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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:
• Draw and Gaussian, uncorrelated, with variances and .
• Compute and according to

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