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Sorry, I asked the original question improperly so I am rephrasing it.

What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ and $f_{Y|X,PA}$ are both gaussian distributions? (The distribution over $X,Y,PA$ is Gaussian.) $Y,PA$ are vectors.

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Hint: Independent random variables are uncorrelated. So the covariance matrix of $Z$ has a block structure: $$\Sigma_Z = \begin{bmatrix}\Sigma_X & \mathbf 0\\\mathbf 0 & \Sigma_Y\end{bmatrix}$$ where $\Sigma_X$ and $\Sigma_Y$ are the covariance matrices of $X$ and $Y$. –  Dilip Sarwate Nov 2 '12 at 18:54
    
Hmm. I see that X and Y are independent because the P(X, Y) = P(X)P(Y) in the way I phrased this question. However, X and Y should not necessarily be independent... I must be interpreting my results incorrectly. I have a distribution over the variables in Z that equals N(X; 0; V_X)*N(Y; mu_Y; V_Y) where XUY = Z. Does this alone require that X and Y are independent? Does it make sense to say N(X; 0; V_X) is not marginal distribution of Z over X? Rather, it is a different distribution over the variables in X? –  user47884 Nov 2 '12 at 20:42

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