To find the expectation of the greater of X and Y

[Suvadip]

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

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?