Let $X$ and $Y$ have the bivariate normal density function

$$f(x, y)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}} \exp \left\{-\frac{1}{2\left(1-\rho^{2}\right)}\left(x^{2}-2 \rho x y+y^{2}\right)\right\}, \quad x, y \in \mathbb{R},$$

for fixed $\rho \in(-1,1)$. Let $Z=(Y-\rho X) / \sqrt{1-\rho^{2}}$. Show that $X$ and $Z$ are independent $N(0,1)$ variables. Hence, or otherwise, determine

$$\mathbb{P}(X>0, Y>0) .$$