If $X_1$ and $X_2$ are independent, having both standard normal distribution, what is $E\left(\max(\left|X_1\right|,\left|X_2\right|\right)$?
Actually, holding similar condition there is no problem when calculating $E(\max(X_1,X_2))$. Since $\max(X_1,X_2)=\frac{1}{2}(X_1+X_2+\left|X_1-X_2\right|)$ and taking expectation from both sides, $E(\left|X_1-X_2\right|)$ can be calculated following the fact that $\left|X_1-X_2\right|$ has half-normal distribution. Using the same fashion for $\max(\left|X_1\right|,\left|X_2\right|)$, yields $E(\left|(\left|X_1\right|-\left|X_2\right|)\right|)$, which seems difficult to calculate. I have seen under What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent? a possible relevant post which does not helped.