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 (X2 < X1) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R = Pr (X2 < X1) 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.