Word Count : 732

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We are given a 4-dimensional multivariate normal vector: $$ \underline{X} = \begin{pmatrix} \underline{X}^{(1)} \\ \underline{X}^{(2)} \end{pmatrix} \sim N_4(\underline{\mu}, \underline{\Sigma}) $$ with $$ \underline{\mu} = \begin{pmatrix} -4 \\ 1 \\ 4 \\ 0 \end{pmatrix} \quad \text{and} \quad \underline{\Sigma} = \begin{pmatrix} 2 & 2 & \big| & 3 & 0 \\ 2 & 1 & \big| & 2 & 0 \\ \hline 3 & 2 & \big| & 2 & 1 \\ 0 & 0 & \big| & 1 & 1 \\ \end{pmatrix} $$ So, we can partition the mean vector and the covariance matrix as follows: ### Step 1: Partition vectors and matrices Let’s write: * $\underline{X}^{(1)} \in \mathbb{R}^2$ * $\underline{X}^{(2)} \in \mathbb{R}^2$ Partition: $$ \underline{\mu}^{(1)} = \begin{pmatrix} -4 \\ 1 \end{pmatrix}, \quad \underline{\mu}^{(2)} = \begin{pmatrix} 4 \\ 0 \end{pmatrix} $$ $$ \Sigma_{11} = \begin{pmatrix} 2 & 2 \\ 2 & 1 \\ \end{pmatrix}, \quad \Sigma_{22} = _____ _________ __________ ________ ______.
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