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$X$ and $Y$ are jointly normal, with the mean vector and covariance matrix given by: $$\mu= \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} \Sigma= \begin{pmatrix} 2 & 0.4 \\ 0.4 & 1 \\ \end{pmatrix}$$ Let $Z_1=X+Y$ and $Z_2=2X-Y$. What is the mean vector and covariance matrix of $Z_1$ and $Z_2$?

What I did so far: I have calculated the mean and variance of $Z_1$ to be 3 and 3.8 respectively. The mean and variance of $Z_2$ are 0 and 7.4 respectively. So all I'm missing is the value for the covariance of $Z_1$ and $Z_2$ which is $\rho\sigma_{Z_1}\sigma_{Z_2}=\rho\sqrt{3.8}\sqrt{7.4}=5.303\rho$. How do I find $\rho$?

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up vote 0 down vote accepted

Upon a linear transformation

$$ \pmatrix{Z_1\\Z_2}=\pmatrix{1&1\\2&-1}\pmatrix{X\\Y}=:A\pmatrix{X\\Y}, $$

the covariance matrix becomes

$$ \Sigma'=A\Sigma A^\top=\pmatrix{1&1\\2&-1}\pmatrix{2&0.4\\0.4&1}\pmatrix{1&2\\1&-1}=\pmatrix{3.8&3.4\\3.4&7.4}\;. $$

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