There are 2 Random Variables X and Y.

X ~ N(a,b) and Y ~ N(c,d). X and Y are correlated with a correlation coefficient of p.

How do I find a closed form solution of E[MAX(X,Y)]?

Whichever way I look at it, I am not able to get rid of the randomness in the solution.

My sol:- E[MAX(X,Y)] = X* P(X>Y) + Y*P(Y>X)

But if I do this I will get an answer in terms of X and Y which are still Random.

Let $X$ and $Y$ be two random variables, with

$X\sim N(a,b)$ and $Y\sim N(c,d)$. Furthermore, $X$ and $Y$ are correlated with a correlation coefficient equal to $p$.

How do I find a closed-form expression for $E[\max(X,Y)]$?

Whichever way I look at it, I am not able to get rid of the randomness in the solution.

My solution.

$E[\max(X,Y)] = XP(X>Y) + YP(Y>X)$

But if I do this I will get an answer in terms of $X$ and $Y$ which is still a random quantity.