Correlation coefficient of normal random random variables

Question

Let $X$ and $Y$ be jointly normal with mean zero. Show that the correlation coefficient equals $\cos(\pi P\{XY<0\})$. This is a problem in Loeve's book. Any hints?

Answer 1

Hint:

Let $U = X/ \sigma_X$ and $V = Y/\sigma_Y$.

Then

$$P\{XY < 0 \} = P\{UV < 0\} \\ = \frac{1}{2 \pi \sqrt{1-\rho^2}}\int_{-\infty}^0\int_0^\infty \exp\left(\frac{-u^2+ 2\rho uv - v^2}{2(1-\rho^2)} \right)\,du \, dv \\ + \frac{1}{2 \pi \sqrt{1-\rho^2}}\int_0^{\infty}\int_{- \infty}^0 \exp\left(\frac{-u^2+ 2\rho uv - v^2}{2(1-\rho^2)} \right)\,du \, dv \\ = \frac{2}{2 \pi \sqrt{1-\rho^2}}\int_{-\infty}^0\int_0^\infty \exp\left(\frac{-u^2+ 2\rho uv - v^2}{2(1-\rho^2)} \right)\,du \, dv \\ =\frac{1}{ \pi \sqrt{1-\rho^2}}\int_{\pi/2}^\pi\int_0^\infty \exp\left(\frac{-r^2(1 - \sin 2 \theta)}{2(1-\rho^2)} \right)r\,dr \, d\theta $$