TITLE: Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?
QUESTION [5 upvotes]: I am interested in finding a closed form solution (wich I suspect does not exist)  to the following integral
$$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x y}{2 \left(1-c^2\right)}\right)}{2 \pi  \sqrt{1-c^2}} dy dx$$
which corresponds to the  integral of the PDF$(x,y)$ of a multiNormalDistribution (of covariance coefficient $c$) over the  quarter plane $x>a$ and $y>b$. Here $a$ and $b$ are positive and $0<c<1$ (and I know a solution exists for $a=b=0$, but this is not sufficient for my purpose).
More generally I would be interested in the $3$D generalization of this problem.
I have tried in Mathematica to no avail.
Regards,

REPLY [7 votes]: Generically, cumulative distribution function of multivariate Gaussian vector is not expressible in terms of cdf of standard normal random variable $\Phi(x)$. The book by Alan Genz and Frank Brentz, "Computation of multivariate Normal and t Probabilities" is good reference on the subject.
For a standard 2D Gaussian vector $(X,Y)$ with correlation coefficient $-1 < \rho <1$, the probability $ \mathbb{P}(X>a,Y>b) $ can be expressed in terms of Owen's T-function.
By the way Mathematica v8 has a built-in support for multi-normal distribution with special efficient cases for 2D and 3D Gussian random vectors, see BinormalDistribution (ref-page), and MultinormalDistribution (ref-page), and OwenT (ref-page).