Multivariate Gaussian Distribution

Definition

Multivariate Gaussian Distribution

A random vector has a multivariate Gaussian distribution with mean and covariance matrix , written , if every linear combination of its coordinates is Gaussian:

Here and . A zero variance means that the corresponding linear combination is constant with probability one.

Density

If is positive definite, the density is

The quadratic expression measures deviation from relative to the covariance. The determinant supplies the normalisation for the volume of that spread. Constant-density surfaces are ellipsoids centred at .

If is singular, is confined to a lower-dimensional affine subspace and has no density with respect to -dimensional volume.