概率论基础习题答案

ROSS 写的著名的《概率论基础》的课后习题答案

A

1

1.67600000;1965600072144

8.120;1260;34650

2.12969.27720

4.24;413.190

5.144;186.24017.720;72;144;

11.720;72;17.60480025.52!/(13!)4

32.1287;

10.40320;10080;1152;2880;384

14.259896021.35

22.1830.564480

16.42;9424.4831.165;35

12.24300000;17100720

19.896;1000;91028.65536;252033.220;572

18.60027.2772014112

20.36;26

29.12600;945

2

9.7410000

10.0.4;0.114.1.057

11.70;212.0.5;0.32;149/19813.20000;12000;11000;68000;

17.9.10947×10 6

33.70/323

35.

15.0.0020;0.4226;0.0475;0.0211;0.00024

30.1/18;1/6;1/2

38.446.5

18.0.048;0.8363

19.5/1820.0.905222.(n+1)/2n23.5/1225.0.426.0.492929

31.2/9;1/939.0.48

37.0.0833;0.544.0.3;0.2;0.1

40.1/64;21/64;

49.0.4329

55.0.2198;0.0343

27.0.0888;0.2477;0.1243;0.2099

36.0.0045;0.0588

41.0.5177

36/64;6/64

48.1.0604×10 354.0.0511

50.2.6084×10 652.0.09145;0.426853.12/35

3

1.1/39.7/110.48481/2

2.1/6;1/5;1/4;1/3;1/2;110.0.2216.0.9835

11.1/17;1/3317.0.0792;0.264

3.0.3395.6/916.1/27.2/38.1/2

15.

12.0.504;0.362922.5/9;1/6;5/54

29.0.0893

14.35/768;210/76823.4/9;1/230.7/12;3/539.46/185

48.2/3

18.0.331;0.383;0.286;48.6219.44.29;41.18

24.1/3;33.5/1140.3/13;50.17.5;

20.0.4;1/2634.27/31

21.0.496;3/14;9/62

36.1/2

26.20/21;40/41

35.3/4

28.3/128;29/1536

42.34.48

37.1/3;1/5;138.12/37

5/13;5/52;15/5238/165;17/333/5;9/25

55.9

41.43/45957.(c)2/3

43.4/945.1/11

51.0.65;56/65;8/65;1/65;14/35;12/35;9/35

60.2/3;1/3;3/4

52.0.11;16/89;12/27;

65.9/13;1/2

79.

71.38/64;13/64;13/64

61.1/6;3/20

69.9;9;18;110;4;4;8;120128

70.1/9;1/1875.3/4;7/12

73.1/16;1/32;5/16;1/4;31/3274.9/19

78.p2/(1 2p+2p2)

ROSS 写的著名的《概率论基础》的课后习题答案

A

40189.

0.555081.0.953083.0.5;0.6;0.884.9/19;6/19;4/19;7/15;53/165;7/33

97/142;15/26;33/102

4

1.p(4)=6/91;p(2)=8/91;p(1)=32/91;p(0)=1/91;p( 1)=16/91;p( 2)=28/914.1/2;5/18;5/36;5/84;5/252;1/252;0;0;0;0p( 3)=1/8;p(1)=p( 1)=3/8p( i)=p(i);p(0)=1p(2000)=0.04537p

1/4;1/6;1/12;1/2

24.p=11/18;

5.n 2i;i=0,···,n

6.p(3)=

12.p(4)=1/16;p(3)=1/8;p(2)=1/16;p(0)=1/2;

17.31.40.

13.p(0)=0.28;p(500)=0.27;p(1000)=0.315;p(1500)=0.09;

20.0.5918;

; 0.108

21.39.28;

14.p(0)=1/2;p(1)=1/6;p(2)=1/12;p(3)=1/20;p(4)=1/519.1/2;1/10;1/5;1/10;1/10

=23/7233.3

26.11/2;17/535. 0.067;1.089

27.A/(p+1/10)37.82.2;84.5

28.3/539.3/8

32.11 10×(0.9)10

42.p 1/2

56.253

11/24345.350.1/10;1/10

51.e 0.2;1 1.2e 0.259.0.3935;0.3033;0.0902

70.p+(1 p)e λt

53.1 e 0.6;60.0.888671.0.1500;

1 e 219.1861.0.4082e 10.1012

57.0.5768;0.6070

63.0.0821;0.242465.0.3935;0.2293;0.3935

68.(1 e 5)80

66.2/(2n+1);2/(2n 2);

78.18×(17)n 1/(35)n

67.2/n;(2n 3)/(n 1)2;e 2

73.5.8125

82.0.3439

74.32/243;4864/6561;160/729;160/729

81.3/10;5/6;75/138

5

2.3.5e 5/210.2/3;2/3(0.9938)10e 1;1/3

3.

4.1/2

5.1 (0.01)1/5

6.4;0;∞7.3/5;6/58.216.34.

11.2/518.22.66

13.2/3;1/315.0.7977;0.6827;0.3695;0.9522;0.1587

20.0.9994;0.75;0.977

29.0.9993

28.0.8363

32.e 1;e 1/2

19.(c)1/2;(d)1/422.9.5;0.0019

23.0.9258;0.1762

38.3/5

26.0.0606;0.052540.1/y

6

2.(a)14/39;10/39;10/39;5/39(b)84/429;70/429;70/429;70/429;40/429;40/429;40/429;15/4291 e a18.7/95/12;7/1251.r/π

3.15/26;5/26;5/26;1/26

8.c=1/8;E[X]=0

12.39.3e 521.2/5;2/532.0.0326

11.0.145819.1/2

4.25/169;40/169;40/169;64/16913.1/6;1/222.

7.p(i,j)=

10.1/2;17.1/325.e 1/i!35.1/3;2/3;

p2(1 p)i+j9.(12x2+6x)/7;15/56;0.8625;5/7;8/7

15.π/4

16.n(1/2)n 1

;1/3

23.1/2;2/3;1/20;1/18

34.0.0829;0.3766

30.e 2;1 3e 2

33.0.3372;0.206147.(1 2d/L)3

36.5/13;8/1352.r

37.1/6;5/6;1/4;3/4

42.(y+1)2xe x(y+1);xe xy;e x

49.1 e 5λa;(1 e λa)5

43.1/2+3y/(4x) y3/(4x3)

48.0.79297

55.(a)u/(v+1)2

ROSS 写的著名的《概率论基础》的课后习题答案

402

A

7

1.52.5/120.6;01/249.214y3/4218

18.4

2.324;198.85.3/26.3522.14.7

7.0.9;4.9;4.223.147/110

8.(1 (1 p)N)/p

14.m/(1 p)

10.15.

11.2(n 1)p(1 p)

12.(3n2 n)/(4n 2);3n2/(4n 2)31.175/638.1/8

33.14

21.0.9301;87.5755

37.0

26.n/(n+1);1/(n+1)

35.21.2;18.929;50.2y265.5.16

51.66.

29.437/35;12;4;123/35

36. n/3653.12

54.8σ2

3240/6137

45.1/2;0

34.20/19;360/361

49.6.06

41.6;112/33

57.12.5

42.100/19;16200/6137;10/19;63. 96/145

47.1/(n 1)48.6;7;5.8192

56.N(1 e 10/N)79.0.176;0.141

67.x[1+(2p 1)2]n

69.1/2;1/16;2/8170.1/2;1/3

72.1/i;[i(i+1)] 1;∞

73.µ;1+σ2;

8

1. 19/207.0.30850.2514

2.15/17; 3/4; 108.0.6932

9.(327)217. 0.2

3. 310.117

4. 4/3;0.842811. 0.057

5.0.14166.0.9431

13.0.0162;0.0003;0.2514;

14.n 2322.0.769;0.357;0.4267;0.1093;0.112184

9

1.1/9;5/915.5.5098

3.0.9735;0.9098;0.7358;0.557810.(b)1/614.2.585;0.5417;3.1267

ROSS 写的著名的《概率论基础》的课后习题答案

B

1

1.(a)C,D,E,F4

4!

,

,

A,B

5

C,D,E,F

,

2

,

A,B

AB

BA

5

,

,

2×5×4!=240

B

A

,

5!

,

B

A

(b)6(c)

,

6!=720

5!

,

,

2×5!=240AB,

A

B

,

,AB

A,B,C

720/2=360

6,

,

720

,

720/6=120,A

B

ABC

(d)A

BB

6!/2=360180

,

C

D

,C

D

(e)

A

,C

D

,

4!=24,D

A,B

,A

B

,

C

4

(f)E2.

,

5!

,

,

4×24=96,

6! 5!=600

3

3!

,

,

3!4!3!3!

3.(a)10×9×8=720

(b)8×7×6+2×3×8×7=672

A,B

,

8×7×6

A,

B,

3×8×7

,A,B,2×3×8×7

(c)8×7×6+3×2×8=384(d)3×9×8=216

(e)9×8×7+9×8=576

10555555

4.(a)(b)++

7344352

7

=2105.

3,2,2

7

=35,(26)3(10)46.

3

,

35×(26)3(10)4

7.n

r

n

(n r)

,

8.(a)10×9×9···9=10×9n 1

ROSS 写的著名的《概率论基础》的课后习题答案

404

B

(b)

nn i

9,i

ni

,

0,

n i

1,2,···,9.

3nn32nn

9.(a)(b)3(c)=3n2(n 1)

331121

3nn3

(e)=3+3n2(n 1)+n3(d)n

33

10.

11.

9×8×7×6×5, 5

×8×7×6,,,2

59××8×7×65,

2

5!5!9

7×7×,,22!2!2!2!

5!59

9×8×7×6×5+9××8×7×6+7×

222!2!

5

,,···,(2n)!/2n

,

,

,

(2n)!/(2n×n!)

,

2n

,

(2n)!/n!

878712.+=3430

3342

3

3

4

2

13.(x1+···+x5=4 891044414.15.

)(x1+···+x5=5

)(x1+···+x5=6

)=

j

j 1n 1

,

k

,

, nk

k

j=n

j 1n 1

,

,

k

,

n

k

k!

,

n

nk!k=0

k

16.

4

20

=48454

5

15

=1365.4

,

{1,2,···,5}

4515

.i=1

i4 i

4845 1365=3480

17.

2,

n(n 1)=k(k 1)+2k(n k)+(n k)(n k 1)

ROSS 写的著名的《概率论基础》的课后习题答案

B

405

k2(1 2+1)+k( 1+2n n n+1)+n(n 1)

n2

,

n

,

k

,

n

k2

2

,

,

k(n k),

n.2

n k2

2

,i)}

1.(a)2×3×4=24(d)AB={(c,2.

(b)2×3=6

(c)3×4=12

,i),(c,

,i),(c,

(e)8,C

(f)ABC={(c,

,i)}

A

,B

,

P(A∪B∪C)=0.22+0.30+0.28 0.11 0.14 0.10+0.06=0.51

(a)1 0.51=0.49

(b)

P(AB∪AC∪BC)=0.11+0.14+0.10 0.06 0.06 0.06+0.06=0.23

,

0.51 0.23=0.28.

3.

,

14

52

,

4/52

52!

,

14

A(

A,

A,···)

p=

A

1

A

4×51×50···2×14

=

(52)!52

14

,

P(A)=

4.

D

70 C

48×47···36×4

=0.0312

52×51×···40×39

P(A∪B)=P(A)+P(B) P(AB)=0.7 P(AB)P(C∪D)=P(C)+P(D) P(CD)=0.2+P(D) P(DC)

,A∪B=C∪D,AB=CD,

0=0.5 P(D)

P(D)=0.5.

(a)

52×39×26×1352×48×44×40

=0.6761(b)=0.1055

52×51×50×4952×51×50×49

5.

ROSS 写的著名的《概率论基础》的课后习题答案

4066.

B

R

,B

,

P(R∪B)=P(R)+P(B)=

3×43×61

+=

6×106×102

8324040 8

(b)=3.3×10 67.(a)1=1.3×10

7188

83240(c)+1.3×10 8+3.3×10 6=1.8×10 4

628

3×4×4×34414814

=0.1439(b)=0.0360(c)=0.06998.(a) 2244414

4 n

9.S=i=1Ai,S,P(A)=N(A)/N(S),

4.3

4.4

10.11.

1

,

5!=120

,N(A)=360,

Ai

,N(B)=120,N(AB)=2×4!=48

9,

N(A∪B)=432.

i,i=1,2,3,4.

P

4 i=1

Ai=

i

P(Ai)

j

i:i<j

P(AiAj)+··· P(A1A2A3A4)

39524265241352=4 +

52255355 395226521352=4 6+4

525555

1

A

4

,

P(A)=P(n,n,n,n,o)+P(n,n,n,o,n)+P(n,n,o,n,n)+P(n,o,n,n,n)

P(n,n,n,o,n)

1

,

23

,

4

(

4

),

5

P(A)=

52×39×26×13×48+52×39×26×36×13

52×51×50×49×48

52×39×24×26×13+52×12×39×26×13+

52×51×50×49×48

52×39×26×13×(48+36+24+12)

=0.2637=

52×51×50×49×48

12.

(10)!/25

,

5

5

,

6 4

2

2

,

,

(10)!/(5!25)

4

ROSS 写的著名的《概率论基础》的课后习题答案

B

4074

,

2

,

3

2

(3=4!/(2!22)),

,

6 4 }=V

2

P{

13.

R

×2×3

=0.5714

(10)!/(5!25)

2

R

,

E

P{

14.

}=P(R)+P(E)+P(V)=

Aj

c

,i>1, ∞

i=1

B1=A1,Bi=Ai

P

2131113×+×+×=78787828

i 1

j=1

∞

i=1

AiBi

=P

Bi

=

∞ i=1

P(Bi)

∞ i=1

P(Ai)

,

Bi Ai.

∞ i=1

15.

P

16.

∞

i=1

Ai

=1 P

,

∞

i=1

Ai

c

=1 P

∞ i=1

Aci

1

P(Aci)=1

{1}

Tk 1(n 1)

,

Tk 1(n 1)

n 1

{2,···,n}

k 1

,

{1},

,

,

11

{2,···,n}

k

,

Tk(n 1)

,

1,

{1,2,···,n}

k

,

,

{1}

kTk(n 1)

,

17.

R

5

,W

,B

,

P(R∪W∪B)=P(R)+P(W)+P(B) P(RW) P(RB) P(WB)+P(RWB)

131812181118718618518=++

555555555555≈0.2933

,5

1 0.2933=0.7067.

3

1.(a)P(

3913

3539

(b)1 P(A) 4

1213

33639

A)=(c)P(i

i13 i13

A)=

3513

2.

Li

10000×i

ROSS 写的著名的《概率论基础》的课后习题答案

408

B

(a)P(L2|L1)=P(L1L2)/P(L1)=P(L2)/P(L1)=1/2(b)P(L3|L1)=P(L1L3)/P(L1)=P(L3)/P(L1)=1/83.101,9104.T,WB

2

,

P(T|W)=

P(W|T)P(T)

P(W|T)P(T)+P(W|Tc)P(Tc)

2/7×2/3

==4/52/7×2/3+1/7×1/3

5.(a)

r

,r+w(b)(c)

r+w

i

6.

Bi

7.

B

(a)

(b)(c)

P(Rj|Ri)=P(R iRj)r =2/ r+w 2r 1

P(Ri)r/(r+w)=

r+w 1

i=j,

i

,

(r+w 1)

j

,(r 1).

i

,

Ri=Bc

i,

P{B1|R2}=

P(R2|B1)P(B1)

P(R2|B1)P(B1)+P(R2|R1)P(R1)

=

[r/(b+r+c)][b/(b+r)]

[r/(b+r+c)][b/(b+r)]+[(r+c)/(b+r+c)][r/(b+r)]

=b/(b+r+c)

A

P{B|

A}

AP{B,

}=

P{A}

1 3 1 51 =1111 52 52 =3/51

22

2

51

,

,

,

3/51.

,

,

(b)

P{B|

P{B,A}

A}=

P{

A}

=

P{B}

P{B}+P{A,

A}

=

(4/52)(3/51)

(4/52)(3/51)+(48/52)(4/51)

=3/51

ROSS 写的著名的《概率论基础》的课后习题答案

B

409

(d)

P{B|

A}=

B

P{

A}

=

8.

(4/52)(3/51)

=1/33

1 (48/52)(47/51)

P(HE)P(H)P(E|H)P(H|E)==

P(G|E)P(GE)P(G)P(E|G)

,H

G

1.5

9.

A

,W

(a)

P(A)=P(A|W)P(W)+P(A|Wc)P(Wc)

=(0.85)(0.9)+(0.2)(0.1)=0.785

(b)

P(Wc|Ac)=

10.

(0.8)(0.1)16P(Ac|Wc)P(Wc)

==

P(Ac)0.21543

,

(a)

P{

(b)

|

}=

P{P{

}

}

=

1/41

=3/43

F

5

,B2

2

,B1

,

P(B2|F)=

11.

1×1/316P(F|B2)P(B2)

==5P(F|B2)P(B2)+P(F|B1)P(B1)1×1/3+(1/2)×(2/3)17

F

A

B,Ci

i

,

cP(F)=P(F|C1)p1+P(F|C1)(1

p1).

P(F|C1)=P(C4∪C2C5)=P(C4)+P(C2C5) P(C4C2C5)=p4+p2p5 p4p2p5

c

P(F|C1)=P(C2C5∪C2C3C4)=p2p5+p2p3p4 p2p3p4p5

,

(a),

P(F)=p1(p4+p2p5 p4p2p5)+(1 p1)p2(p5+p3p4 p3p4p5)

(b),

qi=1 pi,

ccccP(C3|F)=P(F|C3)P(C3)/P(F)=p3[1 P(C1C2∪C4C5)]/P(F)

=p3(1 q1q2 q4q5+q1q2q4q5)/P(F)

ROSS 写的著名的《概率论基础》的课后习题答案

41012.

B

A

1

,F

(a)

P(A|F)=

2P(AF)P(A)1/2

===

P(F)P(F)1 (1/2)231,2

P(F)

1

(b)

P(A|F)=

3P(AF)P(F|A)P(A)3/4×1/2

===

P(F)P(F)(1/2)3+3×(1/2)34

P(F)

3

3

2

13.14.

,

,

10

,

,

P(A

)=P1(1 P2)(1 P3)+(1 P1)P2P3+P1P2P3P(A

+(1 P1)(1 P2)(1 P3)P(A

)

)

P(A

15.

)=

P1(1 P2)(1 P3)+(1 P1)P2P3

P1+P2+P3 P1P2 P1P3 P2P3

A

,B

,E,

,

1=P(A)+P(B)+P(E).

1 P(E)

=2

,P(A)=P(B),

1

n i=1

p2i

P(B)=

2

i

j>i

P(B)=

i

j>i

P{

i,

j}=

pipj

1=

16.

n i=1

pi

n j=1

pj=

i

j

pipj=},

i

p2i

+

i

j=i

pipj=

i

p2i

+2

i

j>i

pipj

E={AB

Aw={nAe={n

,A

B

},},},

Bw={n

,B

A

,A

B

P(E)=P(E|Aw)P(Aw)+P(E|Bw)P(Bw)+P(E|Ae)P(Ae)

=1×P(Aw)+0×P(Bw)+

11

P(Ae)=P(Aw)+P(Ae)22

ROSS 写的著名的《概率论基础》的课后习题答案

B

411

P(Aw)=P(Bw).

1=P(Aw)+P(Bw)+P(Ae)

P(Aw)=

,

P(E)=P(Aw)+

11

P(Ae).22

11111

P(Ae)= P(Ae)+P(Ae)=22222,

17.(a)

E={

7

},F={

4},

G={

3}

,E,F7,

,E,G=

,

P(E|F∪G)=

(b)

P{

{4,3}}

P{

{4,3}}

55/36

==P(E)35/3635

P(E(F∪G))=P(EF∪FG)=P(EF)+P(EG)

EFG= FG=

=P(E)[P(F)+P(G)]=P(E)P(F∪G)

(c)

P(G|EF)=

P(E)P(FG)P(EFG)

=

P(EF)P(EF)P(E)P(F)P(G)=

P(E)P(F)

E

FG

=P(G)

18.(a)

(b)(c)(d)20.

,

,,

0=P(AB)=P(A)P(B)P(AB)=P(A)×P(B)>0P(A∪B)=P(A)+P(B)=1.2

19.(a),(b),(c)

0.5,(0.8)3=0.512,(0.9)7≈0.4783.

Di(i=1,2)B)

i

A(B)

A(

P(D2|D1)=

P(D1D2|A)P(A)+P(D1D2|B)P(B)P(D1D2)=

P(D1)P(D1|A)P(A)+P(D1|B)P(B)

0.052×1/2+0.012×1/213==

0.05×1/2+0.01×1/2300

21.P(A|B)=1

P(AB)=P(B),

P(Bc|Ac)=1

P(AcBc)=P(Ac)

,

,

P(AB)=P(B) P(AcBc)=P(Ac).

P(BcAc)=P((A∪B)c)=1 P(A∪B)

=1 P(A) P(B)+P(AB)=1 P(A)=P(Ac)

22.

n=0

,

Ai

n

,

i

,

P(Ai)=

1

n+1

i=1,···,n+1

ROSS 写的著名的《概率论基础》的课后习题答案

412

B

Bj

n+1

j

,

n+1

1

P(Bj)=P(Bj|Ai)P(Ai)=P(Bj|Ai)

n+1i=1i=1

n+1

=

n

,

1

[P(Bj|Aj 1)+P(Bj|Aj)]n+1

n+2

P(Bj|Aj 1)

n+2

,

j 1

,

,

,n+1

,

i

,

P(Bj|Aj 1)=

j 1n+2

P(Bj|Aj)

j,n+2 j

,

,

P(Bj|Aj)=

n+2 jn+2

P(Bj)

,

P(Bj)=

,

j 1n+2 j11[+]=n+1n+2n+2n+2

23.

Ai

i

‘A’,

2n 2

P(Ai)=1

n

A1A2

1n 13n 12n

=1 ×=n22n 14n 2

2

‘A’

,

2n 2

‘A’

n 1

,

P(A1A2)=

,

12n 22n 1

n2n=n2n 1

P(Ac2|A1)=1 P(A2|A1)=1

n 1P(A1A2)

=

p(A1)3n 1

,

i

i

A,

,

n

,

,

1/2,

,

(=1/3),

,n

,

2b

24.(a)

(b)

n

i1,i2,···,in,

,n

i1,···,ik

n

n!

n

pi1···pin=

n

i=1

pi.

,

i=1

pi.

P(Ei1···Eik)=

n k n

n

ROSS 写的著名的《概率论基础》的课后习题答案

Bn

413

Ei1,···,Eik

i1,···,ik

,,

,

i1,···,ik.

P(Ei1···Ein)

P

1 P

n i=1

Ei=

n

( 1)k+1

n

i=1

P

Ei n

k=1

n k nnkn

i=1

Ei

n

,

(a)

n/nn,

,

n n!k+1n n k1 n=( 1)

knn

nk=1

n!=n

n

n

( 1)k+1

k=1n

n

(n k)nk

n!=

25.

k=0

n

(n k)n( 1)k

k

A=EF,B=FE,C=E∩F,

cc

A,B,C,

E∪F=A∪B∪C

E=A∪C

F=B∪CP(E|E∪F)=

P(A)P(C)P(E∩(E∪F))

=

P(E∪F)P(A)+P(B)+P(C)

P(C)P(EF)

=P(E|F)=

P(F)P(B)+P(C)

,P(E|E∪F) P(E|F)

4

1.

1,

4P{X=3}+0.5=1

P{X=0}=0.375,P{X=3}=0.125,

,

pi=cip0,i=1,2,

E[X]=1×0.3+2×0.2+3×0.125=

1,

1.075.2.

pi=P{X=i}

p0(1+c+c2)=1 p0=

1

1+c+c2

,

c+2c2

E[X]=p1+2p2=

1+c+c2

ROSS 写的著名的《概率论基础》的课后习题答案

4143.

B

X

,X

p2=p2+(1 p)2

p3=1 p2=2p(1 p)

,

E[X]=2p2+3p3=2p2+3(1 p2)=3 p2 (1 p)2

4.

,

i

ni/m,

E[X]=

i

ni,

r i=1

ini/m

ini,

/

r

j=1ini

ini,

ri2niE[Y]=

i =1rini

i=1

,

ri2ni

rinii=1

i=1

r ini

rni

i=1

i=1

rn r rj

i2

n ri

inijnj

j=1

i=1

i=1

j=1

r r2

r rininj

ijninj

i=1j=1

i=1j=1

(i,j),

ninj

i2+j2,

ninj

i2+j2 2ij

1

5.

p=P{X=1},

E[X]=p,Var(X)=p(1 p),

p=3p(1 p)

p=2/3

P{X=0}=1/3.

1.

.

pi= ni

n,

( pp i)(

i2i) ip 2i.

i

2ij,

ROSS 写的著名的《概率论基础》的课后习题答案

B,

415

6.

x,

x

p,

x

1 p

xp x(1 p)=(2p 1)x

p>1/2

,

,

p<1/2,

0.6,

10

(

)

0.3,

,

11

×(1.2 1)×10+×0 C=1 C22

C,

1 111

(2×0.6 1)x+(2×0.3 1)x=×0.2+×( 0.4)x= 0.1x2222

,

,

,

,

C<1

7.(a)

,

x,

,

2x(1/2)+x/2(1/2)=5x/4>x

,

,

(b)

x(

),

2x

x/2,

y

,

x/2.

E[Ry(x)]=5x/4

x/2 y

x/2<y 2x,

,

,

2x,

2x

x/2,,

,

E[Ry(x)]=2x(1/2)+x(1/2)=3x/2

x/2<y 2x

,

<y,

2x,

Ry(x)=x

2x<y

y

,

E[R xx<y/2y(x)]=

3x/2y/2 x<2y 5x/4

x 2y

8.

n

p,

i

n i,

1 p

P{Bin(n,p) i}=P{Bin(n,1 p) n i}

=1 P{Bin(n,1 p) n i 1}

y<x/2,

,2x

,

ROSS 写的著名的《概率论基础》的课后习题答案

4169.

B

E[X]=np,Var(X)=np(1 p)

np=6,np(1 p)=2.4,

10

(0.6)5(0.4)55

p=0.6,n=10,

P{X=5}=

10.

Xi

i

,

,

11.(a)

(b)

12.

13.

C

14.

P{X k}=P{X1 k,X2 k,···,Xm k}

=P{X1 k}P{X2 k}···P{Xm k}=

k mn

P{X=k}=P{X k} P{X k 1}=

k mn

k 1 m

nA

3

,A

B

2

P{A

|A

}=p2+2p2(1 p)+3p2(1 p)2

P{A

P{A

|A

}P{A}

|A

}=

P{A

}

p3

+2p3

(1 p)+3p3

(1 pp3+3p3(1 p)+ )2

=4

2p3

(1 p)2

=

1+2(1 p)+3(1 p)21+3(1 p)+6(1 p)2

,

,

40.5

4 0.4i0.6

4 i

+ 40.5 4 i0.34 i

i=3

i

i=3

i

0.7,

F

4

7P(C)=

7 0.7i0.37 i

i=4

i

7

0.740.33P(C|F)=P(CF)

P(F)=4 7 =0.40.740.33+ 7

73

0.730.34,

3e 5.2(5.2)i/i!

i=0

,

,

ROSS 写的著名的《概率论基础》的课后习题答案

B

417

15.

E[Y]=

16.(a)1/n

D(b)

∞ i=1

iP{X=i}/P{X>0}=E[X]/P{X>0}=

λ

1 e λ

i

j

,

P(GiGj)=P(GiGj|D)P(D)+P(GiGj|Dc)P(Dc)

=P(GiGj|D)P(D)+0

(P(GiGj|Dc)=0)

P(Dc)=P{i,j

=,

n k=1

}

P{i,j

k}=nP{i,j

1}=n×

11

=2nn

1 2 n 11 1

=1 P(GiGj)=P(GiGj|D)(1 )=

nnnn3

(c)(d)

(e)

P(Gi|Gj)=P(GiGj)/P(Gj)=(n 1)/n2.

n,P(Gi|Gj),P(Gi)

n

,,P0～e 1,Pk≈e 1/k!.i=1P(Gi)=1.

k,Dk

.

P(Gi1···Gik)=P(Gi1···Gik|D)P(D)+P(Gi1···Gik|Dc)P(Dc)

1 kn(n 1)···(n k+1)n!

==P(Gi1···Gik|D)P(D)=knn(n k)!n2k

,

i1<···<ik

P(Gi1···Gik)=

n!n!n

P(Gi1···Gik)=k(n k)!(n k)!k!n2k

1 P0=P

17.(a)

(b)

iWj

n i=1

Gi=

n

( 1)k+1

k=1

n!n!

(n k)!(n k)!k!n2k

,

,

i

2n 3

P(Wi)=1/(2n 1).

,

P(Wi|Wj)=1/(2n 3).n,(c)

,

n

i=1

P(Wi)=n/(2n 1)≈1/2.

((d)

,

e

1/2

.

16

).

ROSS 写的著名的《概率论基础》的课后习题答案

418

B

18.(a)

88(9/19)3(10/19)5(9/19)=(9/19)4(10/19)533(b)W,X,

4

,

X 4

,

W=20 5(X 4)=40 5X

E[W]=40 5E[X]=40 5×[4/(9/19)]= 20/9

19.

,

,

1/4

(a)(1/4)2(3/4)=3/64(b)(1/4)4=1/256.

20.

q=1 p,E[1/X]= ∞1∞i 1

p ii=1iqp=

qq/ii=1

=p ∞ q

i p q ∞i 1qx1dx=xdxi=10q 0i=1=

pq1

q01 xdx=p

11qdy= plnppyq

21.

(X b)/(a b)

p

1,

(1 p)

0,

p,

p(1 p(1 p)=Var p)

X b a b=1(a b)2Var(X b)=1

(a b)2

Var(X)

Var(X)=(a b)2p(1 p)

22.

X

,Y

(a)

4

,

,

X 4

,

(1 p),E[X]=E[4+(X 4)]=4+E[X 4]=4+

1

1 p

(b)

Z

4

,

Z

,

(4,1 p)

Y=Z+1,

E[Y]=E[Z+1]=E[Z]+1=4(1 p)+1

23.

m

n

n+m 1

n

3

1

(

4j

)

X

n+m,X

Mn+n+P{X n}=

m 1P N

m 1{X=i}=

in+m 1 ii=n

i=n

N+M n+m 1

,

ROSS 写的著名的《概率论基础》的课后习题答案

B

419

5

1.

X

()

P{X>15}=1 P{X 15}=1 5×0.025=0.875P{20<X<35}=10×0.05+5×0.025=0.625P{X<30}=10×0.025+10×0.05=0.75P{X>36}=4×0.025=0.1

1

2.(a)1=0cxndx=c/(n+1) c=n+1

1

n+1

(b)P{X>x}=(n+1)xxndx=xn+1|1x=1 x3.

(a)(b)(c)(d)

c

1=

2

cx4dx=32c/5 c=5/32

4.

5645 25

=5/3xdx=032326

5 265128

(b)E[X2]=xdx==20/7 Var(X)=20/7 (5/3)2=5/63032327(a)E[X]=

1=

0.6=

1

(ax+bx2)dx=a/2+b/3(ax2+bx3)dx=a/3+b/4

10

a=3.6,b= 2.4.,

1/21/2

(a)P[X<1/2]=0(3.6x 2.4x2)dx=(1.8x2 0.8x3)|0=0.35

1

(b)E[X2]=0(3.6x3 2.4x4)dx=0.42 Var(X)=0.06

5.

i=1,···,n

P{X=i}=P{[Int(nU)]=i 1}=P{i 1 nU<i}

i 1i

=1/n=P U<

nn

(6.

[nU]

nU

)

x,70<x<140,

(140 x)/70x,

,

x 100,

,

,

11

(x 100)(140 x)=(240x x2 14000)7070

0,

240 2x=0

,

120(

),

40/7(

)

7.(a)P{U>0.1}=9/10

(b)P{U>0.2|U>0.1}=P{U>0.2}/P{U>0.1}=8/9

(c)P{U>0.3|U>0.2,U>0.1}=P{U>0.3}/P{U>0.2}=7/8

ROSS 写的著名的《概率论基础》的课后习题答案

420

B

(d)P{U>0.3}=7/10

(a),(b),(c)

(d)

8.

X,Z=(X 100)/15,(a)P{X>125}=P{Z>25/15}≈0.0478(b)

Z

P{90<X<110}=P{ 10/15<Z<10/15}

=P{Z<2/3} P{Z< 2/3}=P{Z<2/3} [1 P{Z<2/3}]

≈0.4950

9.

X

,

x,

P{X>x}=0.05

P

X 40x 40

7>

7=0.05P Z>

x 40

7

=0.05Z

P{Z>1.645}=0.05

x 40

=1.x=51.515

,

12

8.485

764510.

X

(

1000

),

Z=(X 34)/4,

Z

(a)P{X>40}=P{Z>1.5}≈0.0668

(b)P{30<X<35}=P{ 1<Z<0.25}=P{Z<0.25} P{Z>1}≈0.44(c)

P{X>40|X>30}=P{X>40}/P{X>30}

=P{Z>1.5}/P{Z> 1}≈0.079

11.

X

,

Z=(X 40.2)/8.4

(a)P{X>44}=P{Z>3.8/8.4}≈P{Z>0.4524}≈0(b) .3255

7 3(0.3255)3(0.6745)4

12.

Mi

i(

)

Wi

,

.

(a)

P{W25 70}=P{W25 69.=P 5}

√W25 200×0.3469.5 200×0.34 200×0.34×0.66 √

200×0.34×0.66

≈P{Z 0.2239}≈0.4114

Z

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