5
$\begingroup$

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,

$\endgroup$
7
$\begingroup$

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).

$\endgroup$
8
  • $\begingroup$ Is it correct to assume there is no generalization of the Owen's T function which would be applicable to the 3D case? $\endgroup$
    – chris
    Jun 4 '12 at 8:41
  • $\begingroup$ @chris Of course there is. The generalization is called the multinormal probability. Generically it does not reduce to lower-dimensional probability functions though. $\endgroup$
    – Sasha
    Jun 4 '12 at 13:44
  • $\begingroup$ Nor does its cumulative distribution reduces to any analytic form? $\endgroup$
    – chris
    Jun 4 '12 at 14:00
  • 1
    $\begingroup$ @AdelBibi Yes, because $F_{X,Y}\left(x,y)\right) = \Pr\left( X \leqslant x, Y \leqslant y\right) = 1 - \Pr\left(X > x\right) - \Pr\left(Y>y\right) + \Pr\left(X>x, Y>y\right) = F_X\left(x\right) +F_Y\left(y\right) - \left(1- \Pr\left(X>x, Y>y\right)\right)$. The latter bivariate probability can be expressed in terms of Owen's T-function as I claimed in the post. $\endgroup$
    – Sasha
    Jul 4 '17 at 17:33
  • 1
    $\begingroup$ @AdelBibi Please look at OwenT ref-page, last example in the application section. $\endgroup$
    – Sasha
    Jul 5 '17 at 4:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.