Multivariate random variable

Multivariate normal distribution

A random vector has a multivariate normal distribution iff every linear combination of has a normal distribution, #m/def/prob i.e. for some for any . Such a distribution is fully specified by the means and variances of each component, and the covariance of every pair of components. Packaging this information into a mean vector and a covariance matrix

the Joint probability density function is given by

Bivariate case

In the bivariate case, we have

where .

Properties

  1. Any subvector of a multivariate normal vector is multivariate normal.
  2. The concatenation of two independently distributed multivariate normals is multivariate normal.

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