Expectation of maxiumum of two random variables

Question

could you help me or give me a hint on how to start with the following:

Let $X,Y$ ~ $N(0,1)$ with correlation $corr(X,Y) = \rho$. It follows that:

$$E[\max(X,Y)] = \sqrt{\frac{1-\rho}{\pi}}$$

How should I start here?

Answer 1

First, re-write the max: $$\max(X,Y)=\frac{X+Y+|X-Y|}{2}$$ Then, $$\mathsf E \max(X,Y)=\frac{\mathsf E X+\mathsf E Y+\mathsf E |X-Y|}{2}=\frac{0+0+\mathsf E |X-Y|}{2}$$

Now you have the difference of two Normal variables. Hence, the result is Normal and the mean and variance can be easily calculated.

The absolute value converts a Gaussian RV to a folded-Gaussian RV. Again, the mean is easily derived.