My Math Forum Multivariate normal distribution and marginal distribution

 October 3rd, 2013, 08:27 AM #1 Newbie   Joined: Oct 2013 Posts: 1 Thanks: 0 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. Well, with some linear transformation ( $(a^T, b^T)^T*Y=(X_1, X_2)$) 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.

### proof problem prove ae/ac=bd/bc

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