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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
φX(u)=exp(iμTuuTΓu).

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.

Proof?

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.



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