MHB To find the expectation of the greater of X and Y

[Suvadip]

If $$(X, Y)$$ has the normal distribution in two dimensions with zero means and unit variances and correlation coefficient $$\rho$$, then to prove that the expectation of the greater of X and Y is $$\sqrt{(1-\rho)\pi}$$.

How to proceed with it? Help please.

 

[Siron]

What exactly do you mean with the greater of $X$ and $Y$? I suppose $\max\{X,Y\}$?

When $(X,Y)$ has a bivariate normal distribution it follows that $X$ and $Y$ are normally distributed (in this case with zero means and unit variances). The correlation coefficient describes the dependency structure between $X$ and $Y$. If $X$ and $Y$ are independent, in other words $\rho = 0$, then it's not very difficult to derive that
$$\mathbb{E}[\max\{X,Y\}] = \frac{1}{\sqrt{\pi}}$$.

However when $\rho \neq 0$ then it's harder to derive an expression for the expectation but there's a general formula. So before I proceed I have three questions: Do you want a full derivation of the general formula? Where does this problem come from? What have you already tried?