Multivariate Normal Distribution - Definition

Definition

A random vector x = (X1, …, Xk)' is said to have the multivariate normal distribution if it satisfies the following equivalent conditions.

  • Every linear combination of its components Y = a1X1 + … + akXk is normally distributed. That is, for any constant vector a ∈ Rk, the random variable Y = a′x has a univariate normal distribution.
  • There exists a random ℓ-vector z, whose components are independent standard normal random variables, a k-vector μ, and a k×ℓ matrix A, such that x = Az + μ. Here ℓ is the rank of the covariance matrix Σ = AA′. Especially in the case of full rank, see the section below on Geometric interpretation.
  • There is a k-vector μ and a symmetric, nonnegative-definite k×k matrix Σ, such that the characteristic function of x is
\varphi_\mathbf{x}(\mathbf{u}) = \exp\Big( i\mathbf{u}'\boldsymbol\mu - \tfrac{1}{2} \mathbf{u}'\boldsymbol\Sigma \mathbf{u} \Big).

The covariance matrix is allowed to be singular (in which case the corresponding distribution has no density). This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. Note also that the Xi are in general not independent; they can be seen as the result of applying the matrix A to a collection of independent Gaussian variables z.

Read more about this topic:  Multivariate Normal Distribution

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