With the matrix notation set up the bivariate normal distribution is easily extended to higher dimensions. The vector is said to have a -dimensional normal distribution if its pdf is given for all by
where is the vector of expectations and is the variance-covariance matrix with -th element being the pairwise covariance of variables and . As
the variance-covariance matrix can be written
where is the variance of , and is the correlation of and .
We often denote the distribution of by
where stands for multivariate normal distribution of dimensions.
Proposition 10.1.1 showed that a multivariate normal vector can always be written as the sum of an expectation vector, , and a linear combination of iid random variables.
Therefore as in the bivariate case the univariate marginal distributions are all normal
for . In fact, any linear combination of the ’s will again be normal
where the expressions for the expectation and variance follow from Chapter 8; the fact that the distribution is normal is the convolution property of the normal distribution. Using that and the variance can also be written as
The bivariate marginal distribution of any pair of variables is also normal with expectation and variance given by,
In general all the lower order marginal distributions, i.e. the distribution of any subset of the ’s, will be normal with expectation vector and variance matrix given by deleting the rows and columns corresponding to the variables we leave out.
Indeed, any multi-dimensional linear transformation of will be normal
where is an matrix of constants.
It also turns out that all the conditional distributions of a subset of the ’s given the rest is again normal.
Let be multivariate normal with
Find the marginal distribution of .
Solution. The marginal distribution is obtained by deleting rows and of the expectation vector, and rows and and columns and of the variance matrix. Thus
The most important special case is when are i.i.d. random variables. In this case
and
where is the identity matrix in dimensions, i.e. a matrix with ones in the diagonal and zeros everywhere else. The pdf becomes
since .