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October 3rd, 2013, 09:27 AM   #1
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Multivariate normal distribution and marginal distribution

Hi everyone,
I have the following exercise:
Given ,


a) Consider the following decomposition ( omega is supposed to be a matrix).
Show that conditional is , where p is the dimension of .


This one, I have shown.


b) Let . Find the conditional where . In which case this distribution doesn't depend on ?

This one is causing me trouble. Well, with some linear transformation ( ) and question a), I found the conditional distribution for b) but I have some atrocious matrix multiplication to do to find the exact form of my new matrix Omega in terms of a and b and the old Omega. I'm really wondering if there isn't another way. Plus my answer for last part is when sigma_12 * sigma_22_inverse = 0. But this implies a lot of ugly sub cases... what am I missing, I don't think it should be as messy as what I've found.


Thank you in advance for taking time to answer my question.
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