Thread: 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!

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

Oh thank you very much )) You are correct!