Show that for standard bivariate normal variables X and Y with correlation "rho"
E(max(X,Y)) = sqrt((1-rho)/pi)
Thanks in advance
Several approaches are possible.
One approach might start by first proving that the two random variables $\displaystyle X$ and $\displaystyle U = \frac{Y - \rho X}{\sqrt{1 - \rho^2}}$ are independent and standard normal. Then note that $\displaystyle Y > X$ if and only if $\displaystyle (1 - \rho) X < \sqrt{1 - \rho^2} U$.