97年 - 97 國立高雄大學_博士班招生考試試題_統計學研究所：數理統計#147980

1. Suppose that nX is a random sample with variance σ² . Let \( U = \frac{1}{n} \sum_{i=1}^{2n} X_{2i-1} \) and
\( V = \frac{1}{2n} \sum_{i=1}^{2n} X_i \) . Consider Chebyshev inequality and Center Limit Theorem, respectively, tosuggest a value of n which such that \( P(|U - V| \le \sigma) \) . Which method gives a smallersuggested value of n ? Why? (40%)

(a) Find the value of k .

(b) Find \( E(X^2 e^{Y+Y^{-1}} \mid Y = y) \).(25%)

3. Find the value of \( Cov(2X, E(Y|X)) \) in terms of Cov(X,Y)(15%)

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%)