Variance of the population variance.

**kman320** · November 8th 2008, 10:50 AM

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

How do you derive this (the variance part)?

Thanks.

**mr fantastic** · November 10th 2008, 02:55 AM

Originally Posted by kman320

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 $E(S^2) = \sigma^2$ .

**kman320** · November 10th 2008, 09:07 AM

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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