Assume an n-dimensional random vector

has a normal distribution with

where and are two subvectors of respective dimensions and with . Note that , and .

**Theorem 4:**

**Part a** The marginal distributions of and are
also normal with mean vector and covariance matrix
(), respectively.

**Part b** The conditional distribution of given
is also normal with mean vector

and covariance matrix

**Proof:** The joint density of is:

where is defined as

Here we have assumed

According to theorem 2, we have

Substituting the second expression for , first expression for , and into to get:

The last equal sign is due to the following equations for any vectors and and a symmetric matrix :

We define

and

and get

Now the joint distribution can be written as:

The third equal sign is due to theorem 3:

The marginal distribution of is

and the conditional distribution of given is

with