joint probability of correlated normal random variables

Question

Let $Z_1$ and $Z_2$ follow $ N(\delta, \alpha)$ and covariance between them is $\beta$. How to calculate $P(Z_1>0, Z_2>0)?$

Answer 1

The correlation between $Z_1$ and $Z_2$ will be $\beta /(\sigma_1\sigma_2) = \beta/\alpha^2$. Then the probability density function is $$ f(z_1,z_2) = \frac1{2\pi\alpha^2\sqrt{1-\beta/\alpha^2}} e^{-\frac1{2(1-\beta/\alpha^2)}}\left( \frac{(z_1-\delta)^2+(z_2-\delta)^2-2\beta/\alpha^2(z_1-\delta)(z2-\delta)}{\alpha^2}\right) $$ The probability wanted is then $$ \int_{z_1=0}^\infty \int_{z_2=0}^\infty f(z_1,z_2) dz_2\,dz_1 $$ Now for fixed $z_1$ the $z_2$ integral will give an erf result (in fact, it would even if the covariance were zero, because of the non-zero $\delta$) so the answer yo seek is an integral of an erf. Although this is quite tractable numerically, I know of no familiar function that represents it in closed form.