第6章 习题参考答案

习题参考答案

习题6.1 答案

1. ${\hat{\mu }}_3$ 有效性最差． 2. 不是． 3. 略．$4\frac{1}{2(n-1)}$ . 5. $\frac{1}{2}(x_{(1)}+x_{(n)})$ 较有效．

2. $\frac{4}{3}{x}_{(3)}$ 较有效． 7. $a=\frac{n_1}{n_1+n_2},b=\frac{n_2}{n_1+n_2}$ . 8. 略． 9. $a_i=\frac{1}{{\sigma }_i^2}/{\sum }_{j=1}^n\frac{1}{{\sigma }_j^2},i=1,2,\cdots ,n.$

3. 略． 11. $C_1=\sqrt{\frac{n\pi }{2(n-1)}},C_2=\frac{\sqrt{\pi }}{2}$ .

习题 6.2

1. 114 3.75, 96.056 2. 2. 2.68. 3. (1) $2\overline{x}$ ; (2) $\frac{2}{\overline{x}}$ .

2. (1) $3\overline{x}$ ; (2) $\frac{1-2\overline{x}}{\overline{x}-1}$ ; (3) ${\left(\frac{\overline{x}}{1-\overline{x}}\right)}^2$ ; (4) $\hat{\theta }=s,\hat{\mu }=\overline{x}-s.$

3. $u_{k/n}$ 6. (1) $\frac{ab}{c}$ ; (2) $\frac{ab}{c}-a-b+c$ . 7. $\hat{m}=\left[\frac{{\overline{x}}^2}{\overline{x}-s_n^2}\right],\hat{p}=1-\frac{s_n^2}{\overline{x}}$ .

习题 6.3

1. (1) ${\left(\frac{1}{n}{\sum }_{i=1}^n\ln x_i\right)}^{-2}$ ; (2) ${\left(\frac{1}{n}{\sum }_{i=1}^n\ln x_i-\ln c\right)}^{-1}$ .

2. (1) $x_{(1)}$ ; (2) $\hat{\mu }=x_{(1)},\hat{\theta }=\overline{x}-x_{(1)}$ ; (3) $\frac{x_{(n)}}{k+1}$ .

3. (1) $\frac{1}{n}{\sum }_{i=1}^n|x_i|$ ; (2) 可取 $\left(x_{(n)}-\frac{1}{2},x_{(1)}+\frac{1}{2}\right)$ 中的任意值; (3) ${\hat{\theta }}_1=x_{(1)},{\hat{\theta }}_2=x_{(n)}$ .

4. 0.499. 5. $\frac{2(\overline{x}-1)}{\overline{x}}$ . 6. 28.305 3. 7. (1) 略; (2) 不是无偏估计，是相合估计．

8.（1）是相合估计，不是无偏估计；（2）是相合估计，是无偏估计．

9. 15000. 10. 略．

习题 6.4

1—5. 略 6. (1) $-\frac{1}{n}{\sum }_{i=1}^n\ln x_i$ ; (2) $-\frac{1}{n}{\sum }_{i=1}^n\ln x_i$ . 7. $\frac{1}{{\theta }^2}$ . 8. $\frac{1}{{\theta }^2}$ . 9. $\frac{2}{{\theta }^2(1-\theta )}$ .

10. 略． 11. $\frac{m\bar{x}+\frac{1}{2}n\bar{y}}{m+\frac{1}{2}n},\frac{1}{m+n-1}\left({\sum }_{i=1}^m{x}_i^2+\frac{1}{2}{\sum }_{i=1}^n{y}_i^2-\frac{(m\bar{x}+\frac{1}{2}n\bar{y}{)}^2}{m+\frac{1}{2}n}\right)$ . 12. 略．

11. (1) ${\theta }^3$ ; (2) $g(\theta )=a\sqrt{\theta }+b$ ( $a>0,b$ 任意).

12. (1) $\frac{1}{2}+\frac{{\sum }_{i=1}^n{x}_i}{2{\sum }_{i=1}^n{x}_i^2}$ ; (2) $\overline{x}+\frac{1}{2}$ ; (3) $\frac{2\theta (1-\theta )}{n}$ . 15. 略．

习题 6.5

1. $P(\lambda =1.5\mid x=3)=0.3899,P(\lambda =1.8\mid x=3)=\mathrm{0.6101.}$ 2. $U(11.1,11.7)$ .

2. (1) $Be\left(n+1,{\sum }_{i=1}^n{x}_i+1\right)$ ; (2) 0.25. 4—5. 略

3. (1) $\pi (\theta \mid x_1,x_2,\cdots ,x_n)=\frac{2n-1}{{\theta }^{2n}(x_{(n)}^{-2n+1}-1)}$ ; (2) $\pi (\theta \mid x_1,x_2,\cdots ,x_n)=\frac{2n-3}{{\theta }^{2n-2}(x_{(n)}^{-2n+3}-1)}$ .

4. $\frac{n+\alpha }{\lambda -{\sum }_{i=1}^n\ln x_i}$ . 8. (1) 略; (2) $\frac{(n+\beta )\max \{x_{(n)},{\theta }_0\}}{n+\beta -1}$ . 9. $Ga(0.0004,2)$ .

5. 略 11. $\pi (\theta \mid x_1,x_2,x_3)=6\times 8^6{\theta }^{-7},\theta >8.$ 12—13. 略

6. Be(5,297).

习题 6.6

1. [1.916 8,2.583 2]. 2. ${\left(\frac{3.92\sigma }{k}\right)}^2$ .

2. (1) $[-0.9800,0.9800]$ ; (2) $[0.6188,4.3929]$ .

3. (1) [0.148 7, 0.421 5]; (2) [56.073 9, 56.566 1].

4. (1) [432.306 4,482.693 6]; (2) [438.905 8,476.094 2];

(3) [24.223 9,64.137 8]. 6. [0.074 2,0.200 8]. 7. 略．

8. [11.039 2, 12.999 2]. 9. (1) [-0.093 9, 12.093 9]; (2) [-0.206 3, 12.206 3];

(3) $[-0.3288,12.3288]$ ; (4) $[0.3359,4.0973]$ .

10. (1) [0.062 0, 1.007 5]; (2) [-0.285 5, 0.325 5]. 11. $\left(\frac{{\chi }_{\frac{\alpha }{2}}^2(2n)}{2n\overline{x}},\frac{{\chi }_{1-\frac{\alpha }{2}}^2(2n)}{2n\overline{x}}\right)$ .

11. [36.76, 113.64], 98.04, 40.82. 13. $\left[m_{0.5}-\frac{\pi }{2\sqrt{n}}{u}_{1-\frac{\alpha }{2}},m_{0.5}+\frac{\pi }{2\sqrt{n}}{u}_{1-\frac{\alpha }{2}}\right]$ .

12. $\frac{64u_1^2-\frac{\alpha }{2}}{L^2}$ . 15. 略． 16. $\left[\frac{x_{(n)}+x_{(1)}}{2}-\frac{1-{\alpha }^{1/n}}{2},\frac{x_{(n)}+x_{(1)}}{2}+\frac{1-{\alpha }^{1/n}}{2}\right]$ .

13. (1) $\left[\frac{x_{(n)}-x_{(1)}}{B{e}_{1-\alpha /2}(n-1,2)},\frac{x_{(n)}-x_{(1)}}{B{e}_{\alpha /2}(n-1,2)}\right]$ ;

(2) $\left[\frac{x_{(n)}+x_{(1)}}{2}-\frac{c(x_{(n)}-x_{(1)})}{2},\frac{x_{(n)}+x_{(1)}}{2}+\frac{c(x_{(n)}-x_{(1)})}{2}\right]$，其中 $c={\left(\frac{1}{\alpha }\right)}^{\frac{1}{n-1}}-1.$

18. $\left[\frac{x_{(m)}}{y_{(n)}}{\left(\frac{m+n}{2m}\alpha \right)}^{\frac{1}{n}},\frac{x_{(m)}}{y_{(n)}}{\left(\frac{m+n}{2n}\alpha \right)}^{-\frac{1}{m}}\right].$

19. (1) 略; (2) $\left[x_{(1)}+\frac{\ln \alpha }{n},x_{(1)}\right]$ .