Question

Let $X_{1},..., X_{n}$ be i.i.d. $N(\mu ,\sigma ^{2})$ . Define the sample mean and the sample variance by

$\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}$ and $S^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$ .

(i) Find the distribution of $\overline{X}$ and $X_{i}-\overline{X}$ for i = 1, ... , n.

(ii) Show that $\overline{X}$ and $X_{i}-\overline{X}$ are independent for i = 1, ... , n.

(iii) Hence, or otherwise, show that $\overline{X}$ and $S^{2}$ are independent.

Answer 1