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### Marginal and conditional distributions of multivariate normal distribution

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

Next: Appendix B: Kernels and Up: Appendix A: Conditional and Previous: Inverse and determinant of
Ruye Wang 2006-11-14