Question

X\sim N_3\left ( \mu ,\sum \right ) with \mu=\begin{pmatrix} 2\\ 1 \\ 2 \end{pmatrix}  and \sum=\begin{pmatrix} 5 & 3&0 \\ 3 &3 & -2\\ 0 & -2 & 5 \end{pmatrix}.  Then find the joint distribution of   X_1+2X_2,2X_1-X_2 and X_3.

02 Apr 2024
Answer :
Word Count : 584

To find the joint distribution of \( X_1 + 2X_2 \), \( 2X_1 - X_2 \), and \( X_3 \), we first need to express the given multivariate normal distribution \( X \sim N_3(\mu, \sum) \) in terms of its components. Given that:

\[
\mu = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}
\]

and

\[
\sum = \begin{pmatrix} 5 & 3 & 0 \\ 3 & 3 & -2 \\ 0 & -2 & 5 \end{pmatrix}
\]

We have \( X_1 \sim N(2, 5) \), \( X_2 \sim N(1, 3) \), and \( X_3 \sim N(2, 5) \) and they are independent since the covariance matrix \( \sum \) is diagonal.

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