Let $X,Y$ be jointly normal with density $f(x,y)=\dfrac{1}{2\pi\sqrt{1-\rho^2}}\exp(-\dfrac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2))$. Let $Z=\min(X,Y)$. Show that $E[Z]=\sqrt{\dfrac{1-\rho}{\pi}}$ and $E[Z^2]=1$
I am not sure how to go about this. I did something similar before, but the setting included independent random variables, which I don't have this time. I tried computing the CDF of $Z$ using $P(Z\le z)=1-P(X\ge z, Y\ge z)=1-\int_{z}^\infty\int_{z}^\infty f(x,y) dx dy$ but this lead me nowhere. Is there a better aproach?