Mean and Variance of ${S ^ 2}$ when $\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$

Question

I am looking for the detailled proof for this case, if

$$\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$$

Prove that :

$$E[{S ^ 2}] = \sigma ^ {2}$$ $$Var[{S ^ 2}] = \frac{2 \cdot {\sigma ^ {4}} }{n-p}$$

Know that :

$$E[{\chi ^ 2 }(n-p)] = n-p$$ $$Var[{\chi ^ 2 }(n-p)] = 2 \cdot (n-p)$$

Thanks for your help

Answer 1

Variance can be computed in the same way