In probability theory and statistics, a random vector X = (X1, ..., Xn) follows a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (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×m matrix A such that X = A Z + μ.
- there is a vector μ and a symmetric, positive semi-definite matrix Γ such that the characteristic function of X is
φX(u)=exp(iμTu − (½) uT Γ u).
The following is not quite equivalent to the conditions above, since it fails to allow for a singular matrix as the variance:
- there is a vector μ=(μ1, ..., μn) and a symmetric, positive definite 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.
