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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