Let X and Y be jointly distributed random variables. Define $W = 2X+3Y$.
(a) What is $E(W)$? (b) What is $Var (W)$? (c) What is $Var W$ when $X$ and $Y$ are independent?
For this, I used the property $E(aX+bY) = aE(X)+b(Y)$ to get the answer for (a). For (b) and (c) I used the property that $Var(aX+bY)= a^2Var(X) + b^2Var(Y)$, however, I think it only applies for (c) because X and Y are independent. Is this correct? If not what theorem should I use for (b)? Should I also indicate the discrete and continuous cases for (a) (b) and (c)? What does it mean also to 'Define $W= 2X +3Y$? Thanks!
You are right. $$Var(2X+3Y) = 4Var(X) + 9Var(Y) + 12Cov(X,Y).$$
So, if $X$ and $Y$ are independent, $Cov(X,Y) = 0$. then $Var(W) = 4Var(X) + 9Var(Y)$.