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
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Daniel Lemire 11.20.09 at 09:00
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.)
John 11.20.09 at 09:04
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.
Gregor Gorjanc 11.20.09 at 11:19
Have you looked for bivariate normal distribution of X and Y? That would be a usefull result!
John 11.20.09 at 14:32
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.
Gregor Gorjanc 11.21.09 at 07:17
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.
John 11.21.09 at 09:02
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.