# Math Help - Variance paradox

1. ## Variance paradox

Hello guys, how are you?
Say we have to find Var(((a-1)/a) x) which equals with ((a-1)^2/a^2) Var (x)

But if we try Var(((a-1)/a) x)= Var((1-(1/a))x) =Var(x)+(1/(a^2))Var (x)
= (1+1/(a^2)) Var (x) is a different result from ((a-1)^2/a^2) Var (x).

With an arithmetic example we see that the first way is correct but what is really going wrong with the second way? Thank you very much!

2. ## Re: Variance paradox

Hello,

The problem with the second one is that you get the sum of two random variables : X and -(1/a)X.
But Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y).
your formula will be okay if your two variables are independent (and hence cov(X,Y)=0), but obviously X and -(1/a)X are not independent

3. ## Re: Variance paradox

Oh thank you very much )) You are correct!