Just posted: Exact calculation of inequality probabilities. This report summarizes previous results for calculating P(X > Y) where X and Y are random variables.

Previous posts on random inequalities:

Introduction

Analytical results

Numerical results

Cauchy distributions

Beta distributions

Gamma distributions

Three or more random variables

Folded normals

Cool. I’ll know where to look if I need such inequalities!

(Alas, in my research, I always end up with unknown or custom distributions.)

Thanks. For custom distributions, you can always simulate. But numerical integration may be much faster. I give some general techniques for numerically evaluating random inequalities in the first section of this report.

Have you looked for bivariate normal distribution of X and Y? That would be a usefull result!

Gregor, I talk briefly about the bivariate normal in my tech report on folded normals. For correlated normals, you can compute P(X > Y) in closed form, but you can reduce it to a well-known form. I give a link to a paper on how to solve the numerical problem.

But there the correlation between U and V rises due to sums, U=X+Y and V=X-Y, while X and Y are assumed uncorrelated. I am talking about P(X > Y), where P(X,Y) ~ Normal(mu, Sigma) with “general” mu and Sigma.

If (X, Y) is bivariate normal, then P(X > Y) is just the probability above the line x = y. By rotating the coordinates, this reduces to computing P(W < 0) where (W, Z) is bivariate normal. The folded normal tech report has a reference to a paper by Alan Genz that explains how to compute such probabilities numerically.