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
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:
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (18031882)