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Math Help - Variance paradox

  1. #1
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    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!
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  2. #2
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    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
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  3. #3
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    Re: Variance paradox

    Oh thank you very much )) You are correct!
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