Random inequalities between survival and uniform distributions

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. Random inequalities between survival and uniform distributions John Cook September 14, 2011 Abstract 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 computed 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.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 computed 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. 1 Analytical results For any distributions on independent random variables X and Y , P (X > Y ) = ∫ ∞ −∞ fX(x)FY (x) dx. Here we use fW and FW for the PDF and CDF functions of W respectively. For more information on such inequalities, see [1]. Assume Y is uniformly distributed on [a, b] with 0 ≤ a < b <∞. Then FY (x) =  0 if x < a (x− a)/(b− a) if a ≤ x ≤ b 1 if x > b and it follows that ∫ ∞ −∞ fX(x)FY (x) dx = ∫ b a x− a b− a fX(x) dx+ ∫ ∞