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I was wondering if we have two normal distributions of X,Y~N(0,1), why is then X-2Y~N(0,5)? I understand the mean of the X-2Y distribution, but why is the variance 5?

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If $X$ and $Y$ are independent, we can do $$Var(X-2Y) = Var(X)+Var(2Y)=Var(X)+4 Var(Y) = 1 +4(1) = 5.$$

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