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In my probability class I was given this problem that truly has me stumped:

Let $ X=(X_1,X_2,X_3) $ be a Gaussian random vector with mean vector zeros, with the 3x3 co variance matrix:

$ \Sigma $

which is invertible so a joint density function exists. Now we define the following: $ p_{13} = {{\Sigma}^{-1}}_{13} = \frac{1}{|\Sigma|} \begin{bmatrix} \Sigma_{12} & \Sigma_{13} \\ \Sigma_{22} & \Sigma_{23} \end{bmatrix} $

that is the {1,3} entry of the inverse of the covariance matrix $ \Sigma $. We are to prove for the general 3x3 covariance matrix case that given $ X_2 $ the random variables $ X_1,X_3 $ are statistically independent if and only if $ p_{13}=0 $.

As a hint we are advised to look at the conditional expectation of $ X_1 $ given $ X_2,X_3 $ is given by $ \mu = \Sigma_{12} {\Sigma_{22}}^{-1} \left( \begin{array}{c} x_2 \\ x_3 \\ \end{array} \right) $

where we define $ \Sigma_{12} = \begin{bmatrix} E\{X_1X_2\} & E\{X_1X_3\} \end{bmatrix} $

and we define $ \Sigma_{22} = \begin{bmatrix} E\{{X_2}^2\} & E\{X_2X_3\} \\ E\{X_3X_2\} & E\{{X_3}^2\} \end{bmatrix} $

I have no clue as to how to use this clue to show the necessary and sufficient condition for the conditional independence we are given, so I would appreciate anyone giving me help on this here as this clue seems quite vague and unapproachable to me. I thank all kind helpers.

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