Multivariate normal distribution and marginal distribution

• Oct 2nd 2013, 04:57 PM
sunmalus
Multivariate normal distribution and marginal distribution
Hi everyone,
I have the following exercise:
Given $Y \sim \mathcal{N}_p(\mu,\Omega )$,

a) Consider the following decomposition $Y=(Y_1,Y_2)^T, \mu=(\mu_1, \mu_2)^T, \Omega=( \Omega_{11}, \Omega_{12};\Omega_{21},\Omega_{22} )$ ( omega is supposed to be a matrix).
Show that conditional $Y_1 |(Y_2=y_2)$ is $\mathcal{N}_p ( \mu_1+\Omega_{12}\Omega_{22}^{-1}(y_2-\mu_2),\Omega_{11}-\Omega_{12}\Omega_{22}^{-1}\Omega_{21})$, where p is the dimension of $Y_1$.

This one, I have shown.

b) Let $a,b \in \mathbb{R}^n$. Find the conditional $X_1|X_2=x_2$ where $X_1=a^TY,X_2=b^TY$. In which case this distribution doesn't depend on $x_2$?

This one is causing me trouble. I stated by writing explicitly the product in $f_{X}(a^TY|b^TY=x_2)$ but it gets me nowhere.

Well, with some linear transformation ( $(a^T, b^T)^T*Y=(X_1, X_2)$)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.