Multinormal conditional distribution

In Section 4.3, which deals with the multinormal distribution, an important property was demonstrated, namely that correlation coefficients are parameters of the multinormal distribution. In the same way, it will be shown here that partial correlation coefficients, described in the previous section, are parameters of a distribution, derived from the multinormal distribution, which is called the conditional distribution of multinormal random variables (or multinormal conditional distribution). The fact that p is a parameter of the multinormal distribution is the basis for testing the significance of simple correlation coefficients. Similarly, the fact that partial correlation coefficients are parameters of a distribution is the basis for testing their significance, using the approach explained in the previous section.

In the multidimensional context, a set of random variables is sometimes partitioned into two subsets, so as to study the distribution of the variables in the first set (y1,y2,.--,yp) while maintaining those in the second set (yp+1,...,yp+q) fixed. These are the conditions already described for partial correlations (eqs. 4.34 to 4.37). Such a probability distribution is called a conditional distribution. It can be shown that the conditional distribution of variables of the first set, given the second set of fixed variables, is:

where /(y1,y2, ..., yp+q) is the joint probability density of the (p + q) variables in the two subsets and h(yp+1, ., yp+q) is the joint probability density of the q fixed variables (second subset). When the two subsets are independent, it has already been shown

(Section 4.3) that their joint probability density is the product of the densities of the two subsets, so that:

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