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

Famous quotes containing the word definition:

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)