Question

Let equation, where equation and equation.
Then find equation and equation.

24 Mar 2026
Answer :
Word Count : 670
We are asked to find the conditional expectation and conditional covariance of a partitioned multivariate normal. Let's solve it carefully, step by step. We have [ X = \begin{pmatrix} X^{(1)} \ X^{(2)} \end{pmatrix} \sim N_4(\mu, \Sigma), ] with [ \mu = \begin{pmatrix} -4 \ 1 \ \hline 4 \ 0 \end{pmatrix}, \quad \Sigma = \begin{pmatrix} 2 & 0 & 1 & 0 \ 0 & 2 & 2 & 0 \ \hline 1 & 2 & 6 & 1 \ 0 & 0 & 1 & 1 \end{pmatrix}. ] Let’s denote the partitions: [ X^{(1)} = \begin{pmatrix} X_1 \ X_2 \end{pmatrix}, \quad X^{(2)} = \begin{pmatrix} X_3 \ X_4 \end{pmatrix}. ] --- ### Step 1: Partition (\Sigma) The covariance matrix can be partitioned as [ \Sigma = \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} \end{pmatrix}, ] where [ \Sigma_{11} = \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix}, \quad \Sigma_{12} = \begin{pmatrix} 1 & 0 \ 2 & 0 \end{pmatrix}, \quad \Sigma_{21} = \Sigma_{12}^T = \begin{pmatrix} 1 & 2 \ 0 & 0 \end{pmatrix}, \quad \Sigma_{22} = \begin{pmatrix} 6 & 1 \ 1 & 1 \end{pmatrix}. ] --- ### Step 2: Conditional expectation formula For multivariate normal, [ E[X^{(2)} \mid X^{(1)} = x^{(1)}] = \mu_2 + \Sigma_{21}\Sigma_{11}^{-1}(x^{(1)} - \mu_1), ] where [ \mu_1 = \begin{pmatrix}-4 \ 1\end{pmatrix}, \quad \mu_2 = \begin{pmatrix} 4 \ 0 \end{pmatrix}. ] --- ### Step 3: Compute (\Sigma_{11}^{-1}) [ \Sigma_{11}
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