112年 - 112 國立臺北大學_碩士班一般入學考試試題_金融與合作經營學系：統計學#146963

(1)(10%) Show that the maximum likelihood estimator of \( \theta \) is \( \hat{\theta} = -(1 / n)\sum{i = 1}^{n}\ln (Xi) \)

(2)(5%) Show the \( \operatorname{E}(\hat{\theta}) = \theta \) and thus \( \hat{\theta} \) is an unbiased estimator of \( \theta \).

2. (10%) Let \( X \) have an exponential distribution with mean \( \theta > 0 \). Show that \( P(X > x + y | X > x) = P(X > y) \).

(1)(10%) E(Y). (Round the answer to 4 decimal places)

(2)(5%) \( \operatorname{Var}(Y) \). (Round the answer to 4 decimal places)

4. (10%) A die was cast $n = 120$ independent times and the following data resulted:

If we use a chi-square test, for what values of b would the hypothesis that the die is unbiased be rejected at the 0.025 significance level? (\( \chi^2_{(5),\alpha=0.025} = 12.8 \); \( \chi^2_{(6),\alpha=0.025} = 14.4 \); \( \chi^2_{(5),\alpha=0.05} = 11.1 \); \( \chi^2_{(6),\alpha=0.05} = 12.6 \))

(1)(5%)The regression line \( \hat{Y} = \hat{\alpha} + \hat{\beta}X \)

(2)(5%)The variance of residuals \( \hat{\sigma}^2 \)。

(3)(5%)The t-statistic of the test \( H_0: \beta = 0 \)

(1) \( (10\%)\mathrm{E}(\mathrm{Y}) \)

(2) \( (10\%)\mathrm{Var}(\mathrm{Y}) \)

(1) \( e^{-x}\geq 1 - x \) (2) \( e^{-1} = 0.3678;\ln (0.1) = -2.30259 \)

(3) \( m \times (m + 1) \times (m + 2) \ldots (m + k) \leq \left(m + \frac{k}{2}\right)^{k + 1} \)