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##

3.2.5 Multivariate Gaussian Distribution

or

with the *covariance matrix* of the

is the determinant of this matrix.
and
are symmetric, their eigenvalues are
called
and (sorry!).

__EXAMPLE:__
Assume that two Gaussian variates have the variances
,
,
and the covariance
:

The quadratic form in the exponent is then
, and
the lines of equal density (that is, of equal ) are ellipses
which are inclined with respect to the
coordinate axes:

Rotate the axes of the ellipsoids
to coincide with the coordinate axes:
cross correlations vanish!

Having found
, we arrive at the following prescription for
the production of
correlated Gaussian variables:

Let's try it out:

__EXAMPLE:__
Once more, let

**Principal axis transformation:**
The eigenvalues of are
,
and the corresponding eigenvectors are

**Generator:** To produce pairs of Gaussian
random numbers with the given covariance matrix:

__EXERCISE:__
Write a program that generates a sequence of bivariate Gaussian random
numbers with the statistical properties as assumed in the foregoing
example. Determine
,
,
and
to see if they indeed approach the given
values of , , and .

** Next:** 3.2.6 Homogeneous distributions in
**Up:** 3.2 Other Distributions
** Previous:** 3.2.4 Rejection Method
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001