As stated in the title, I am looking for the solution to this expectation, possibly as a function of $\rho$. One relevant solution I find is Lord Shark the Unknown's answer, from which I can derive $E(max(X, Y)) = \frac{1}{2}E(|V|)$. But I am not sure what to do next. Any suggestions?
Expectation of max(X, Y) where X and Y follow a joint normal distribution with correlation of $\rho$
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If $X$ and $Y$ have zero means and unit variances, then you might follow this answer:https://math.stackexchange.com/questions/382237/expected-value-of-ordered-statistics#382260. – StubbornAtom Aug 01 '18 at 19:09
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Actually, in your linked post the OP has answered your question. – StubbornAtom Aug 01 '18 at 19:16
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@StubbornAtom Indeed! My fault. Thank you. – Yi Bao Aug 01 '18 at 19:28