# Expectation for a $\chi^2_n$ distributed random variable.

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?

• Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)\sigma^2$, which is enough for your purposes. – J.G. Nov 30 '18 at 23:38