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. I stated by writing explicitly the product in but it gets me nowhere.

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

Re: Multivariate normal distribution and marginal distribution

Hey sunmalus.

One suggestion I have is to use your results in a) and find the situation where the conditional distribution is free from any value of x2. (Think of when sigma_12 * sigma_22_inverse = 0)

Re: Multivariate normal distribution and marginal distribution

Well, with some linear transformation ( )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, as chiro said, 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.