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Math Help - Variance of the population variance.

  1. #1
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    Variance of the population variance.

    Xi's are iid with ~N(μ,σ^2). I am given the population variance.



    How do you derive this (the variance part)?

    Thanks.
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  2. #2
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    Quote Originally Posted by kman320 View Post
    Xi's are iid with ~N(μ,σ^2). I am given the population variance.



    How do you derive this (the variance part)?

    Thanks.
    The very easy way is to know that the random variable \frac{(n-1)S^2}{\sigma^2} has a \chi^2 distribution with (n-1) degrees of freedom. Therefore:

    Var \left( \frac{(n-1)S^2}{\sigma^2} \right) = 2(n-1)

    \Rightarrow \left(\frac{(n-1)}{\sigma^2}\right)^2 Var (S^2) = 2(n-1)

    \Rightarrow \frac{(n-1)^2}{\sigma^4} \, Var (S^2)= 2(n-1)

    and the result is obvious.


    This also provides a very easy way of proving E(S^2) = \sigma^2.
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  3. #3
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    Thanks so much Mr. Fantastic. I was doing it the long and difficult way of Var(S^2)= E(S^4)-σ^4. The E(S^4) was taking forever.
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