I have a problem with this exercise.
Let $Y_1$ and $Y_2$ be independent random variables with $Y_1∼N(1,3)$ and $Y_2∼N(2,5)$. If $W_1=Y_1+2Y_2$ and $W_2=4Y_1−Y_2$, what is the joint distribution of $W_1$ and $W_2$?
So I know that $W_1∼N(5,23)$ and $W_2∼N(2,53)$. But I also need the correlation coefficient for $W_1$ and $W_2$ for the variance-covariance matrix, how do I calculate this?
Do I need to calculate the covariance first, and in that case, how do I do that?
First note that $(Y_1,Y_2)'$ is bivariate normal because $Y_1,Y_2$ are independent.
Next observe that $$\left(\begin{array}{c} W_{1}\\ W_{2} \end{array}\right)=A\left(\begin{array}{c} Y_{1}\\ Y_{2} \end{array}\right)$$ $$A=\left(\begin{array}{cc} 1 & 2\\ 4 & -1 \end{array}\right),$$
and use the fact that affine transforms of normal random vectors are normal, i.e. $$X\sim N(\mu,\Sigma)\implies AX+b\sim N(A\mu+b,A\Sigma A').$$