第2章 习题参考答案

习题参考答案

习题2.1 答案

$$1.(1)\frac{X}{P}|\begin{array}{lll}3& 4& 5\\ 0.1& 0.3& 0.6\end{array};(2)F(x)=\{\begin{array}{ll}0,& x<3,\\ 0.1,& 3⩽x<4,\\ 0.4,& 4⩽x<5,\\ 1,& 5⩽x.\end{array}$$$$2.(1)\begin{array}{ccccccc}X& 1& 2& 3& 4& 5& 6\\ P& \frac{11}{36}& \frac{9}{36}& \frac{7}{36}& \frac{5}{36}& \frac{3}{36}& \frac{1}{36}\end{array};$$$$\begin{array}{ccccccc}Y& 0& 1& 2& 3& 4& 5\\ P& \frac{3}{18}& \frac{5}{18}& \frac{4}{18}& \frac{3}{18}& \frac{2}{18}& \frac{1}{18}\end{array}.\text{\hspace{1em}(2)}$$$$3.(1)\begin{array}{ccccc}X& 1& 2& 3& 4\\ P& \frac{7}{10}& \frac{7}{30}& \frac{7}{120}& \frac{1}{120}\end{array};$$$$\begin{array}{ccccc}X& 1& 2& 3& 4\\ P& \frac{7}{10}& \frac{6}{25}& \frac{27}{500}& \frac{3}{500}\end{array}.\text{\hspace{1em}(2)}$$$$4.(1)\begin{array}{ccccc}X& 0& 1& 2& 3\\ P& \frac{5}{30}& \frac{15}{30}& \frac{9}{30}& \frac{1}{30}\end{array};(2)\frac{1}{3}.$$$$\begin{array}{cccccc}X& 0& 1& 2& 3& 4\\ P& 0.4823& 0.3858& 0.1157& 0.0154& 0.0008\end{array};$$$$6.P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{39}{5-k}\right)\left(\genfrac{}{}{0ex}{}{13}{k}\right)}{\left(\genfrac{}{}{0ex}{}{52}{5}\right)},k=0,1,2,3,4,5.$$

7. (1) $P(X=k)=\frac{C_{90}^{5-k}{C}_{10}^k}{C_{100}^5},k=0,1,2,3,4,5;$ (2) 0.416 2.

8. $\frac{X}{P}$ $\frac{1}{4}$ $\frac{1}{12}$ $\frac{1}{6}$ $\frac{1}{2}$ . $\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}$ . 9. ln 2, 1, ln 1.25.

9. $1-\alpha -\beta$ 11. (1) $F(x)=\{\begin{array}{ll}0,& x<2,\\ \frac{3}{10},& 2⩽x<3,\\ \frac{7}{10},& 3⩽x<4,\\ 1,& 4⩽x;\end{array}$

10. $F(x)=\{\begin{array}{ll}0,& x<-1,\\ \frac{x^2}{2}+x+\frac{1}{2},& -1⩽x<0,\\ -\frac{x^2}{2}+x+\frac{1}{2},& 0⩽x<1,\\ 1,& 1⩽x.\end{array}$ 13.0.875.

11. (1) $A=\frac{1}{2}$ ; (2) $\frac{\sqrt{2}}{4}$ . 15. (1) $A=1$ ; (2) 0.4; (3) $p(x)=2x$ , $0<x<1$ .

12. (1) $c=21$ ; (2) $F(x)=\{\begin{array}{ll}0,& x<0,\\ 7x^3+\frac{1}{2}{x}^2,& 0⩽x<0.5,\\ 1,& 0.5⩽x;\end{array}$ (3) $\frac{17}{54}$ ; (4) $\frac{103}{108}$ .

13. 46 千升. 18. $\sqrt[3]{4}$ . 19. 略.

习题 2.2

1. $E(X)=-0.2,E(3X+5)=4.4$ . 2. 1.9. 3. 1.201.

X	0	1	2	3	4	5
P	0.051 1	0.255 4	0.397 3	0.238 4	0.054 2	0.003 6

5. 乙组砝码. 6. $\frac{2}{9}$ . 7. $\frac{1-(1-p{)}^a}{p}$ . 8. 先回答问题 2. 9. 正确. 10. $0.1+p$ .

6. 50. 12. 0.2. 13. 7. 14. 1.

7. $a=\frac{1}{3},b=2$ . 16. (1) 11; (2) 100; (3) 20. 17. 5. 18. $\frac{3}{4}$ . 19—21. 略.

习题 2.3

1. 1. 2. 0.217 3. 3. 9. 4. $\frac{1}{3}$ 5. 6.5. 6. 1.5. 7. 0.05.

2. $\frac{\sqrt{\pi }}{2},1-\frac{\pi }{4}$ . 9—13. 略. 14. $\frac{8}{9}$ .

习题 2.4

1. 0.972. 2. 0.993 3. 3. 0.784. 4. 0.000 127 9. 5. $n=6,p=0.4$ .

2. $\frac{80}{81}$ . 7. (1) 0.1905; (2) 0.1912. 8. 0.0902. 9. 略. 10. 0.889.

3. (1) 0.1875; (2) 0.0156. 12. $\frac{n}{m}$ . 13. 1.

4. $\frac{9}{64}$ . 15. 1.055 6. 16. $p>\frac{1}{2}$ . 17. $E(X^3)={\lambda }^3+2{\lambda }^2$ .

18—20. 略.

习题 2.5

1. $\frac{20}{27}$ . 2. $\frac{3}{4}$ . 3. $\frac{4}{5}$ . 4. 4. 5. $\frac{602}{3}$ . 6. $\frac{\pi (a^2+b^2+ab)}{12}$ . 7. 21.

2. 0.152 3. 9.4. 10.33.64 元. 11.0.516 7. 12. $1-e^{-1}$ .

3. $k=\frac{\ln 2}{\lambda }$ . 14. [1,3]. 15. $-2,\frac{1}{\sqrt{2}};0,\frac{1}{2};0,\frac{1}{\sqrt{2}}$ . 16. 0.593 4.

4. (1) $\mu =70,\sigma =14.81$ ; (2) 0.94. 18. 0.045 6. 19. 0.841 3.

5. (1) 0.5328; (2) 0.6977; (3) 3.

6. (1) 0.9544; (2) 0.6304; (3) $d⩽0.154$ .

7. 0.869 8. 23. (1) 0.158 7; (2) 0.691 5; (3) 0.682 6. 24. 78.75.

8. $a=55.56,b=58.5,c=61.5,d=\mathrm{64.44.}$ 26.一样大小. 27.0.95.

9. 不变. 29. 24.3. 30. $\sigma \sqrt{\frac{2}{\pi }}$ . 31. 由上题即可得. 32. 0.594 0.

10. 0.2639, 0.1818. 34. 0.4096.

习题 2.6

1.	Y	0	1	4	9	Z	0	1	2	3	2.	Y	-1	1
	P	$\frac{1}{5}$	$\frac{7}{30}$	$\frac{1}{5}$	$\frac{11}{30}$	P	$\frac{1}{5}$	$\frac{7}{30}$	$\frac{1}{5}$	$\frac{11}{30}$		P	0.5	0.5

3. $\frac{Y}{P}|\begin{array}{cc}-1& 1\\ \frac{1}{3}& \frac{2}{3}\end{array}$ 4. $1-X\sim U(0,1)$ .5. $p(y)=\frac{2}{\pi \sqrt{1-y^2}},0<y<1.$

4. $p(y)=\frac{1}{\sqrt{\pi y}}$ ， $0<y<\frac{\pi }{4}.$ 7. $p(y)=\frac{1}{2y},{\mathrm{e}}^2<y<{\mathrm{e}}^4.$

5. (1) $p(y)=\frac{1}{4\sqrt{y}}$ , $0<y<4$ ; (2) $\frac{1}{\sqrt{2}}$ . 9. (1) $\frac{1}{2}$ ; (2) $p(y)=y,0<y<1$ .

6. (1) $p(y)=0.5e^{-0.5y}$ , $y>0$ ; (2) $p(y)=\frac{1}{3}$ , $1<y<4$ ; (3) $p(y)=\frac{1}{y}$ , $1<y<e$ ;

(4) $p(y)={\mathrm{e}}^{-y},y>0.$

11. (1) $p(y)=\frac{y^2}{18}$ , -3 < y < 3; (2) $p(y)=\frac{3(3-y{)}^2}{2}$ , 2 < y < 4; (3) $p(y)=3\frac{\sqrt{y}}{2}$ , 0 < y < 1.

12. $p(y)=\frac{1}{\sqrt{2\pi y}\sigma }{\mathrm{e}}^{-\frac{y}{2{\sigma }^2}},y>0.$

13. $p(y)=\frac{1}{\sqrt{2\pi }y\sigma }\exp \left\{-\frac{(\ln y-\mu {)}^2}{2{\sigma }^2}\right\},y>0.$

$$E(Y)=\exp \left\{\mu +\frac{{\sigma }^2}{2}\right\},\mathrm{Var}(Y)={\mathrm{e}}^{2\mu +{\sigma }^2}({\mathrm{e}}^{{\sigma }^2}-1).$$

14. (1) $p(y)=\sqrt{\frac{2}{\pi }}{e}^{-\frac{y^2}{2}},y>0;$ (2) $p(y)=\frac{1}{2\sqrt{\pi (y-1)}}{e}^{-\frac{y-1}{4}},y>1.$

15. (1) $p(y)=\frac{1}{2}{e}^{-\frac{y-1}{2}}$ , $y>1$ ; (2) $p(y)=\frac{1}{y^2}$ , $y>1$ ; (3) $p(y)=\frac{1}{2\sqrt{y}}{e}^{-\sqrt{y}}$ , $y>0$ .

16—17. 略. 18. 0.977 2.

习题 2.7

$$1.{\mu }_1=\frac{a+b}{2},{\mu }_2=\frac{a^2+ab+b^2}{3},{\mu }_3=\frac{a^3+a^{2}b+a{b}^2+b^3}{4},{\mu }_4=\frac{a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4}{5};$$$${\nu }_1=0,{\nu }_2=\frac{(b-a{)}^2}{12},{\nu }_3=0,{\nu }_4=\frac{(b-a{)}^4}{80};{\beta }_s=0,{\beta }_k=-\mathrm{1.2.}$$

2. $\frac{\sqrt{3}}{3}$ . 3. (1) $\frac{a+b}{2}$ , (2) $\mu$ ; (3) ${\mathrm{e}}^{\mu }$ .

$${\mu }_1=\frac{\alpha }{\lambda },{\mu }_2=\frac{\alpha (\alpha +1)}{{\lambda }^2},{\mu }_3=\frac{\alpha (\alpha +1)(\alpha +2)}{{\lambda }^3};{\nu }_1=0,{\nu }_2=\frac{\alpha }{{\lambda }^2},{\nu }_3=\frac{2\alpha }{{\lambda }^3}.$$$${\mu }_1=\frac{1}{\lambda },{\mu }_2=\frac{2}{{\lambda }^2},{\mu }_3=\frac{6}{{\lambda }^3},{\mu }_4=\frac{24}{{\lambda }^4};{\nu }_1=0,{\nu }_2=\frac{1}{{\lambda }^2},{\nu }_3=\frac{2}{{\lambda }^3},{\nu }_4=\frac{9}{{\lambda }^4};C_v(X)=1,{\beta }_s=2,{\beta }_k=6.$$

6. $x_{0.1}=6.16,x_{0.9}=\mathrm{13.84.}$

7. $x_p=\eta [-\ln (1-p){]}^{\frac{1}{m}}$ ；当 $m=1.5,\eta =1000$ 时， $x_{0.1}=223.08,x_{0.5}=783.22,x_{0.8}=\mathrm{1373.36.}$

8. $x_{0.1}=0.211,x_{0.5}=1.386,x_{0.8}=\mathrm{3.219.}$ 9—10. 略.

9. (1) $\exp \{\mu +\sigma u_p\}$ ; (2) 62; (3) 341. 12. $\mu =9.799,\sigma =1.527$ . 13. $4099\mathrm{kg}$ .