Encyclopedia > Multivariate normal distribution

  Article Content

Multivariate normal distribution

A random vector X=(X1,...,Xn) follows a multivariate normal distribution, also sometimes called a multivariate Gaussian distribtuion (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution), if it satisfies the following equivalent conditions:
  • every linear combination Y=a1X1+...+anXn is normally distributed;
  • there is a random vector Z=(Z1,...,Zm), whose components are independent standard normal random variables, a vector μ=(μ1,...,μn) and an n-by-m matrix A such that X=A Z + μ.
  • there is a vector μ=(μ1,...,μn) and a symmetric, positive semidefinite matrix Γ such that X has density
fX(x1,...,xn)dx1...dxn = (det(2πΓ))-1/2 exp ½((X-μ)TΓ-1(X-μ)) dx1...dxn

The vector μ in these conditions is the expected value of X and the matrix Γ=ATA is the covariance matrix of the components Xi. It is important to realize that the covariance matrix must be allowed to be singular. That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the Xi are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z.


Multivariate Gaussian density

Recall characteristic function of a random vector.

Recall characterizations of gaussian random variables.

Calculate characteristic function of Z in terms of characteristic function of X.

Deduce characteristic functional of X in terms of mean vector and covariance matrix.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Grateful Dead

... half of the U.S. On February 14, 2003, reflecting the reality what was, the band renamed itself The Dead[?], keeping 'Grateful' retired out of respect for Jerry. Th ...

This page was created in 37.9 ms