<s>[INST] Let $X$ ~ Geometric with $p=\frac{1} {4}$ and $Y$ ~ Uniform on $(1,2,3,4)$. Given X and Y are independent, I need to find $VAR(2X + 2Y)$.
Does independence mean that $VAR(2X + 2Y) = VAR(2X) + VAR(2Y)$? If so, then $VAR(X)=\dfrac{1-p} {p^2} = \dfrac{\dfrac {3} {4}} {\dfrac{1} {16}}=12$. And $VAR(Y)= \dfrac {1} {4}(1^2 +2^2 +3^2 +4^2)-[\dfrac{1} {4}(1+2+3+4)]^2=\dfrac{5} {4}.$ Then $VAR(2X+2Y)=2VAR(X)+2VAR(Y)=24+10/2=\dfrac{53} {2}$?
Did I do this right or am I off somehow?
Thanks!
 [/INST] $$Var(2X+2Y) =Var(2(X+Y)) = 4Var(X+Y) = 4[Var(X) + Var(Y)]$$
Or:
If X and Y are independent, then 2X and 2Y are independent.
$$Var(2X+2Y)=Var(2X)+Var(2Y)=4Var(X)+4Var(Y)$$
Assuming your $Var(X)$ and $Var(Y)$ are correct, we have $Var(2X+2Y)=53.$
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