# Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?

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,

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## 1 Answer

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).

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 Is it correct to assume there is no generalization of the Owen's T function which would be applicable to the 3D case? – chris Jun 4 '12 at 8:41 @chris Of course there is. The generalization is called the multinormal probability. Generically it does not reduce to lower-dimensional probability functions though. – Sasha Jun 4 '12 at 13:44 Nor does its cumulative distribution reduces to any analytic form? – chris Jun 4 '12 at 14:00 No, I am afraid it does not – Sasha Jun 4 '12 at 15:11