Let be a multivariate normal random vector with mean and covariance matrix . In this lecture we present some useful facts about partitionings of , that is, about subdivisions of into two sub-vectors and such thatwhere and have dimensions and respectively and .
In what follows, we will denote by:
the mean of ;
the mean of ;
the covariance matrix of ;
the covariance matrix of ;
the cross-covariance between and .
Given this notation, it follows thatand
The following proposition states a first (trivial) fact about the two sub-vectors.
Proposition Both and have a multivariate normal distribution, i.e.,
The random vector can be written as a linear transformation of :where is a matrix whose entries are either zero or one. Thus, has a multivariate normal distribution, because it is a linear transformation of the multivariate normal random vector and multivariate normality is preserved by linear transformations (see the lecture entitled Linear combinations of normal random variables). By the same token, also has a multivariate normal distribution, because it can be written as a linear transformation of :where is a matrix whose entries are either zero or one.
This, of course, implies that any sub-vector of is multivariate normal when is multivariate normal.
The following proposition states a necessary and sufficient condition for the independence of the two sub-vectors.
Proposition and are independent if and only if .
and are independent if and only if their joint moment generating function is equal to the product of their individual moment generating functions (see the lecture entitled Joint moment generating function). Since is multivariate normal, its joint moment generating function isThe joint moment generating function of isThe joint moment generating function of and , which is just the joint moment generating function of , isfrom which it is obvious that if and only if .
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