Identifying joint distribution

Question

Let $Y_1$ and $Y_2$ be independent random variables with $Y_1\sim N(1,3)$ and $Y_2 \sim N(2,5).$

If $W_1=Y_1+2Y_2$ and $W_2=4Y_1-Y_2$ what is the joint distribution of $W_1$ and $W_2$?

Is my procedure correct?

$E(W_1)=E(Y_1+2Y_2)=E(Y_1)+2E(Y_2)=1+2\cdot2=5$

$Var(W_1)=Var(Y_1+2Y_2)=Var(Y_1)+4 Var(Y_2)=3+4\cdot 5=23$

$f_{W_1}(w_1)=\frac{1}{\sqrt{2\pi}23}\epsilon^{\frac{-1}{2}(\frac{x-5}{23})^2}$

I can't conclude.

Answer 1

Being a linear transformation of independent Gaussian variables, they are jointly Gaussian. So to specify the distribution completely, you have to compute $E[W_1]$, $Var[W_1]$ (which you have already done), $E[W_2]$, $Var[W_2]$, and $cov(W_1,W_2)$.