Let $(W,Z)$ be the gaussian random variable vector that we want to find its covariance matrice and characteristic function.
We have $(X,Y)$ a guassian random variable vector with a mean of $m=(1,2)$ and a covariance matrix: $$ \begin{bmatrix} 4 & 1 \\ 1 & 4\\ \end{bmatrix} $$
We have that $Z=2X+Y-2$ and $W=\alpha$X+Y.
How do I calculate the covariance matrix of Z and W?
I calculated the following :
The expected value of W is $\alpha +2$.
The expected value of Z is $2$
and both of their characteristics functions.
How to calculate the characteristic function of $(W,Z)$ given both of their characteristic function and we don't know anything about their independency?
$E(ZW)=E(2X+Y-2) (\alpha X+Y)=2\alpha E(X^{2}+2E(XY)+\alpha (XY)+E(Y^{2})-2\alpha EX-2EY$ You know $EX$ and$EY$. Now $E(XY)=4 $, $EX^{2}=5$ and $EY^{2}=8$. Independence is not required in this calculation. Finally, the covariance of $Z$ and $W$ is $E(ZW)-(EZ)(EW)$.