Question

If equation with equation and equation. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.

24 Mar 2026
Answer :
Word Count : 663
Let ( \mathbf{X} = \begin{pmatrix} X_1 \ X_2 \ X_3 \end{pmatrix} \sim N_3(\boldsymbol{\mu}, \boldsymbol{\Sigma}) ) with [ \boldsymbol{\mu} = \begin{pmatrix} 2 \ 1 \ 2 \end{pmatrix}, \quad \boldsymbol{\Sigma} = \begin{pmatrix} 5 & 3 & 0 \ 3 & 3 & -2 \ 0 & -2 & 5 \end{pmatrix}. ] We are asked to find the joint distribution of [ \mathbf{Y} = \begin{pmatrix} Y_1 \ Y_2 \ Y_3 \end{pmatrix} = \begin{pmatrix} X_1 + 2X_2 \ 2X_1 - X_2 \ X_3 \end{pmatrix}. ] --- Step 1: Express (\mathbf{Y}) as a linear transformation of (\mathbf{X}) [ \mathbf{Y} = A \mathbf{X}, \quad A = \begin{pmatrix} 1 & 2 & 0 \ 2 & -1 & 0 \ 0 & 0 & 1 \end{pmatrix}. ] --- Step 2: Find the mean vector of (\mathbf{Y}) [ \boldsymbol{\mu}_Y = A \boldsymbol{\mu} = \begin{pmatrix} 1 & 2 & 0 \ 2 & -1 & 0 \ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 2 \ 1 \
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