Discussion
Let \(X \sim N_{3}(\mu, \Sigma)\) with
\[ \Sigma = \begin{bmatrix} 1 & \rho & \rho^{2} \\ \rho & 1 & 0 \\ \rho^{2} & 0 & 1 \end{bmatrix} \]
Then the conditional distribution of \((X_{1}, X_{2})\) given \(X_{3} = x_{3}\) is \(N_{2}(\mu^{*}, \Sigma^{*})\) where \(\mu^{*}\) and \(\Sigma^{*}\) are respectively given by :
Answer: C
The conditional parameters are \(\mu^{*} = (\mu_{1} + \rho^{2}(x_{3} - \mu_{3}),\ \mu_{2})\) and \(\Sigma^{*}\) with rows \((1 - \rho^{4}, \rho)\), \((\rho, 1)\).
Partition \(\Sigma\) with \(\Sigma_{12} = (\rho^{2}, 0)'\) (the covariances of \(X_{1}, X_{2}\) with \(X_{3}\)) and \(\Sigma_{22} = 1\). Then
\[ \begin{aligned} \mu^{*} &= \mu_{(1)} + \Sigma_{12}(x_{3} - \mu_{3}) \\ \Sigma^{*} &= \Sigma_{11} - \Sigma_{12}\Sigma_{12}' \end{aligned} \]
The mean of \(X_{1}\) shifts by \(\rho^{2}(x_{3} - \mu_{3})\) and \(X_{2}\), uncorrelated with \(X_{3}\), keeps mean \(\mu_{2}\). Only the (1, 1) entry of the covariance changes, to \(1 - \rho^{4}\). Options A and B keep the unconditional covariance, and options A and D the unconditional mean.
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