Reliability for some bivariate beta 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 < Y). The algebraic form for R = Pr(X < Y) has been worked out for the vast majority of the well-known distributions when X and Y are independent random variables belonging to the same univariate family. In this paper, we consider forms of R when (X, Y) follows a bivariate distribution with dependence between X and Y. In particular, we derive explicit expressions for R when the joint distribution is bivariate beta. The calculations involve the use of special functions.