Suppose I have four normal r.v (X,Y,W,Z) and the variance-covariance matrix is know. If I create a new r.v J=aX+bY (a and b are scalar), what is the new variance-covariance matrix? Thank you
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$\begingroup$ I mean, the variance-covariance matrix of (J,W,Z) $\endgroup$– AndreaCommented Aug 3, 2015 at 16:18
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$\begingroup$ This question has been answered on a very great number of threads on this site, but it's hard to find the answers with a site search. For a duplicate I chose a thread that clearly asks (and answers) the same question. $\endgroup$– whuber ♦Commented Aug 3, 2015 at 17:57
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$$\mbox{var}(ax + by) = a^2\mbox{var}(x) + b^2\mbox{var}(y) + 2ab \mbox{cov}(x, y)$$
for $x, y$ not mutually independent,
$$\mbox{cov}(ax_1 + by_1, ax_2 + by_2) = a^2 \mbox{cov}(x_1, x_2) + b^2 \mbox{cov}(y_1, y_2) + ab \left( \mbox{cov}(x_1, y_2) + \mbox{cov}(x_2, y_1 )\right)$$
