Word Count : 670

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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} __________ _____ ____ _____ ______ ____ ______ _________ __________ ____ _____ ________.
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