Use Theorem 8.11 to show that, for random samples of size n from a normal population with the variance 2, the sampling distribution of S2

Use Theorem 8.11 to show that, for random samples of size n from a normal population with the variance σ2, the sampling distribution of S2 has the mean σ2 and the variance 2σ4/n–1. (A general formula for the variance of S2 for random samples from any population with finite second and fourth moments may be found in the book by H. Cramer listed among the references at the end of this chapter.)
Theorem 8.11
If  and S2 are the mean and the variance of a random sample of size n from a normal population with the mean µ and the standard deviation s, then
1.  and S2 are independent;
2. The random variable (n–1)S2/σ2 has a chi-square distribution with n – 1 degrees of freedom.