Moments of maximum of bivariate standard normal

Question

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$,

and let $Z=max\{X,Y\}$.

I'm looking for the first two moments of $Z$. I know it is near impossible to write the density of $Z$ in a closed form, so I believe calculating the moments from density would be extremely hard? Are there easier ways of calculating the moments?

Also, does $Z$ has the skew-normal distribution?

Answer 1

Let $U = X+Y$ and $V = X - Y$. Note that $U$ and $V$ are jointly normal random variables with means $0$, and independent since $E[UV] = E[X^2 - Y^2] = 0$. Now $Z = (U + |V|)/2$ so $E[Z] = E[|V|]/2$, and $E[Z^2] = E[U^2]/4 + E[V^2]/4$.