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!