Finding the covariance matrix of two random gaussian vectors and their characteristic function

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 $$E(ZW)-(EZ)(EW)$$.
• @wageeh Yes., $EZ^{2}$ and $EW^{2}$ can be calculated is a similar way and then you can use the equations $var (Z)=EZ^{2}-(EZ)^{2}, var (W)=EW^{2}-(EW)^{2}$ Commented Jan 23, 2021 at 23:15