Word Count : 652

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We are given that $$ \underline{X} \sim N_3(\underline{\mu}, \underline{\Sigma}),\quad \text{where} \quad \underline{\mu} = \begin{pmatrix}2\\1\\2\end{pmatrix}, \quad \underline{\Sigma} = \begin{pmatrix} 5 & 3 & 0 \\ 3 & 3 & -2 \\ 0 & -2 & 5 \end{pmatrix} $$ We need to find the joint distribution of the transformed variables: $$ Y_1 = X_1 + 2X_2,\quad Y_2 = 2X_1 - X_2,\quad Y_3 = X_3 $$ --- ### Step 1: Define transformation Let: $$ \underline{Y} = A \underline{X} $$ where $A$ is the transformation matrix such that: $$ \underline{Y} = \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 0 \\ 2 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} $$ So the transformation matrix is: $$ A = __________ __________ _____ ______ _______ ___.
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