$Var[W]$ when $W = X - 2Y$

Question

I am trying to solve $Var[W]$ when $W = X - 2Y$ for independent discrete RVs $X$ & $Y$:

$$E[X] = 9, Var[X] = 4; E[Y] = 2, Var[Y] = 1.$$

I understand

$$ Var[W] = Var[X] + Var[2Y]$$ $$= Var[X] + 4Var[Y]$$ $$ = 4 + 4*1 = 8$$

But I am trying to answer the same using:

$$ Var[W] = E[W^2] - E[W]^2$$

I think

$$E[W] = 1 \implies E[W]^2 = 1$$

But when I calculate

$$ E[(X-2Y)^2] = E[X^2 - 4XY + 4Y^2]$$ $$ = E[X^2] - 4E[X]E[Y] + 4[Y^2]$$ $$ = 85 -72 + 20 = 33$$ $$ \implies Var[W] = 32$$

Which I don't think is correct.

Where have I gone wrong the second calculation? Thanks

Answer 1

Your mistake is $E[W]=1$. Note that $$E[W]=E[X-2Y]=E[X]-2E[Y]=9-2\cdot 2=5.$$ The rest is correct except $$Var(W)=33-5^2=8.$$ Alternatively, use $$Var(X-2Y)=Var(X)+Var(-2Y)+2Cor(X,-2Y).$$ Since $X$ and $Y$ are independent, $Cor(X,-2Y)=-2Cor(X,Y)=0$. Hence, $$Var(X-2Y)=Var(X)+Var(-2Y)=Var(X)+4Var(Y)=4+4\cdot 1=8.$$