第3章习题参考答案 · 《概率论与数理统计教程（第三版）》

习题参考答案

习题3.1 答案

(1) $p_{ij} = \frac{\binom{50}{i}\binom{30}{j}\binom{20}{5 - i - j}}{\binom{100}{5}},\quad i + j\leqslant 5;$

(2) $p_{ij} = \frac{5!}{i! j! (5 - i - j)!}(0.5)^i (0.3)^j (0.2)^{5 - i - j}, i + j \leqslant 5.$

$\frac{9}{35}.$

$X_1$	$X_2$	$X_3$	P
0	0	0	0.195 8
0	0	1	0.163 2
0	1	0	0.163 2
1	0	0	0.163 2
0	1	1	0.093 2
1	0	1	0.093 2
1	1	0	0.093 2
1	1	1	0.035 0

(2)

$X_1$	$X_2$
$X_1$	0	1
0	$\frac{14}{39}$	$\frac{10}{39}$
1	$\frac{10}{39}$	$\frac{5}{39}$

1. 1. (1) $\frac{1}{8}$ ; (2) $\frac{3}{8}$ ; (3) $\frac{27}{32}$ ; (4) $\frac{2}{3}$ .
(1) 12; (2) $F(x, y) = (1 - e^{-3x})(1 - e^{-4y})$ , $x > 0, y > 0$ ; (3) $1 - e^{-3} - e^{-8} + e^{-11}$ .
(1) $\frac{15}{64}$ . (2) 0; (3) 0.5; (4) $F(x) = \begin{cases} 0, & x < 0\text{, 或 } y < 0, \\ x^2y^2, & 0 \leqslant x < 1, 0 \leqslant y < 1, \\ x^2, & 0 \leqslant x < 1, 1 \leqslant y, \\ y^2, & 1 \leqslant x, 0 \leqslant y < 1, \\ 1, & x \geqslant 1, y \geqslant 1. \end{cases}$ 8. $\frac{\pi}{4}$ .
(1) 6; (2) $\frac{1}{2}$ , 0.664 2.
(1) $\frac{1}{8}$ ; (2) $\frac{7}{8}, \frac{1}{2}$ ; (3) $\frac{3}{4}$ . 11. $X_{1}$ $\begin{array}{c|cc} & X_{2} \\ \hline & 0 & 1 \\ \hline 0 & 1 - e^{-1} & 0 \\ 1 & e^{-1} - e^{-2} & e^{-2} \end{array}$
$\frac{65}{72}$ . 13. 0.580 9. 14. $\frac{5}{8}$ . 15. 0.044.

习题 3.2

X	-1	0	1	Y	0	1	2
P	$\frac{5}{12}$	$\frac{1}{6}$	$\frac{5}{12}$	P	$\frac{7}{12}$	$\frac{1}{3}$	$\frac{1}{12}$

$F_{x}(x) = 1 - \mathrm{e}^{-\lambda_ {1} x},\quad x > 0,\quad F_{y}(y) = 1 - \mathrm{e}^{-\lambda_ {2} y},\quad y > 0.$
$p_{X}(x) = \frac{2\sqrt{1 - x^{2}}}{\pi}, - 1 < x < 1,\quad p_{Y}(y) = \frac{2\sqrt{1 - y^{2}}}{\pi}, - 1 < y < 1.$

p (x, y) = \frac {1}{2}, \quad 1 < x < e ^ {2}, \quad 0 < y < \frac {1}{x}, \quad p _ {X} (x) = \frac {1}{2 x}, 1 < x < e ^ {2}.

(1) $p_{x}(x) = \mathrm{e}^{-x}, \quad x > 0, \quad p_{y}(y) = y\mathrm{e}^{-y}, \quad y > 0;$

(2) $p_{x}(x)=\frac{5(1-x^{4})}{8},\quad-1<x<1,\quad p_{y}(y)=\frac{5\sqrt{1-y}(1+2y)}{6},\quad0<y<1;$

(3) $p_{x}(x) = 1, \quad 0 < x < 1, \quad p_{y}(y) = -\ln y, \quad 0 < y < 1.$

$p_{X}(x) = 6(x - x^{2})$ ， $0 < x < 1$ ， $p_{Y}(y) = 6(\sqrt{y} -y)$ ， $0 < y < 1.$ 7.略.8.0.5.
0.89. 10. $a = \frac{1}{18}, b = \frac{2}{9}, c = \frac{1}{6}$ .
(1) $p(x, y) = \mathrm{e}^{-y}$ , $0 < x < 1$ , $y > 0$ ; (2) $\mathrm{e}^{-1}$ ; (3) $\mathrm{e}^{-1}$ .
(1) $p_{x}(x) = 3x^{2}$ , $0 < x < 1$ , $p_{y}(y) = 3(1 - y^{2}) / 2$ , $0 < y < 1$ ; (2) 不独立．
(1) $p_X(x) = \begin{cases} 1 + x, & -1 < x < 0, \\ 1 - x, & 0 < x < 1, \\ 0, & \text{其他}, \end{cases} \quad p_Y(y) = \begin{cases} 2y, & 0 < y < 1, \\ 0, & \text{其他}; \end{cases}$ (2) 不独立．
(1)(2)(5) 独立，(3)(4)(6) 不独立．
$\frac{2}{9}$ . 16. 略．

习题 3.3

1.	U	1	2	3	V	0	1	2
1.	P	0.12	0.37	0.51	P	0.40	0.44	0.16

$P(Z = 1) = \frac{\lambda}{\lambda + \mu}, P(Z = 0) = \frac{\mu}{\lambda + \mu}$ . 3. $\frac{Z}{P} \begin{array}{c|cc} & 0 & 1 \\ \hline 0.25 & 0.75 \end{array}$ .
(1) $\frac{Z}{P}\left|\begin{array}{cc}0 & 1\\ 0.25 & 0.75\end{array}\right.;$ (2) $(1 - p)^{i - 1}p[2 - (1 - p)^{i - 1} - (1 - p)^{i}],i = 1,2,\dots .$
$\frac{5}{7}$ . 6. (1) $p_{Z}(z) = 4ze^{-2z}$ , $z > 0$ ; (2) $p_{Z}(z) = \frac{e^{-|z|}}{2}$ , $-\infty < z < \infty$ .
$p_{z}(z) = \frac{3(1 - z^{2})}{2}, 0 < z < 1.$
(1) $p_{2}(x) = x^{3} \mathrm{e}^{-x} / 6, \quad x > 0;$ (2) $p_{3}(x) = x^{5} \mathrm{e}^{-x} / 120, \quad x > 0.$
(1) $p_{z}(z)=\left\{\begin{aligned}z, & 0 \leqslant z<1, \\ 2-z, & 1 \leqslant z<2, \\ 0, & \text{其他};\end{aligned}\right.$ (2) $p_{z}(z)=\left\{\begin{aligned}1-\mathrm{e}^{-z}, & 0<z<1, \\ (\mathrm{e}-1)\mathrm{e}^{-z}, & z>1, \\ 0, & \text{其他}.\end{aligned}\right.$
(1) $p_{Z}(z) = 1 - \left(1 + \frac{1}{z}\right)\mathrm{e}^{-1 / z}, z > 0;$ (2) $p_{Z}(z) = \frac{\lambda_{1}\lambda_{2}}{(\lambda_{1}z + \lambda_{2})^{2}}, z > 0.$
0.5. 12. $p_{z}(z) = \frac{4}{3} z^{3} - 4z + \frac{8}{3}, 0 < z < 1$ . 13. $p_{T}(t) = 3\lambda e^{-3\lambda t}, t > 0$ .
$p_{z}(z) = (\ln 2 - \ln z) / 2, \quad 0 < z < 2.$ 15. $p(r,\theta) = \frac{r}{\pi}, 0 < r < 1, 0 < \theta < 2\pi.$
(1) $p(u, v) = u e^{-u}$ , $u > 0$ , $0 < v < 1$ ; (2) 独立． 17—21. 略．

习题 3.4

$\frac{91}{36}$ . 2. $E(X) = \frac{7n}{2}$ , $\operatorname{Var}(X) = \frac{35n}{12}$ . 3. $\frac{n + 2}{3}$ .
$\frac{n - 1}{n + 1}$ . 5. $\frac{2n - 1}{n - 1}$ . 6. 0.25. 7. $\frac{4}{3}, \frac{1}{18}$ . 8—9. 略． 10. $2\sigma^2$ .
$\frac{5}{8}$ . 12. $p_{Y}(y) = 10y^{9}, 0 < y < 1, E(Y) = \frac{10}{11}, \operatorname{Var}(Y) = \frac{5}{726}$ .
(1) $\frac{1}{n\lambda}$ ; (2) $\frac{n}{\lambda} - C_n^2\left(\frac{1}{2\lambda}\right) + C_n^3\left(\frac{1}{3\lambda}\right) + \cdots + (-1)^{n-1}\left(\frac{1}{n\lambda}\right)$ .
$\frac{1}{\sqrt{\pi}}$ . 15. $E(Y) = \frac{n\theta}{n + 1}, E(Z) = \frac{\theta}{n + 1}$ .
1. 1. 14 166.67 元． 18. 略． 19. -0.02. 20. $-\frac{n}{36}, -\frac{1}{5}$ . 21. 0. 22. $-\frac{2}{3}$ .
$-\frac{n}{4}, -1.$ 24. $\frac{3}{5}$ . 25. $1, 1$ . 26. $\frac{1}{12}$ . 27. $\frac{2}{3}, 0, 0$ .
$\frac{3}{\sqrt{57}}$ . 29. $\rho$ . 30. 0.569 2. 31. $\frac{a^2 - b^2}{a^2 + b^2}$ . 32. (1) $\sqrt{\frac{(1 - \rho)}{\pi}}$ ; (2) 0,0.
$\frac{1}{36}, \frac{1}{2}$ . 34. $-\frac{1}{147}, -\frac{\sqrt{5}}{23}$ . 35. $\frac{1}{\sqrt{3}}$ . 36. (1) $\left( \begin{array}{cc} \frac{1}{18} & 0 \\ 0 & \frac{3}{80} \end{array} \right)$ ; (2) $\left( \begin{array}{cc} \frac{11}{36} & -\frac{1}{36} \\ -\frac{1}{36} & \frac{11}{36} \end{array} \right)$ .
a=0.5 时， $X$ 与 $Y$ 不相关．
当 $d \neq 0$ 时， $\rho_{12} = \rho_{13} = \rho_{23} = 1$ ，当 $d = 0$ 时， $\rho_{12} = \frac{c^2 - a^2 - b^2}{2ab}, \rho_{13} = \frac{b^2 - a^2 - c^2}{2ac}, \rho_{23} = \frac{a^2 - b^2 - c^2}{2bc}$ . 39—49. 略．

习题 3.5

$\frac{n!}{m!(n-m)!}\left(\frac{7.14}{14}\right)^{m}\left(\frac{6.86}{14}\right)^{n-m},\quad m=0,1,\cdots,n.$
$P(X=m,Y=n)=(1-p)^{n-2}p^{2}, m=1,2,\cdots,n-1, n=2,3,\cdots,$

P (Y = n \mid X = m) = (1 - p) ^ {n - m - 1} p, m = 1, 2, \dots , n - 1, n = 2, 3, \dots .

(1) $\frac{Y=1}{P}$ $\begin{array}{c|cc}X & & X \\ \hline 1 & 2 & \\ \hline \frac{1}{2} & \frac{1}{2} & \\ \end{array}$ , $Y=2$ $\begin{array}{c|cc}X & & \\ \hline 1 & 2 & \\ \hline P & \frac{1}{3} & \frac{2}{3} \\ \hline \end{array}$ ; (2) 不独立．
(1) $P(X = k \mid X + Y = m) = \frac{1}{m - 1}, \quad k = 1, 2, \cdots, m - 1;$

P (X = k \mid X + Y = m) = \frac {\binom {n} {k} \binom {n} {m - k}}{\binom {2 n} {m}}, \quad k = 0, 1, \dots , \min \{n, m \}.

当 $0 < x < 1$ 时， $p(y|x) = \frac{1}{x}$ , $0 < y < x$ . 6. 当 $-1 < y < 1$ 时， $p(x|y) = \frac{1}{(1 - |y|)}$ , $|y| < x < 1$ .
$\frac{7}{15}$ . 8. $\frac{47}{64}$ . 9. 略． 10. $E(X|Y=2)=\frac{78}{25}, E(Y|X=0)=2$ .
$\frac{n\lambda_1}{(\lambda_1 + \lambda_2)}$ . 12. $\frac{7}{12}$ . 13. $\frac{(2y + 1)}{3}$ . 14—15. 略． 16. $\frac{(\lambda + 9)}{(2\lambda)}$ . 17—18. 略．