This note will look at ways of computing P(X>Y)where X is a distribution modeling survival (gamma, inverse gamma, Weibull, log-normal) and Y has a uniform distribution. Each of these can be computer in closed form in terms of common statistical functions. We begin with analytical calculations and then include software implementations in R to make some of the details more explicit. Finally, we give a suggestion for using simulation to compute random inequalities that cannot be computed in closed form.
Numerical Analysis and Computation
Cook, John D., "Random inequalities between survival and uniform distributions" (September 2011). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 71.