概率论基础教程第8版参考答案.pdf

?A? ? 1 ? 1. 67 600 000; 19 656 0002. 12964. 24; 45. 144; 186. 24017. 720; 72; 144; 728. 120; 1260; 34 6509. 27 72010. 40 320; 10 080; 1152; 2880; 38411. 720; 72; 14412. 24 300 000; 17 100 72013. 19014. 2 598 96016. 42; 9417. 604 800 18. 60019. 896; 1000; 91020. 36; 2621. 3522. 1824. 4825. 52!/(13!)4 27. 27 72028. 65 536; 252029. 12 600; 94530. 564 48031. 165; 3532. 1287; 14 11233. 220; 572 ? 2 ? 9. 7410. 0.4; 0.111. 70; 212. 0.5; 0.32; 149/19813. 20 000; 12 000; 11 000; 68 000; 10 00014. 1.05715. 0.0020; 0.4226; 0.0475; 0.0211; 0.000 2417. 9.109 47 106 18. 0.048;19. 5/1820. 0.905222. (n + 1)/2n23. 5/1225. 0.426. 0.492 929 27. 0.0888; 0.2477; 0.1243; 0.209930. 1/18; 1/6; 1/231. 2/9; 1/933. 70/32336. 0.0045; 0.058837. 0.0833; 0.538. 439. 0.4840. 1/64; 21/64; 36/64; 6/6441. 0.517744. 0.3; 0.2; 0.146. 548. 1.0604 10349. 0.432950. 2.6084 106 52. 0.091 45; 0.426853. 12/3554. 0.051155. 0.2198; 0.0343 ? 3 ? 1. 1/32. 1/6; 1/5; 1/4; 1/3; 1/2; 13. 0.3395. 6/916. 1/27. 2/38. 1/2 9. 7/1110. 0.2211. 1/17; 1/3312. 0.504; 0.362914. 35/768; 210/76815. 0.484816. 0.983517. 0.0792; 0.26418. 0.331; 0.383; 0.286; 48.6219. 44.29; 41.18 20. 0.4; 1/2621. 0.496; 3/14; 9/6222. 5/9; 1/6; 5/5423. 4/9; 1/224. 1/3; 1/2 26. 20/21; 40/4128. 3/128; 29/153629. 0.089330. 7/12; 3/533. 0.76, 49/76 34. 27/3135. 0.62, 10/1936. 1/237. 1/3; 1/5; 138. 12/3739. 46/18540. 3/13; 5/13; 5/52; 15/5241. 43/45942. 34.4843. 4/945. 1/1148. 2/350. 17.5; 38/165; 17/3351. 0.65; 56/65; 8/65; 1/65; 14/35; 12/35; 9/3552. 0.11; 16/89; 12/27; 3/5; 9/2555. 957. (c) 2/360. 2/3; 1/3; 3/461. 1/6; 3/2065. 9/13; 1/269. 9; 9; 18; 110; 4; 4; 8; 120?128?70. 1/9; 1/1871. 38/64; 13/64; 13/64 73. 1/16; 1/32; 5/16; 1/4; 31/3274. 9/1975. 3/4; 7/1278. p2/(12p+2p2)79. 2?A? 0.555081. 0.953083. 0.5; 0.6; 0.884. 9/19; 6/19; 4/19; 7/15; 53/165; 7/3389. 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) = 28/91 4. 1/2; 5/18; 5/36; 5/84; 5/252; 1/252; 0; 0; 0; 05. n 2i; i = 0, ,n6. p(3) = p(3) = 1/8; p(1) = p(1) = 3/812. p(4) = 1/16; p(3) = 1/8; p(2) = 1/16; p(0) = 1/2; p(i) = p(i); p(0) = 113. p(0) = 0.28; p(500) = 0.27; p(1000) = 0.315; p(1500) = 0.09; p(2000) = 0.04514. p(0) = 1/2; p(1) = 1/6; p(2) = 1/12; p(3) = 1/20; p(4) = 1/517. 1/4; 1/6; 1/12; 1/219. 1/2; 1/10; 1/5; 1/10; 1/1020. 0.5918;?; 0.10821. 39.28; 3724. p = 11/18;?= 23/7225. 0.46, 1.326. 11/2; 17/527. A/(p + 1/10) 28.3/531.p32.11 10 (0.9)1033.335.0.067; 1.08937.82.2; 84.539.3/840.11/24342.p ? 1/245.350.1/10; 1/1051.e0.2; 1 1.2e0.253. 1 e0.6; 1 e219.1856. 25357. 0.5768; 0.607059. 0.3935; 0.3033; 0.090260. 0.888661. 0.408263. 0.0821; 0.242465. 0.3935; 0.2293; 0.3935 66. 2/(2n + 1); 2/(2n 2); e167. 2/n; (2n 3)/(n 1)2; e268. (1 e5)8070. p + (1 p)et71.0.1500; 0.101273.5.812574.32/243; 4864/6561; 160/729; 160/72978. 18 (17)n1/(35)n81. 3/10; 5/6; 75/13882. 0.343983. 1.5 ? 5 ? 2. 3.5e5/23.?4. 1/25. 1 (0.01)1/56. 4; 0; 7. 3/5; 6/58. 2 10. 2/3; 2/311. 2/513. 2/3; 1/315. 0.7977; 0.6827; 0.3695; 0.9522; 0.158716. (0.9938)1018. 22.6619. (c)1/2; (d)1/420. 0.9994; 0.75; 0.97722. 9.5; 0.0019 23. 0.9258; 0.176226. 0.0606; 0.052528. 0.836329. 0.999332. e1; e1/234. e1; 1/338. 3/540. 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; 15/4293. 15/26; 5/26; 5/26; 1/264. 25/169; 40/169; 40/169; 64/1697. p(i,j) = p2(1p)i+j8. c = 1/8; EX = 09. (12x2+6x)/7; 15/56; 0.8625; 5/7; 8/710. 1/2; 1 ea11. 0.145812. 39.3e513. 1/6; 1/215. /416. n(1/2)n117. 1/3 18. 7/919. 1/221. 2/5; 2/522.?; 1/323. 1/2; 2/3; 1/20; 1/1825. e1/i! 28. 1/2et; 1 3e229. 0.032630. 0.3372; 0.206131. 0.0829; 0.376632. e2; 1 3e235. 5/13; 8/1336. 1/6; 5/6; 1/4; 3/441. (y + 1)2xex(y+1); xexy; ex 42. 1/2+3y/(4x)y3/(4x3)46. (12d/L)347. 0.7929748. 1e5a; (1ea)5 52. r/53. r56. (a) u/(v + 1)2 ?A?3 ? 7 ? 1. 52.5/122. 324; 199.63. 1/2; 1/4; 04. 1/6; 1/4; 1/25. 3/26. 357. 0.9; 4.9; 4.28. (1(1p)N)/p10. 0.6; 011. 2(n1)p(1p)12. (3n2n)/(4n2); 3n2/(4n 2)14. m/(1 p)15. 1/218. 421. 0.9301; 87.575522. 14.723. 147/11026. n/(n + 1); 1/(n + 1)29. 437/35; 12; 4; 123/3531. 175/633. 14 34. 20/19; 360/36135. 21.2; 18.929; 49.21436. n/3637. 038. 1/841. 6; 112/3342. 100/19; 16 200/6137; 10/19; 3240/613745. 1/2; 047. 1/(n 1)48. 6; 7; 5.819249. 6.0650. 2y251. y3/453. 1254. 856. N(1e10/N)57. 12.563. 96/14565. 5.1666. 21867. x1 + (2p 1)2n69. 1/2; 1/16; 2/81 70. 1/2; 1/372. 1/i; i(i + 1)1; 73. ; 1 + 2;?279. 0.176; 0.141 ? 8 ? 1. ? 19/202. 15/17; ? 3/4; ? 103. ? 34. ? 4/3; 0.84285. 0.14166. 0.9431 7. 0.30858. 0.69329. (327)210. 11711. ? 0.05713. 0.0162; 0.0003; 0.2514; 0.251414. n ? 2316. 0.013; 0.018; 0.69118. ? 0.223. 0.769; 0.357; 0.4267; 0.1093; 0.112 184 ? 9 ? 1. 1/9; 5/93. 0.9735; 0.9098; 0.7358; 0.557810. (b)1/614. 2.585; 0.5417; 3.1267 15. 5.5098 ?B? ? 1 ? 1.(a) C,D,E,F?4?4!?.?,?A,B ?5? ?C,D,E,F?,?2?,? ? A,B?AB?BA?5?,?,?254! = 240 ? ?B?%?A?,?5!?,? B%?A?,?5!?,?2 5! = 240? (b) 6?6! = 720?,?A?B?,?A?B?,? ?, A?B?720/2 = 360? (c)?A, B, C?6?,?,?720?,?720/6 = 120 ?A B C? (d) A?B?6!/2 = 360?,?C?D? ?, A?B?, C?D?180? (e)?B%?A?, C%?D?,?4! = 24?A,B ?#?,?A?%?B?,?, D?%?C? ? 4?,?,?4 24 = 96? (f) E?5!?,?,?6! 5! = 600? 2.?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 4. (a) ?10 7 ? (b) ?5 3 ?5 4 ? + ?5 4 ?5 3 ? + ?5 5 ?5 2 ? 5. ? 7 3,2,2 ? = 210 6.? ?7 3 ? = 35?,?(26)3(10)4?,? 35 (26)3(10)4? 7. n?r?n?(n r)? ?,? 8. (a) 10 9 99 = 10 9n1 ?B?5 (b) ?n i ? 9ni,? ?n i ? ?,?0,?n i? ?1,2, ,9. 9. (a) ?3n 3 ? (b) 3 ?n 3 ? (c) ?3 1 ?2 1 ?n 2 ?n 1 ? = 3n2(n 1) (d) n3(e) ?3n 3 ? = 3 ?n 3 ? + 3n2(n 1) + n3 10.?9 8 7 6 5?,? ?,? ?5 2 ? 8 7 6?,?,? ?9 ?5 2 ? 8 7 6?5?,?,? ?7 5! 2!2! ? ?,? ?9 2 ? 7 5! 2!2! ?,? ?,? ? 9 8 7 6 5 + 9 ?5 2 ? 8 7 6 + ?9 2 ? 7 5! 2!2! ?5? 11.(a)?7?.?10?6?,? ? ?10 6 ? ?.?6?1?,? ?6?.?,? ?10 6 ? 26?. (b)?10?6?,? ?10 6 ? ?.?,?3? ?,?3?3 1 2 = 6? ?#?;?5 2 1 = 10?;? 7 2 2 = 28?#?;?6 2 3 = 36? ?#?.?,?80?. 19.?,?.? ?8 3 ? 26 25 24 23 22 10 9 8?.?,? ?6?,?6 26 25 24 23 22 10 9 8?. ? 2 ? 1. (a) 2 3 4 = 24(b) 2 3 = 6(c) 3 4 = 12 (d) AB = (c,?,i),(c,?,i),(c,?$,i)(e) 8(f) ABC = (c,?,i) 2.?A? ?, B? ?, C? ?,? P(A B C) = 0.22 + 0.30 + 0.28 0.11 0.14 0.10 + 0.06 = 0.51 (a) 1 0.51 = 0.49 (b)? P(AB AC BC) = 0.11 + 0.14 + 0.10 0.06 0.06 0.06 + 0.06 = 0.23 ?,?%?0.51 0.23 = 0.28. 3.?,?14#?52#?#,?4/52? ?52!?,?14#? ?A?(?$ ?A?,? ?A?, )? p = 4 51 502 1 (52)! = 4 52 ?B?7 ?A? ?1#?A?14#?,? P(A) = 48 4736 4 52 51 40 39 = 0.0312 4.?D? ?70C? ? ? 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. 5. (a) 52 48 44 40 52 51 50 49 = 0.6761 (b) 52 39 26 13 52 51 50 49 = 0.1055 6.?R? ?$? ?, B? ?“? ?,? P(R B) = P(R) + P(B) = 3 4 6 10 + 3 6 6 10 = 1 2 7. (a) 1 -?40 8 ? = 1.3 108(b) ?8 7 ?32 1 ?-?40 8 ? = 3.3 106 (c) ?8 6 ?32 2 ?-?40 8 ? + 1.3 108+ 3.3 106= 1.8 104 8. (a) 3 4 4 3 ?14 4 ? = 0.1439(b) ?4 2 ?4 2 ?-?14 4 ? = 0.0360(c) ?8 4 ?-?14 4 ? = 0.0699 9.?S = .n i=1Ai,? S?,?P(A) = N(A)/N(S),? ?4.3?4.4? 10.?1?,?5! = 120?, N(A) = 360,? ?, N(B) = 120,N(AB) = 2 4! = 48? ?9,?N(A B) = 432. 11.?#? ? ?Ai? ? ?i?, i = 1,2,3,4.? P ? 4 / i=1 Ai ? = ? i P(Ai) ? j ? i:i 1,? P ? / i=1 Ai ? = P ? / i=1 Bi ? = ? i=1 P(Bi) ? ? i=1 P(Ai) ?Bi?,?Bi Ai. 15. P ? 0 i=1 Ai ? = 1 P ? 0 i=1 Ai ?c? = 1 P ? / i=1 Aci ? ? 1 ? i=1 P(Aci) = 1 16.?1?,?Tk1(n 1)?,?Tk1(n 1)?-? n 1?2, ,n?k 1?,?1,?, ?1? ? ?2, ,n?k?,?Tk(n1)? ?,?1,?1,2, ,n?,?1? ?k?,?,?1?kTk(n 1)?,? 17.?R? ?5?$?, W? ?, B? ?,? P(R W B) = P(R) + P(W) + P(B) P(RW) P(RB) P(WB) + P(RWB) = ?13 5 ?-?18 5 ? + ?12 5 ?-?18 5 ? + ?11 5 ?-?18 5 ? ?7 5 ?-?18 5 ? ?6 5 ?-?18 5 ? ?5 5 ?-?18 5 ? 0.2933 ?, 5?%?1 0.2933 = 0.7067. ?B?9 18. (a) 8 7 6 5 4 17 16 15 14 13 = 2 221 (b)?9?,? 9 8 7 6 5 17 16 15 14 13 = 9 442. (c)?%?4 8 5?,?3!?. ? 3! 4 8 5 17 16 15 = 4 17. (d) 4?$?,? ?14?,? 4? ?4!?.?, 4?$? 14 4! 17 16 15 14 = 1 170. 19. (a) 10#?4#“ ?3#$ ?2#?#? #13 4 $#13 3 $#13 2 $#13 1 $ #52 10 $. ?4?!?“?,? 24 #13 4 $#13 3 $#13 2 $#13 1 $ #52 10 $. (b)?,?#?,?,? ?4#?.? ?4 2 ? 2 = 12?.? 12 #13 3 $#13 3 $#13 4 $ #52 10 $. 20.$?.?.?,$ ?10/30. ? 3 ? 1.(a) P(? ?A?) = ?35 13 ?-?39 13 ? (b) 1 P(? ?A?) 4 ?35 12 ?-?39 13 ? (c) P(i? ?A?) = ?3 i ? 36 13 i ?-?39 13 ? 2.?Li? ?(?10000 i? ? (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) = 1/8 3.?1?0?“?1,?9?10?“?2? 4.?T? ? ?, W? ?B?,? P(T|W) = P(W|T)P(T) P(W|T)P(T) + P(W|Tc)P(Tc) = 2/7 2/3 2/7 2/3 + 1/7 1/3 = 4/5 5. (a) r r + w ,?r + w?i? 10?B? (b)(c) P(Rj|Ri) = P(RiRj) P(Ri) = #r 2 $/#r+w 2 $ r/(r + w) = r 1 r + w 1 ? ?i ?= j,?i?$?,?(r + w 1)? ? ?j?,?$?(r 1). 6.?Bi? ?i?“?,?Ri= Bc i,? PB1|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) 7.?B? ?#?A? ? ? (a) PB|?#?“ ?A? = PB,?#?“ ?A? P?#?“ ?A? = ?1 1 ?3 1 ? ?52 2 ? 1 ?1 1 ?51 1 ? ?52 2 ? = 3/51 (b)?2#?51#?#,?#?, ?,?3/51. (c)?,?,?(b)? ? PB|?#? ?A? = PB,?#? ?A? P?#? ?A? = PB PB + P?#? ?A?,?#? ?A? = (4/52)(3/51) (4/52)(3/51) + (48/52)(4/51) = 3/51 (d) PB|?#? ?A? = B P?#? ?A? = (4/52)(3/51) 1 (48/52)(47/51) = 1/33 8. P(H|E) P(G|E) = P(HE) P(GE) = P(H)P(E|H) P(G)P(E|G) ?, H?G?1.5$? 9.?A? ?%? ?, W? ?%?%? ? ?B?11 (a) P(A) = P(A|W)P(W)+ P(A|W c)P(Wc) = (0.85)(0.9) + (0.2)(0.1) = 0.785 (b) P(W c|Ac) =P(Ac|W c)P(Wc) P(Ac) = (0.8)(0.1) 0.215 = 16 43 10. (a) 1 P(?$?)= 1 #22 6 $ #30 6 $ (b)?$?,?-?22?$?6? ?.? P(2?|?$?) = #10 2 $#12 4 $ #22 6 $ 11.?W?“?”,?C?D?C?D? (a) P(W) = P(W|C)P(C) + P(W|D)P(D) = 0.7 (8/14) + 0.4 (6/14) = 4/7 (b) P(C|W c) =P(CW c) P(Wc) = P(Wc|C)P(C) 3/7 = 0.3 (8/14) 3/7 = 0.4 12.?Li?“Maria?i”,i = 1,2.? P(L2|Lc 1) = P(Lc1L2) P(Lc 1) = P(Lc1L2) 0.4 ?L2?L1L2?Lc1L2?,? 0.5 = P(L2) = P(L1L2) + P(Lc1L2) = 0.4 + P(Lc1L2) ?, P(L2|Lc1) = 0.1/0.4 = 0.25 13. (a)? ?.?30? ?,?1/3. (b)? ?$?,?.?30 ?($?)?.? ?(?-?).?1/3. (c)?B1,R2,G3?,?,?$?,? ?.? P(B1R2G3) = P(G3)P(R2|G3)P(B1|R2G3) = 8 38 20 30 = 8 57 ?P(G3)?G3?“?”,? ?8/38. P(R2|G3)? ?,?20? $?10?.? ?20/30.?P(B1|R2G3),?1. 12?B? (d) P(B1) = P(B1G2R3) + P(B1R2G3) = 20 38 8 18 + 8 58 = 64 171 14.?H?, Th?“B? ?(? ?)”, F?“A!?”, C?B?.? (a) P(Th) = P(Th|F)P(F) + P(Th|F c)P(Fc) = 0.5 0.4 + P(H) 0.6 = 0.68 (b) P(C) = P(C|F)P(F) + P(C|F c)P(Fc) = 0.5 0.4 + 1 0.6 = 0.80 (c)? P(H|Th) = P(HTh) P(Th) ? P(HTh) = P(HTh|F)P(F) + P(HTh|F c)P(Fc) = P(H|F)P(Th|HF)P(F) + P(H)P(F c) = 0.8 0.5 0.4 + 0.8 0.6 = 0.64 ?P(H|Th) = 0.64/0.68 = 16/17. 15.?A?, B?, E? ?,?1 = P(A) + P(B) + P(E).?, P(A) = P(B),?, P(B) = 1 P(E) 2 = 1 n ? i=1 p2 i 2 ? P(B) = ? i ? ji P?i,?j = ? i ? ji pipj ? 1 = n ? i=1 pi n ? j=1 pj= ? i ? j pipj= ? i p2 i+ ? i ? j?=i pipj= ? i p2 i+ 2 ? i ? ji pipj ?B?13 16.?E = A?B?, Aw= n?, A?B?, Bw= n?, B?A?, Ae= n?, A?B?,? P(E) = P(E|Aw)P(Aw) + P(E|Bw)P(Bw) + P(E|Ae)P(Ae) = 1 P(Aw) + 0 P(Bw) + 1 2P(Ae) = P(Aw) + 1 2P(Ae) ?P(Aw) = P(Bw).?1 = P(Aw)+P(Bw)+P(Ae)?P(Aw) = 1 2 1 2P(Ae). ?, P(E) = P(Aw) + 1 2P(Ae) = 1 2 1 2P(Ae) + 1 2P(Ae) = 1 2 17. (a)?,?E = ?7?, F = ?4, G = ?3? ?, E,F?, E,G?,? P(E|F G) = P?7,?4,3 P?4,3 = 5/36 35/36 = 5 35 ?= P(E) (b) P(E(F G) = P(EF FG) = P(EF) + P(EG)?EFG = = P(E)P(F) + P(G) = P(E)P(F G)FG = (c) P(G|EF) = P(EFG) P(EF) = P(E)P(FG) P(EF) ?E?FG? = P(E)P(F)P(G) P(E)P(F) ? = P(G) 18. (a)? ?,?0 = P(AB) ?= P(A)P(B)? (b)? ?,?P(AB) = P(A) P(B) 0? (c)? ?,?P(A B) = P(A) + P(B) = 1.2? (d)? 19. (a), (b), (c)?0.5,(0.8)3= 0.512,(0.9)7 0.4783. 20.?Di(i = 1,2)?i?A(B)?A(? B)? ? P(D2|D1) = P(D1D2) P(D1) = P(D1D2|A)P(A) + P(D1D2|B)P(B) P(D1|A)P(A) + P(D1|B)P(B) = 0.052 1/2 + 0.012 1/2 0.05 1/2 + 0.01 1/2 = 13 300 14?B? 21. P(A|B) = 1?P(AB) = P(B),?P(Bc|Ac) = 1?P(AcBc) = P(Ac)?,? ?,?P(AB) = P(B) P(AcBc) = P(Ac).? P(BcAc) = P(A B)c) = 1 P(A B) = 1 P(A) P(B) + P(AB) = 1 P(A) = P(Ac) 22.?n = 0?,? ?Ai? ?n?,?i?$?,? ? P(Ai) = 1 n + 1 i = 1, ,n + 1 ?Bj? ?n + 1?j?$? ?,? P(Bj) = n+1 ? i=1 P(Bj|Ai)P(Ai) = 1 n + 1 n+1 ? i=1 P(Bj|Ai) = 1 n + 1P(Bj|Aj1) + P(Bj|Aj) ?n?,?n + 2?P(Bj|Aj1)?n + 2?,? j 1?$?,?$?,?, n+1?,?i? $?,? P(Bj|Aj1) = j 1 n + 2 ?P(Bj|Aj)?j?$?, n+2j?,? ?,? P(Bj|Aj) = n + 2 j n + 2 ?P(Bj)?,? P(Bj) = 1 n + 1 j 1 n + 2 + n + 2 j n + 2 = 1 n + 2 ?,? 23.?Ai? ?i?A?,? P(Ai) = 1 ?2n 2 n ?-?2n n ? = 1 1 2 n 1 2n 1 = 3n 1 4n 2 A1A2? ?2#A?#,?2n2#? A?n1#? ? ? ?, P(A1A2) = ? 1 2 ?2n 2 n 1 ?-?2n n ? = n 2n 1 ?, P(Ac2|A1) = 1 P(A2|A1) = 1 P(A1A2) p(A1) = n 1 3n 1 ?,?i?i#?A?,?n ?,?,?1/2,?,? ?,?(= 1/3),?, n?,?2b? ? ?B?15 24. (a)?n?i1,i2, ,in,?pi1pin= +n i=1pi. ?, n?n?n! +n i=1pi. (b)?i1, ,ik?,? P(Ei1Eik) = ?n k n ?n ?Ei1, ,Eik?i1, ,ik?,?n?,? ?i1, ,ik.?,?P(Ei1Ein)? ? ? P ? n / i=1 Ei ? = n ? k=1 (1)k+1 ?n k ?n k n ?n ?1P ?. n i=1Ei ? ?n?,?(a)?n/nn, ?P #. n i=1Ei $ ?,? 1 n! nn = n ? k=1 (1)k+1 ?n k ?#n k n ?n ? n! = nn n ? k=1 (1)k+1 ?n k ? (n k)n ? n! = n ? k=0 (1)k ?n k ? (n k)n 25.?A = EFc,B = FEc,C = E F,?A,B,C?,? E F = A B C E = A CF = B C P(E|E F) = P(E (E F) P(E F) = P(A)P(C) P(A) + P(B) + P(C) P(E|F) = P(EF) P(F) = P(C) P(B) + P(C) ?, P(E|E F) ? P(E|F)? ? 4 ? 1.?1,? 4PX = 3 + 0.5 = 1 ?PX = 0 = 0.375,PX = 3 = 0.125,?EX = 1 0.3 + 2 0.2 + 3 0.125 = 1.075. 16?B? 2.?,?pi= cip0,i = 1,2,?pi= PX = i? ?1,? p0(1 + c + c2) = 1 p0= 1 1 + c + c2 ?, EX = p1+ 2p2= c + 2c2 1 + c + c2 3.?X?, X? p2= p2+ (1 p)2p3= 1 p2= 2p(1 p) ?, EX = 2p2+ 3p3= 2p2+ 3(1 p2) = 3 p2 (1 p)2 4.?,?i?!?ni/m ,? EX = r ? i=1 ini/m ?i?!?ni,?!?ini,?!? ?ini/ ?r j=1ini? ?, EY = r ? i=1 i2ni r ? i=1 ini ?,? r ? i=1 i2ni r ? i=1 ini ? r ? i=1 ini r ? i=1 ni ? r ? j=1 nj r ? i=1 i2ni? r ? i=1 ini r ? j=1 jnj r ? i=1 r ? j=1 i2ninj? r ? i=1 r ? j=1 ijninj ?(i,j),?ninj?i2+j2,?ninj?2ij,? ? i2+ j2? 2ij ? 1 1. ?. ? pi= ni ? ni , ? (?pi)(?i2pi) ? ? ipi ?2 . ? ? ? ? ? ?B?17 5.?p = PX = 1,?EX = p,Var(X) = p(1 p),? p = 3p(1 p) ?p = 2/3? ?PX = 0 = 1/3. 6.?&?x,?x?p,?x?1 p? ?,? xp x(1 p) = (2p 1)x ?p 1/2?,?,?p 0 = 1 e 16. (a) 1/n (b)?D?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) = Pi,j? = n ? k=1 Pi,j?k = nPi,j?1 = n 1 n2 = 1 n ?, P(GiGj) = P(GiGj|D)(1 1 n) = ?1 n ?2? 1 1 n ? = n 1 n3 ?P(Gi|Gj) = P(GiGj)/P(Gj) = (n 1)/n2. (c)(d)?n?, P(Gi|Gj)?,?P(Gi)? ? ?,? ?n i=1P(Gi) = 1. ?, P0 e1,Pk e1/k!. (e)?k?,?D?k? ?. P(Gi1Gik) = P(Gi1Gik|D)P(D) + P(Gi1Gik|Dc)P(Dc) = P(Gi1Gik|D)P(D) = ?1 n ?kn(n 1)(n k + 1) nk = n! (n k)!n2k ?, ? i1 x = (n + 1) “1 x xndx = xn+1|1 x= 1 x n+1 3.?c? 1 = ? 2 0 cx4dx = 32c/5 c = 5/32 (a) EX = 5 32 “2 0 x5dx = 5 32 64 6 = 5/3 (b) EX2 = 5 32 “2 0 x6dx = 5 32 128 7 = 20