

In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliability R = Pr (X-2 < X-1) when X-1 and X-2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R = Pr (X-2 < X-1) has been worked out for the majority of the well-known distributions including normal, uniform, exponential, gamma, Weibull, and Pareto. However, there are still many other distributions for which the form of R is not known. We have identified at least some 30 distributions with no known form for R. In this paper, we consider the class of lift-time distributions (in particular, exponential and gamma) and derive the corresponding forms for the reliability R. The calculations involve the use of various special functions. (C) 2003 Elsevier Science Ltd. All rights reserved.