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 $\displaystyle \frac{(n-1)S^2}{\sigma^2}$ has a $\displaystyle \chi^2$ distribution with (n-1) degrees of freedom. Therefore:
$\displaystyle Var \left( \frac{(n-1)S^2}{\sigma^2} \right) = 2(n-1)$
$\displaystyle \Rightarrow \left(\frac{(n-1)}{\sigma^2}\right)^2 Var (S^2) = 2(n-1)$
$\displaystyle \Rightarrow \frac{(n-1)^2}{\sigma^4} \, Var (S^2)= 2(n-1)$
and the result is obvious.
This also provides a very easy way of proving $\displaystyle E(S^2) = \sigma^2$.