所屬科目：研究所、轉學考(插大)◆數理統計學
(a) \( (5\%) \) Compute \( P[X_3 > X_1 + X_2] \), where \( \sqrt{2} = 1.414 \), \( \sqrt{3} = 1.732 \), \( \sqrt{7} = 2.646 \), \( \sqrt{11} = 3.167 \).
(b) \( (10\%) \) Use the moment generating function technique to find the joint distribution of \( W_{1} = X_{1} + 2X_{2} + X_{3} \) $W_2 = 2X_1 - X_3$, and $W_3 = -2X_2 + X_3$.
(c) (5%) Compute $P[W_1 > W_2 + W_3]$, where $\sqrt{2} = 1.414$, $\sqrt{3} = 1.732$, $\sqrt{7} = 2.646$, $\sqrt{11} = 3.167$.
(a) \( (5\%) \) Let \( Y_{n} = \max \{X_{1},\dots,X_{n}\} \). Find the limiting distribution of \( Z_{n} = n(1 - Y_{n}) \).
(b) \( (5\%) \) Let \( Y_{2} \) denote the second smallest item from this sample. Find the limiting distribution of \( W_{n} = nF(Y_{2}) \).
(c) (5%) Find the maximum likelihood estimator (MLE) \( \theta_{n} \) of \( \theta \).
(d) (5%) Prove that \( \theta_{n} \) is a consistent estimator of \( \theta \).
(e) (5%) Find an unbiased estimator \( \tilde{\theta}_n \) of \( \theta \).
(f) (5%) Find the MLE of the median of the distribution for this sample.
(a) (7%)
(b) (7%) Find the uniformly most accurate (UMA) \( 1 - \alpha \) confidence interval. Prove your claim.
(c) (7%) Is your finding in Problem b the shortest length confidence interval (C.I.)? Prove your claim.
(a) (7%) Find a two-dimensional sufficient statistic for \( (\beta, \sigma^2) \).
(b) (7%) Find the MLE of \( \beta \), and show that it is an unbiased estimator of \( \beta \).
(c) (7%) Find the UMVUE of \( \beta \).
C. (8%) Let F be a symmetric distribution and suppose $1 - F(x) = \frac{e}{2x \ln x}, \quad x \geq e.$ Suppose that $\{X_n, n \geq 1\}$ is an iid sequence of random variables with comment distribution F. Show that the sample mean converges in probability to 0.
阿摩線上測驗
登入
阿摩線上測驗
登入
