4. Let \( X_1, \dots, X_n \). be i.i.d random variables which has distribution \( N(\mu, \sigma^2) \). Prove that thesample mean \( \overline{X} = \frac{1}{n} \sum_{i=1}^{n} X_{i} \)  and sample variance \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X})^2 \) are independentto each other. (20%)