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Consider $X_1,\ldots, X_n\sim\mathcal{N}(\mu,\sigma^2)$ iid, where $\mu$, $\sigma^2$ are unknown and $c>0$. Define

$$S_n^2:=\sum_{k=1}^n\Big(X_k-\frac{1}{n}\sum_{i=1}^nX_i\Big)^2$$

Now I want to calculate

$$E[cS_n^2-\sigma^2]$$ Actually I was reading that $S_n^2$ might be $\chi_n^2$-distributed, but I do not know why. So does someone has a hint on this problem?

Thanks in advance!

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  • $\begingroup$ Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)\sigma^2$, which is enough for your purposes. $\endgroup$ – J.G. Nov 30 '18 at 23:38
  • $\begingroup$ I was able to show it by simple calculation. Thanks for your comment! $\endgroup$ – user408858 Dec 1 '18 at 12:08

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