A vector is said to have a ‐dimensional multivariate normal distribution (denoted , where is ‐dimensional multivariate normal distribution) with mean vector and covariance matrix if its density can be written as
where we used the usual notations for the determinant, transpose, and inverse of a matrix. The vector of means may have any elements in , but, just as in the one‐dimensional case, the standard deviation has to be positive. In the multivariate case, the covariance matrix has to be symmetric and positive definite.
The multivariate normal defined thus has many nice properties. The basic one is that the one‐dimensional distributions are all normal, that is, and . This is also true for any marginal. For example, if are the last coordinates, then
So any particular vector of components is normal.
Conditional distribution of a multivariate normal is also a multivariate normal. Given that is a and using the vector notations above assuming that and , then we can write the vector and matrix as
where the dimensions are accordingly chosen to match the two vectors ( and ). Thus, the conditional distribution of given , for some vector is
Furthermore, the vectors and are independent. Finally, any affine transformation , where is a matrix and is a ‐dimensional constant vector, is also a multivariate normal with mean vector and covariance matrix Скачать книгу
