. : : .

. : : 1385 1386 1387 1388 1389 5 1 11112622% 2 1100027% 3 0000000% 4 21210622% 5 232331348% 6655527100%

: 1 . A B 1613. : ) 74 (1 ( ( ) ( )1 1PA , PB2 3= = 2 ( ( ) ( )1 1PA , PB3 2= =3 ( ( ) ( ) PA PB 4 ( ( ) ( ) PA PB = : 3 . : A B A B .( ) ( ) ( ) ( ) ( )1 1PA B P A B P AP B I6 6 = = = .( ) ( ) ( ) ( )1 1PA B PA B PA PB3 3 = = = .( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )11 PA 1 PB 1 PA PB PAPB3 - - = - - + ( ) ( ) ( )161 5PA PB II3 6= + =B , A B, A : ( ) PA B : : n ni ii 1 i 1P A P A= = = ( ) ( ) ( ) ( ) ( ) ( ) PA B PA B 1 PA B 1 PA PB PA B = =- =- - + ( ) ( )161 5PA PB3 6= + = ( ) I ( ) II 16 56 12 13. 1312 ( ) I ( ) II :12 ( )1 1PB3 3 = ( )1PA2= 1 2 3 . 2 . ( ) ( )3 3P B , P A4 4= = ) 76 (1 ( ( )3PA B8< 2 ( ( )3PA B4< 3 ( ( )3P A B4 4 ( ( )3P A B4 : 4 .( )( )( )( ) ( ) ( ) ( )PA3 6PA B PA B PA PB PA B4 4 PB +

3 . ( )3PA4= ( )3PB8= ( ) PA B ) 76 (1 ( ( )3PA B8= 2 ( ( )3PA B4 3 ( ( )3PA B4 4 ( ( ) PA B . : 3 .

: ( )( )( )( ) ( ) ( )( )( )( )( )( )( )PAPA B PA B PA PBPB3PA B34PA B3 4PA B83PA B3 94PA B3 3 9 4 8PA B4 8 8+ + =

4 . A B W ( ) ( ) PB 0.4 , PA 0.2 = = ( ) PA B 0.06 = ( )PA B ) 80 (1 ( 0.1 2 ( 0.12 3 ( 0.14 4 ( 0.17 : 3 . :

( ) ( ) ( ) ( )( ) ( )PA B PA B PA PA B 0.2 0.06 0.14PA 0.2 , PA B 0.06 = - = - = - = = =

5 . A B . a B b . A ) 80 (1 ( ( )aP A1 b=-2 ( ( )b aP A1 b-=- 3 ( ( )a bP A1 b-=-4 ( ( )1 b aP A1 b- -=- : 4 .( ) ) PA B a = = A ) P(B = A B P(( ) ( ) PB b , PA ? = = : : A B B , A .( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )PA B PA .PB PA 1 PB PA 1 b aa a 1 b aPA PA 11 b 1 b 1 b = = - = - =- - = = - =- - -

: : : n ni ii 1 i 1P A P A= = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )PA B PA B 1 PA B 1 PA PB PAPB aPA PB PAPB 1 a PA b PAb 1 a1 a bPA 1 b 1 a b PA1 b = = - = - + - = + - = - + - = -- - - = - - =-

6 . A B : 0.8 0.9 ) 82 (1 ( 0.7 2 ( 0.72 3 ( 0.02 4 ( 0.98 : 4 .) 0.8 = P(A) 0.9 = P(B :( ) ( ) ( ) ( ) ( ) ) PA B PA PB PA B 0.8 0.9 0.8 0.9 0.98 = = + - = + - = P( :( ) ( ) ( ) PA B PA PB = :( )( ) ( )( ) ) 1 1 0.8 1 0.9 1 0.2 0.1 0.98 =- - - =- = ) 1 P( = - P( 7 . I II A B C . 0.9 0.8 0.7 ) 86 (1 ( 0.892 2 ( 0.902 3 ( 0.908 4 ( 0.994 : 2 .) 0.398 0.504 0.902 = + = 3 2) P( + ) P( = 2 P( : :( ) ( ) ( ) 2) P A B C P A B C P A B C = + + . P(0.216 0.126 0.056 0.398 = + + = 0.9 0.8 0.3 0.9 0.2 0.7 0.1 0.8 0.7 = + + ( ) ) PA B C 0.9 0.8 0.7 0.504 = = = 3 P(( ) ( ) ( ) PA 0.9 PB 0.8 PC 0.7 = = =

8 . 4 13 . ): 86 (1 ( 0.2 2 ( 0.3 3 ( 0.6 4 ( 0.5 : 1 .42 16) 0.197 0.23 81 = = = = 4 ) P( = P( 13 23 4 4 423 .9 . . . ) 84 (1 (3111 2 (133 (3114 (518 : 4 .) ) P( + ) P( + ) P( = P(3 1 2 2 1 3 3 1 2 2 1 3 10 53 2 1 1 2 3 3 2 1 1 2 3 3 2 1 1 2 3 6 6 6 6 6 6 36 18= + + = + + = =+ + + + + + + + + + + + 10 . 52 1 52 . j 1, 2, 3, 4 = . 4k j + k 0,1, 2,....,12 = j . 41, 31,11 ,1 ) ( ) 80 (1 (( )31317 25 49 4 2 (( )31317 25 49 3 (( )31317 50 49 4 4 (( )31351 50 49 4 : : 2 . : n k 1n 2n k kn :1 2 k1 2 k1 2 knn!; n n ... n nn n ...nn !n !... n ! = + + + = 52!13! 13! 13! 13!= 52 4 ( ) nS =4! 48!1! 1! 1! 1! 12! 12! 12! 12!= _ _ ( ) n A = 48 ( )( )( )( )348!4!nA 1312! 12! 12! 12!PA52!nS 17 25 4913! 13! 13! 13!= = = k 0,1, 2, ...,12 , 4k j = + j . 4 4 ( ) 4! 48 4 . 52 4 .j 1 4k 1 1 5 9 13 17 ... 494k j j 2 4k 2 2 6 10 14 18 ... 50k 0,1, 2, ...12 j 3 4k 3 3 7 11 15 19 ... 51j 4 4k 4 4 8 12 16 20 ... 52= + + = + = = + = +

13 .11 . 10 ...., 2,1, 0 9 . . 3 ) 83 (1 ( 0.15 2 ( 0.8 3 ( 0.2 4 ( 0.4 : : 3 .

{ } 0,1, 2,..., 9 10 = = = ( ) nS{ } 1, 3, 5, 7, 9 = B ={ } 3, 6, 9 = C 3 ={ } 3, 9 2 = = B C = ( ) n A 3 =( )( )( )nA 2PA 0.2nS 10= = =12 . 10 . ) 79 88 (1 (12102 (12523 (13304 (1420 : 4 . : n n! . 5! 3! 2! 3! 110! 420= = ( ) PA= : 10 10! . :P5 5! .P3 3! .P2 2! .P 3 ) ( 3! .13 . ( ) 0,1 . 5 ) 85 (1 ( 0.01 2 ( 0.07 3 ( 0.10 4 ( 0.50 : 3 . 0 9 10 :10 10 100 = = : 5 5 . 10 : 510 1 10 = =

100.1100= = = 14 .12 . 8 . ) 76 (1 (1281248 - 2 (812281248 - 3 (812812428 - 4 ( : 2 .81228) 1248 = - ) 1 P( =- P( :248 = 8 12 24) ( ) nS: ( : ) ( 8 12 128 .812 2 2 2 1228 1 1 1 8 =

: 15 .n 10 = . ) : 74 (1 (10 10103 23-2 (10312 - 3 (1023 4 (13 : 1 .) ) 1 P( = - P(10 10 10 1010 102 2 2 2 2 3 21 .... 1 13 3 3 33 3- = - = - = - = _10 : 23.16 .n N N 3 2 1u ,...., u , u , u . 1u k ) )(. 77 (1 (( )n knN 1N--2 (( )n knnN 1kN- - 3 (N n k 2n kN n 1n+ - - - + - 4 (n N n k 2k n kN n 1n+ - - - + -

: 2 . n N nN . k nk 1U . ( ) n k - N 1 - . ( )n kN 1-- ) .( ( )n knnN 1kN- - = ( ) PA=17 . 10 10 . 2 . ) (1 (( )210!20!2 (( )2 102 10!20!3 (( )102 10!20!4 (( )( )21010!2 20 ! : : 2 .( ) ( )2 2 102 2 21010 10 9 9 1 110! 2 10! 1 1 1 1 1 1 10 9 1P ... ...20 19 18 17 2 1 20! 20 18 2 20!2 2 22 2 2 2 = = = =

1 10 101 1 10 101 2 20 102 1 9 91 1 9 91 2 18 182 . .18 . M M n . . ) 74 (1 (1 2 n 1P 1 1 .... 1M M M- = - - - 2 (nPM=3 (( ) M n!PM!-= 4 (1 2 nP 1 1 .... 1M M M = - - - : 1 . M .( ) ( ) M n 1 n 1 M M1 M 2 1 2) ... 1 1 1 ... 1M M M M M M M- - - - - = = - - - P( MM M . M1 - M1M- . 2 M 2 - M 2M- n ) n ( n 1 - n 1 - ( ) M n 1 - - ( ) M n 1M- -. : 19 . 4 2 . ) 78 88 (1 (1102 (3103 (4104 (15 : 4 . : .) ) P( + ) P( + P(2 4 3 1 4 2 3 1 4 3 2 1 1 1 1 3 16 5 4 3 6 5 4 3 6 5 4 3 15 15 15 15 5 = + + = + + = = . : 1 2 1 2 13 .2 41 2 1 2 6 1 1P6 3 20 3 53 = = =

) . (20 . 3 4 5 . ) ( ). 84 (1 ( 0.288 2 ( 0.310 3 ( 0.272 4 ( 0.324 : : 1 . : 32 42 52 . 2 12 122 .( )( )( )3 4 5n A 2 2 2 3 6 10 19PA 0.28812 nS 66 662 + + + + = = = = = :3 2 4 3 5 40.28812 11 12 11 12 11 = + + = _ _ _

2 2 2 21 . A B C D 0.8 0.9 0.9 0.7 . A D B C :) (1 ( 0.5544 2 ( 0.5454 3 ( 0.4545 4 ( 0.4455 : 1 . . ( ) ( ) ( ) ( ) B)] PA B C D PA PB C PD 0.8 0.99 0.7 0.5544 = = = = (C D) ) P[ (A = P(( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B) PB C PB PC PB C PB PC PBPC 0.9 0.9 0.9 0.9 0.99 = = + - = + - = + - = P(C : ( ) ( ) ( ) ) 1 PB C 1 PB PC 1 0.1 0.1 0.99 =- =- =- = B B) 1 P(C = - P(C :1 ( .2 ( B C B C . : 22 . A B . A B ): 77 (1 ( 0.5796 2 ( 0.5043 ( 0.378 4 ( 0.882 : 1 . 4 4 3 2 1N, N, N, N A B 1N 4N 2N 3N )2N 3N :(

( ) ( )4 1 2 3 4N] PN N N N 0.7 0.92 0.9 0.5796 = = = 2N) 3(N 1) P[N = P(( ) ( ) ( ) ( ) ( )2 3 2 3 2 3P N N P N P N PN N 0.8 0.6 0.8 0.6 0.92 = + - = + - = ( )( ) ) 1 1 0.8 1 0.6 0.92 =- - - = ) 1 P( = - 3 2N, N P( .23 . A B C . A B C 100 %90 %80 %70 . B C 100 %60 . A B C . A B C . 100 : ) 84 (1 ( %68 2 ( %75 3 ( %81 4 ( %86 : 3 . . A B C :

( ) B)] P A B C = (C ) P[A = P( : A B C :( ) ( ) ( ) P A B C P A P B C 0.9 0.9 0.81 = = = ( ) ( ) ( ) ( )( ) ( ) ( )P B C PB PC P B C 0.8 0.7 0.6 0.9P B , C P B C 0.6, PA 0.9 = + - = + - == = = .24 . p 1B 3B :) 79 (1 ( ( )n 1p 1 p-- -2 (2np2pn-3 (n2p p -4 ( ( )n1 1 p p - - : 4 . 1B 3B n 1B 2B 2B 3B n 1B 2B :( )n) 1 1 p p = - - 2B 3) P(B n 1B 2) P(B = 1B 3P(B( )n) 1 1 p =- - n ) 1 P( =- n 1B 2P(B p 1 p - n . 25 . 3 . A B . A A ) 79 (1 (122 (493 (144 (29 : 1 .

1A)4= P(

3 2 1 1A) A B A4 3 2 4 = = = ) ( P(

) P(A

: 1 1 1)4 4 2= + = ) P( + ) P( = P(A . 4 A ) ( . 26 . n ( ) n 365 ) (1 (122 (1365 3 ( ( )1n 1 !365- 4 (1 2 n 11 1 ... 1365 365 365- - - - : 4 . 365 n 365 :( ) 365 n 1365 365 1 365 2 1 2 n 1... 1 1 1 ... 1365 365 365 365 365 365 365- -- - - = - - - 365 . 365 1 - 365 2 - n n 1 - ( ) 365 n 1 - - . 27 . A B . A 0.4 B 0.5 A B ) 72 (1 (162 (563 (264 (36 : 2 . A B A B :

( ) ( )( ) ( )PA B PAPB A PB = =

:

( )( )( )( )( )PB A PB 0.5 5) PB| APA PA 0.6 6 = = = = = B A P(( ) ( ) ( ) P A 0.4 P A 1 0.4 0.6 , P B 0.5 = = - = = : A B ) ( A B . A B A B . 28 . ( ) ( ) P E P F 0.6 = = ( ) P E| F 0.8 = ) 73 (1 (( )P E| F 0.3 = 2 (( )PE| F 0.2 = 3 (( )PE| F 0.5 = 4 (( )P E| F 0.5 = : 1 . ( ) P E | F ( ) P E | F :( )( )( )( )( )( )P E F P E FP E| F ; P E | FP F P F = =

( ) ( ) P E P F 0.6 = = ( ) ( ) P E P F 0.4 = = ( ) P E F ( ) P E F :( ) ( ) ( ) ( ) ( )( )( )( )( )( ) ( )( )( )P E P E F P E F 0.6 P E F 0.48 P E F 0.12PE F PE F PE F0.12PE| F 0.8 0.8 PE F 0.48 PE| F 0.3PF 0.6 PF 0.4 = + = + = = = = = = = = P ( )( )( )( ) ( ) ( )( ) ( ) ( ) ( )P E F0.28P E | F 0.7P F 0.4P E F P E F 1 P E F 1 0.72 0.28P E F P E P F P E F 0.6 0.6 0.48 0.72 = = = = = - = - == + - = + - =

29 . A B ) : A A ( ) : 82 (( ) ( ) ( )5 45 1PA B , PA B , PA B100 100 10= = = 1 (( ) ( ) ( )1PB| A 0.1 , PA| B 0.2 , PB| A2= = =2 ( ( ) ( ) ( )1PA| B 0.2 , PB| A 0.1 , PB| A3= = =3 ( ( ) ( ) ( )1PA| B , PB| A 0.1 , PB| A 0.23= = =4 ( ( ) ( ) ( )1PB| A 0.2 , PB| A , PA| B 0.12= = = : : 3 .

( )( )( )( )( )( )( )( )( )P A B0.05 1P A| BP B 0.15 3P A B0.05P B| A 0.1P A 0.5P B A0.1 0.1 1PB| A 0.2P A 1 0.5 0.5 5= = == = = = = = = = -

( ) ( ) ( )( ) ( ) ( )PA PA B PA B 0.05 0.45 0.5PB PA B PA B 0.05 0.1 0.15 = + = + = = + = + = ( ) ( ) ( ) PA B 0.45; PA B 0.05; PA B 0.1 = = = 30 . A B ( ) ( ) PA B 0.2 , PA B 0.5 = = ( ) ( )PA B 0.2 , PA B 0.1 = = :) (1 ( A B ( ) PA| B 0.83 =2 ( A B ( ) PA| B 0.7 =3 ( A B ( ) PA| B 0.83 =4 ( A B ( ) PA| B 0.7 = : 1 . A B .( ) ( ) ( )( ) ( ) ( ) ( )P A B P AP BP A | B P A , P B| A P B == =

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )PA PA B PA B 0.5 0.2 0.7PA B 0.5 0.7 0.6 PAPBPB PA B PA B 0.5 0.1 0.6 = + = + = = = = + = + = A B .( )( )( )( )PA B0.5PA | B 0.83 0.7 PAPB 0.6= = = =

31 . 4 3 . . ) 85 (1 (362 (463 (374 (47 : : 4 . : : = 47. : :) ) P( | ) P( + ) P( | ) P( = P(3 4 4 3 46 7 6 7 7 = + = . 32 . A B ( ) ( ) PA PB 1 + > . ( ) PB| A ) 85 (1 (( )( )PB1PA- 2 ( ( ) ( )PA PB 1 + -3 (( )( )1 PBPA-4 ( ( ) 1 PB - : 1 . :( )( )( )( ) ( )( )( )( )( )( )( )( )PBPB A PA PB 1 PB 1 1 PBPB| A 1 1 1PA PA PA PA PA+ - - -= = + = - = -

( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 PA B 1 PA PB PA B 1 PA B PA PB 1 + - + - 33 . . . . ) (1 (392 (4103 (494 (310 : : 2 . : : 410. :) ) P( + ) P( + P(4 4 2 4 4 3 36 410 9 10 9 10 9 90 10 = + + = = 34 . p "0" . e "0" ) (1 ( pe2 ( p+e 3 ( p 2p +e - e 4 ( 1 p 2p - -e + e : 3 . p ( ) 1 p - 1 e 1-e :

1-e 1 e :( ) ( ) ) p 1 1 p p p p p 2p = -e + - e = - e +e - e = +e - e 0 P(35 . 3 . %4 %5 %6 . ) 86 (1 (4752 (10753 (65754 (7175 : : 1 .: E

: ) k = P() 2k = P(1k 2k 3k 1 k6 + + = =) 3k = P(

( ) ( ) ( ) ( ) ( ) ( ) ( )1 4 2 5 3 6 32 4PE PAPE | A PBPE | B PCPE | C6 100 6 100 6 100 600 75 = + + = + + = = 36 . ) 1 ( . ) 2 ( . . . ) 80 (1 (152 (253 (124 (23 : 2 .: E : H : T

:( ) ( ) ( ) ( ) ( )3 1 2 1 2PE PE | HPH PE | TPT5 2 10 2 5= + = + =37 . A 3 7 B 4 6 C 5 5 . ) 85 (1 ( 0.4 2 ( 0.533 3 ( 0.467 4 ( 0.6 : : 4 .: E :( ) ( ) ( ) ( ) ( ) ( ) ( )7 1 6 1 5 1PE PE | APA PE | BPB PE | CPC 0.610 3 10 3 10 3= + + = + + = 13) 3 .(38 . 3 n . . n 12 ) (1 ( 1 2 ( 4 3 ( 3 4 ( 2 : 3 .

:1)2= P() ) P( | ) P( + ) P( | ) P( = P(( )( )21 3 3 5 n 1 9 5n18 10n n 8n 152 5 n 3 n 5 n n 3 2 5 n 3 n+ = + = + = + ++ + + + + + ( )( )2n 1n 2n 3 0 n 1 n 3 0n 3= - - - = + - = =

: 33 n + 2 3 n 2 + 3n 2 3 + +. nn 3 + 2 5 n 55 n +. 39 . m . p 1 p - . 1m. ) 81 (1 (11pm +2 (( )p11 1 pm+ -3 (( )11p 1 pm+ -4 (( )mp1 m 1p + - : 4 .: E : A

( )( ) ( )( ) ( ) ( ) ( )( )( )P E | AP A1 p mpP A | E1P E | AP A P E | A P A 1 m 1p1 p 1 pm= = = + + - + -40 . b r . c . ) 84 (1 (rb r c + +2 (bb r c + +3 (b cb r c++ +4 (r cb r c++ + : : 2 .1: b 1: r : E :( )( )( )( ) ( )( ) ( ) ( ) ( )1 1 111 1 1 1r bPb E PE | b Pb b r b c b rPb | Er b r c rPE PE | b Pb PE | r Pr r b cr b c b r r b c b r+ + += = = =++ + + + + + + + + +

b r rb r + bb r +. c ) ( b r c + r cr b c++ +. c ) ( b c + r rr b c + +.41 . 0.5 . 80 98 . ) 75 (1 ( 0.005 2 ( 0.199 3 ( 0.976 4 ( 0.795 : 3 .: E : A : A

( )( ) ( )( ) ( ) ( ) ( )PE | APA0.2 0.995PA| E 0.976PE | APA PE| A PA 0.2 0.995 0.98 0.005= = = + + : .42 . A B C . 1614 13 . A : ) 77 (1 (31722 (6313 (10314 (1531 : 2 .: E . :( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )P E | AP AP A| EP E | APA PE | BPB PE | CPC=+ + ( )1 3 266 4 3PA | E1 2 3 5 1 2 5 3 1 316 3 4 6 4 3 6 4 3 = = + +

43 . %60 . %70 18 %50 18 . 18 :) (1 ( %33 2 ( %503 ( %67 4 ( . : 3 .: A .: E 18 .

:( )( ) ( )( ) ( ) ( ) ( )P E| AP A0.7 0.6P A| E 0.67 %67P E| A P A P E | A PA 0.7 0.6 0.5 0.4= = = = + +

: 44 . A B C %70 A %20 B %10 C . %5 %6 %4 A B C . B ) 87 (1 ( %1.2 2 (12513 ( %5.1 4 (3951 : 2 .: E

:( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )PE| BPB0.06 0.2 12 12PB| EPE| APA PE| BPB PE| CPC 0.05 0.7 0.06 0.2 0.04 0.1 35 12 4 51= = = =+ + + + + + 45 . 0.4 . 0.3 . 0.7 . .) 87 (1 ( 0.21 2 (2303 ( 0.28 4 (2830 : 4 .: E : A ( )( )( )( ) ( )( ) ( )( )( ) ( ) ( ) ( )PA E PE | APA0.4 0.7 28PA | E 0.3 PEPE PE PE 30PE | A 0.4 , PA | E 0.3 , PA 0.7 , PE ?= = = == = = =

46 . %60 . %50 . %30 . ) 88 (1 ( 30 2 ( 50 3 ( 60 4 ( 100 : : 4 .: E : A

0.60.40.51 x -x0.5

0.3 :( ) ( ) ( ) ( ) ( ) PE 0.3 PE | APA PE | A PA 0.3 0.5 0.6 x 0.4 0.3 x 0 = + = + = = :( )( )( )( ) ( )( )PE A PE | APA0.5 0.6PA | E 1 %100PE PE 0.3= = = = =

47 . . . ) 87 (1 (342 (233 (134 (14 : 1 . :: E : A : B ( )( ) ( )( ) ( ) ( ) ( )1 1 1PE | APA12 2 2PA | E1 1 1 1PE | APA PE | BPB 51 12 2 2 2 = = =+ + : 48 . 1A 2A 3A 1412 34. 6 . 3 . 3A ) 88 (1 (32324 + 2 (31423 + 3 (31324 + 4 (32423 +

: 2 .3: H 3

3 31p461p23 33p461 1 33 3 4 46 1 13 3 26 1 3 13 3 4 4=== 1A2A3A6 3 6 3 6 3

:( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )3 3 33 33 1 1 3 2 2 3 3 3PH A PAPA HPH A PA PH A PA PH A PA=+ + 3 33 33 3 6 3 3 3 3 6 3 33 3363 1 13 13 4 4 34 46 6 61 3 1 1 1 3 1 1 1 3 1 3 13 3 3 4 4 3 2 3 4 4 3 4 4 2 4 43 14 414 = = + + + + = 33 3 3 3 3 3 331 43 1 3 1 3 3 41 24 4 4 4 4 4 3 = = + + + + +

: (Discrete Probability Function) X ( ) f x X : ( ( ) ( ) x : 0 f x Px 1 " = ( ( ) ( ) f x Px 1 = = ( ) ( ) ( ) ( ) ( ) ( )1 n1 nx x xPX x f x 0 f x 1 0 f x 1 f x Px 1 = = = = .. (Continuous Probability Density Function)( ) f x X a b : ( ( ) f x 0 ( ( ) f xdx 1b=+a=-= : : a b a b .( ) ( )baPa X b f xdx < < = : x a = a .( ) ( )aaPX a f xdx 0 = = = :( ) ( ) ( ) ( ) Pa X b Pa X b Pa X b Pa X b = < = < = < = - . ( )XF x ) ( X :1 - ( ) ( ) xXF PX x = ( ) xXF x 1 :( ) ( )( ) ( )( )( )F PX 0 F 0F PX 1 F 1-= -= -= += += += 2 - 1 ( )XF x 0 1 .( )X0 F x 1 3 - ( ) xXF .( ) ( ) a b Fa Fb < ( ) ( ) xXF PX x = x x ( ) xXF ) ( ) xXF 1 ( ( ) xXF ) ( .4 - ( ) xXF .( )( )( ) xX X Xx alimF F a F a++= = : 9 . X ( )XF x . F ( ) Y FX = [ ]1P Y EY4 - < ) 82 (1 (142 (343 (124 (23 : 2 . X ( )XF x ( )XY F x = ( ) 0,1 ( ) ( )X0 y F x 1 = .

( ) ( ) ( )a 0b11 1 0 1Y ~ U0,1 f y 1 , EY1 0 2 2==+ = = = =- ( ) ( ) [ ]3 3 34 4 4001 1 1 3 3P Y EY P Y P Y f ydy 1.dy y4 2 4 4 4 - < = - < = < = = = =

X ( ) ( ) f x Px = ) ( ( ) ( )xXF :( ) ( ) ( ) ( )XXxF x PX x Px f x= = = ( ) f x X ( ) ( )xXF :( ) ( ) ( )xxXF PX x f xdx = =

: ( )f x ( )XF x ( ) f x ( ) xXF : :( ) ( )+0 Fx Fx-- fX(x) ) ( :( ) F xx) (X(x) F ) ( ( ) ( ) X XF x F x+- :( )( ) ( ) X Xf x F x F x -= -+ . ( ) ( )XPX x = F x ) ( X ( ) x fx :( )( ) ( )( )( )( ) ( )( )( ) ( )X XXXX X1) PX a F a F a2) PX a F a3) PX a F a4) Pa X b F b F a+ --- += = -< = =< < = -

10 . ) ( X :( )2X0 x 0x0 x 141F x1 x 22x2 x 331 x 3< Y=1 Y : ( ) PX = Y= 0 . X Y ) ( :( ) ( )X Yf t = f t ; t R ... . X Y :( )( )f x 2x ; 0 x 1f y 2y ; 0 y 1= < Y) :1 ( X Y ( ) iid :( ) ( )( ) ( )( ) ( )PX < Y+ PX > Y=1PX < Y= PX > Y1P X < Y = P X > Y =2 2 ( X Y ( ) iid :( ) ( ) ( )( ) ( )( ) ( )( )PX < Y+ PX = Y+ PX > Y=1PX < Y= PX > Y1- P X = YP X < Y = P X > Y =2 3 ( ) 1 ( ) 2 ( X Y ( ) iid :( ) ( ) 1P X < Y = P X > Y2 ! X Y ( ) iid : ( X Y : ( ) PX = Y= 0 ( X Y : ( ) ( ) ( )2XY XPX = Y= Px, y Px== 5 . :( )( )1PX x , x 1, 2,...,1PY y , y 1, 2,...,= = = qq= = = qq

( ) PX Y ) 88 (1 (1q2 (1 q -q3 (11 q -4 (21 q- q

: : 2 . X Y :( )22 2 2x 1 x 1 x 11 1 1 1PX Y 1q q q= = =q = = = = = = q q q q q ( ) ( )1 1PX Y 1 PX Y 1q- = - = = - =q q 6 . X Y ( ) iid ( ) PX Y > :) 85 (1 ( 0 2 ( 13 (124 ( X Y . : 4 .( ) ( ) ( )( ) ( )( ) ( )PXY PXYP X Y P X Y PX Y 1 2PX Y 1 PX Y< = >< + = + > = > =- =( ) ( )( ) ( )( )P X Y 0P X Y 011 P X Y2P X Y P X Y1 22= == =- => = < = 12 X Y 12 12 .X Y X Y X Y : (Conditional Probability Function) ( ) f x, y ( ) f x ( ) f y :( )( )( )( )( )( )f x, y f x, yf x | y , f y | xf y f x= =7 . ( ) X, Y 3 1P X Y4 2 = 3x 0 y x 1 0( )x, yf x, y= ) 80 (1 ( 0 2 (143 (124 (34 : 4 .( )( )( )( )( ) ( )( )2 22112 2x yyf x, y3x 2x 1 2x 8f x | y f x | y x3f y 2 31 y 11 y1223 3f y f x, ydx 3x dx x 1 y2 2 = = = = = = - -- = = = = - 33 32424 40 003 1 1 8 8 1 8 1 3 3P X | Y f x | Y dx xdx x4 2 2 3 3 2 3 2 4 4 = = = = = = = 8 . X Y ( )10 x y 1y f x , y0< < 0 0( )X,Yf x, y= ( )EYX x = ) 88 (1 (1 x2-2 (1 x2+3 (1x2 - 4 (12 : 1 .( ) ( )( )( )1 x 1 x220 y 01 1 1 1 1 1 xEY| X yf y | xdy y. dy y . 1 x1 x 1 x 2 1 x 2 2- -- = = = = - = - - -

( )( )( ) ( )f x, y2 1f y | xf x 2 1 x 1 x= = =- - ( ) ( ) [ ] ( )1 x1 x0y 0f x f x, ydy 2dy 2y 21 x--= = = = - 11 . X Y :ye 0 x y-< < = - = - = - =) 1 n ( ( ) ( ) ( )nPX 1 1 PX 1 1 PX 0 1 q = - < = - = = -1 . ) ( 3 ) 82 (1 (656 2 (516-3 (6516 - 4 (2 3 61 1 1 1....6 6 6 6 + + + + : 3 . 1p6 = ) n 6 = ( :5) , n 66= = 1) ; q P(36= = p P(3 =( ) ( )0 6 66 n 1 5 5n x x) P X 1 1 P X 0 1 p q 1 1x 0 6 6 6-= = - = = - = - = - 3 P( 3 n 6 = X:

2 . . 2 5 150 ) (( )20 x 100f x 100x 100x =>1 (1202432 (802433 (602434 (40243 : : 2 . 150 ) 150 ( :( )1501502 100100100 100 100 00 1PX 150 dxx 150 100 3x- - 1 < = = = + = ) ( ) 150 ( ) n ( 1 2n 5 , p , q3 3= = = .( )2 3x n x5n 51 2 10 8 80P X 2 p qx 2 3 3 2433- = = = = = n 5 = X:

3 . 5 . 10 3 ) 81 (1 ( 1.75 2 ( 0.75 3 ( 0.375 4 ( 0.175 : 3 . ) ( ( ) p 0.05 = ( ) n 10 = p 0.05 , q 0.95 , n 10 = = = . 3 :( ) ( ) ( ) ( ) ( ) ( )3x n xx 0nPX 3 1 PX 3 1 PX 0 PX 1 PX 2 PX 3 1 pqx-= > =- = - = - = - = - = = - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )10 1 9 2 8 3 7 010 10 10 101 0.05 0.95 0.05 0.95 0.05 0.95 0.05 0.95 0.00102850 1 2 3 =- - - - = 3 : . ) n 365 = ( ) ( :( )np 365 0.0010285 0.375n 365, p PX 3 0.0010285m = = = = = > =

: 4 . X [ ] 0,1 . 5 3 X [ ] 0.3 , 0.8 ) : 85 (1 (352 (232 3 (10324 (0.80.351dx3 2 : 3 . X ( ) 0.3, 0.8 : :1 ( X ( ) a , b : ( )1f x ; a x bb a= < n 0.05N > N nN 1-- . : : ) ( n n0.05N

N X 1m p Y 2m p ( )Z X vX Y t = = + = v t n t = k v = 1 2N m m = + . ( ) ( )1 2X Binm , p, Y Binm, p ~ ~X Y ( ) Z X vX Y t = = + = ( )1 2Z HGN m m , k v , n t = + = = ~8 . X Y n p X X Ym + = ( )PX k X Y m = + = ) 85 (1 (nk2nm 2 (nm k2nm - 3 (n nk m k2nm - 4 (km : : 3 .( ) ( )( ) ( )x n xy n ynX ~ Bn, p PX x p qxnY ~ Bn, p PY y p qy-- = = = =

:1 ( X Y : ( ) ( ) ( ) 1) PX x , Y y PX x . PY y = = = = =2 ( n 1X ,..., X in p iX inp .( ) ( )ni ii 1X~ B n, p X Y ~ B 2n , p= + ( )i.i.d1 n i2) X... X ~B n, p( )( )( )( )( )( ) ( )( )PX k, X Y m PX k, Y m k PX kPY m kPX k| X Y mPX Y m PX Y m PX Y m= + = = = - = = -= + = = = =+ = + = + = ( ) n m k k n k m k m 2n mm 2n m m 2n mn n n n n np q p q p qk m k k m k k m k2n 2n 2np q p qm m m- - - - -- - - - - = = =

N 2n , n m , k K = = = .9 . 100 4 . 10 . ) (1 (35 242 25 2 (1073 253 (32425 4 (10725 : . ( ) n 10 = ( ) N 100 = : 4 961 91) 0.29910010 = = P( : n 5N 100 : 1 9104 961) 0.2771 100 100 = = P( n 10N 100= 5100 . . (Poisson Distribution) : X ( ) X 0,1, 2, = . . X l ( ) X Pl ~ . x l( )xePxx!x 0,1, 2,...-ll== ( ) EX= l ) ( 2Xs = l

Xs = l

( )( )t1 eXM t e-l -= : ( ) l . l . 0 10 l ( ) 10 l> . : ( l ) ( l . 2 l = 20 : ( ) l 21 60 = 2 20 260 3 l = =?20 5 : ( ) l 21 2 5101 l = =?5 (( )xePxx!x 0,1, 2,...-ll == x . ( ) Px 1 =) 1 ( ( )xx 0 x 0ePx 1x!-l = =l= = :( ) ( ) ( )0 1 2PX0 PX1 PX2e e e10! 1! 2!-l -l -l= = =l l l+ + + =

( )0ePX 0 e0!-l-ll= = = = ( )1ePX 1 e1!-l-ll= = = = l 1 ( ) ( ) ( ) ( ) PX 1 PX 0 PX 1 1e-l= = = + = = l + 1 ( ) ( ) PX 1 1 PX 0 1 e-l= = - = = - 1 ( ) ( ) ( ) ( ) PX 1 1 PX 0 PX 1 1 1e-l= > = - = - = = -l + 1

: n p ( ) n ( ) p :I) n 20 , p 0.05II) n 100 , np 10 . np l = . n p np l = .( )( )( )( )I : n 20, p 0.05X Bin n, p X P npII : n 100 , np10 l = ~ ~10 . 120 . 9 7 ) : 83 (1 (e 3e-2 (ee 3 +3 (22ee 3 +4 (22e 3e- : 4 . 9 7 ) 10 ( 10 120 m = l= 10 . ( ) l 12010 600 = 120 102600 l = =?10 2 l = 10 :( ) ( ) ( ) ( )x 2 0 2 1 2 122x 0e e 2 e 2 e 3PX 2 1 PX 2 1 PX 0 PX 1 1 1 1 3ex! 0! 1!e-l - --=l - = - < =- = - = = - =- - =- = , 2 l = 10 X:

11 . 6 . 2 ) (. ) 88 (1 (1e-2 (12e-3 (2e-4 (22e-

: : 4 . : ( )xePX x ; x 0,1, 2,...x!-ll= = =6 l = 3 1 : ( ) l 63 6 123 l = =?1 2 l = 1 :( ) ( )x 2 2x 2 22e e 2Px PX 2 2ex! 2!-l -= -l=l= = = =, 2 l = X: (Exponential Distribution) : l : X ( ) x 0 1l. X l ( ) X El ~ .X ( )xf x e ; 0 x-l= l < ( )1EX= m =l ) ( 2X21s =l

X1s =l ( )XM ttl=l-

: 1 ( ) ( )1 1 n nX Exp , , X Exp l l ~ . ~{ }1 2 nY min X, X, , X = . ( )1 n...Y Expl + + l ~ (Gamma) : r r ( ) r , l . X , r l ( ) X Gr, l ~ ( ) X r , G l ~ .: X r ( )( )( ) ( )rr 1 xf x x e ; 0 x , r 1rr r 1 !- -ll= < = > = > = -j s s s s ( )a X b a b b aPa X b P P Z-m -m -m -m -m -m -m < < = < < = < < = j - j s s s s s s s a z - < < + :( ) ( )( ) ( )PZ PZPZ PZ> a = < -a< a = > -a A B z -< < + :( ) ( )( ) ( )PZ A PZ BA BPZ A PZ B> = < = -< = > ( ) ( )( ) ( )PZ A PZ BA BPZ A PZ B> = > =< = < . : q :1 - 2 - 1 - (Method of Moment Estimation) : ( : 1 ( ) EX :( ) ( )EX xf xdx = 2 ( ) EX ( )2EX :( ) ( )( )( )2 2EX , EX xf xdx xf xdx = = : n ( ) EX ( )nEX . ( : 1 X :iXXn= 2 X 2X :2ii 2XXX , Xn n= = n X nX . ( ) ( ) ( : 1 :( ) EX X = 2 :( )( )2 2EX X, EX X = = n n :( )( )n nEX X, ... , EX X = = ! X 2X .2 - (MLE: Maximum Likelihood Estimation) ( )xf x; q X q q : ( ( ) Lq n ( )n 2 1X,..., X, X :( ) ( ) ( ) ( ) ( )n1 2 n 1 n ii 1L f x, x ,..., x ; f x; ... f x ; x;=q = q = q q = q 1 : ) ( ( ) Ln . ( ( ) Lq q . : 2 : ) ( : ( ) ( )ln L 0q = q =( ) L 0 q = : :( )( )n ni 1 2 n 1 2 n ii 1 i 1u1) ln uu2) ln X ln x x x ln x ln x ln x ln xx3) ln ln x ln yy= = = = = + + + = = - . . : x q ) ( : ( x q : ( )1 2 nminX, X,..., X = q ( x q : ( )1 2 nmaxX, X,..., X = q : X m 2s MLE :( )n2i2 2 2 i 1 XX X X Xn=m =-s = = - : ( ) m X m .( )n2i2 2 2 i 1X Xn=-ms = = -m

1 . ( )1f x , 0 x = < < qq. n q ) 86 (1 ( X 2 (X23 ( 2X 4 (2X : : 3 . : ( ) EX X =( ) ( )2001 1 1EX x.f x dx x. dx x X 2X2 2 2q+ q-q q = = = = = q= q q b = q a 0 = :( )a b 0EX2 2 2+ + q q= = =2 . 0.3, 0.7, 0.2, 0.4, 0.9 :x 1 q ( )1f x1q=-q ( ) MLE q ) (1 ( 0.9 2 ( 0.4 3 ( 5.0 4 ( 0.2

: . x q :( ) ( ) ( )1 2 nMLE min X, X,..., X min0.3, 0.7, 0.2, 0.4, 0.9 0.2 q = = =3 . X :( )x 1 2 3 41 11 1 1 2 1 2f x8 84 4 4 4q- q ( ( ) Lq :( ) ( ) ( )ii iX 1 1nX Xn1 nni 11 1L f X; ... f X; e e e- - --q q q=q = q q = = = qqq Ln :( )iXLn L n ln q = - q -q ( ( ) ( )ln L 0q = :( ) ( )i i 02 2iiX Xn nln L 0 0X X n X 5nq>-q = + = = q qq q= q q = q = = : X 4 X 5 = :X 5 q = = :( )( ), 5, 5 qq =

5 . q 2s 5 2,1, 0, 1, 2 - - . ( )2, s q ) : 87 (1 ( ( ) 2, 0 2 ( ( ) 2,1 3 ( ( ) 2, 0 - 4 ( ( ) 2,1 - : 1 . : m 2s :( )2i2X X X,n-m = s = m = q 2s :iX2 1 0 1 2X 0n 5- - + + +q = = = = ( ) ( ) ( ) ( ) ( ) ( )22 2 2 2 2i2X X2 0 1 0 0 0 1 0 2 010 2n 5 5-- - +- - + - + - + -s = = = = :( )( )2 , 2, 0 s q = ) 1 ( .6 . ( ) U0, q 4 3 15 14 14 . q :) (1 ( 3 2 ( 10 3 ( 14 4 ( 15 : 4 .( ) ( )1X ~ U0, f x ; 0 x q = < < qq : X q :( )( )n1x Xx X< q q => q q = X q :15 = X ( ) n: X q = = 0 x < < q q : (Unbias) q q q q . : q ) ( q ( )E = : ) = E( ( )Eq = qq q (Bias) ) q ( E q ( )E ) q > q ( )(Eq < q q :( ) ( ) E E = q -q = q-q ) (7 .n 2 1X ,..., X , X n X ) ( ) : 84 (1 (( )n2kk 1X X=-2 (( )n2kk 11X Xn=- 3 (( )n2kk 11X Xn 1=-+4 (( )n2kk 11X Xn 1=-- : 4 . : :1 - m ( )( )2i2 2 21 1X XS ESn 1-= = s- 2 - m ( )( )2i2 2 22 2XS ESn- m= = s 22S 2s m 21S 2s .8 . X m . 2m :) 85 (1 (X22 (2X X + 3 (2X24 (2X X - : : 4 . :1 - ( ) l . ( ) ( ) EX Var X = = l2 - q q :( )Eq = q X m :( ) ( )( )( )( )( )( ) ( ) ( )2 2 2 2 2EX Var X mVar X EX EX EX Var X EX m m = == - = + = + 2m 2) m = E( 2m : ( )2X 1 mE E X m2 2 2 = = ) 1( ) ( )( )2 2 2 2 2E X X EX EX m m m 2m m m + = + = + + = + ) 2( ) ( )22 2 2X 1 1E EX m m m2 2 2 = = + ) 3P ( ) ( )( )2 2 2 2E X X EX EX m m m m - = - = + - = ) 4

9 . n 1X ,..., X :( )( )x231f x .xe3.-q=G q q ) (1 ( 3X 2 (13X3 (X34 (3X : 3 . : T q ( ) ET = q ( ) EX 1EX . : 1l =q3 a = : ( )3EX 31a= = = qlq

: ( ) ( )EX EX = :( )EX 3 = q ( ) EX :( ) ( )( ) ( )x x2 33 30 01 1EX xf xdx x xe dx xe dx3 3 - -q q= = =G q G q ( )x x x x3 2 2 3 431x e 3x e 6x e 6 e3 1 !0- - - -q q q q -q - q + -q - q - q X3q . ( ) ( )431 X.6 3 EX EX 3 E32 = q = q = = q = q q 1EX :( )( ) ( )x x23 30 01 1 1 1 1E f xdx xe dx xe dxX x x3 3 1 ! - -q q = = = G q - q x x2 23 31 1 1 1 1 1 2 1x e 1 e E E E2 X 2 X X2 20- -q q -q - q = q = = = = q q q q q 2X 1q .

10 . 1, 2, 3, 4, 5 5 ( ) N , q q . q ) (1 ( 2 2 ( 2.5 3 ( 3 4 ( 2.5 , 3 : 4 . : q q ( )Eq = q . :( ) ( )2 2ES , EX = s = m : ( ) m ( )2s iXXn= ( )n2ii 12X XSn 1=-=- 2s = q m = q q :( )( )2 21) EX2) ES= m = q= s = q iX:1, 2, 3, 4, 5 :( )( ) ( ) ( ) ( ) ( )in2i2 2 2 2 2i 12x 1 2 3 4 5 151) x 3n 5 5x x1 3 2 3 3 3 4 3 5 3102) S 2.5n 1 5 1 4= + + + += = = =-- + - + - + - + -= = = =- -

) ( m 2s n X ) ( m ) ( : X e e . :1 - 2s n 1 X XX2 2 2X Z X Z , X Zn na a a s s s - + 2 - 2s ( n 30 >X XX2 2 2S SX Z S X Z , X Zn na a a - + ( n 30 ( ) ( )X XX n 1 , n 12 2 2S SX t S X t , X t ,n na a a - - - + : 3 - 2s ( n 30 > ) (X XX2 2 2X Z X Z , X Zn na a a s s s - + ( n 30 ) (X XX1 1 1X X , Xn n s s s - + a a a :2: e Znas= 2Za 2ta 1a s S . :/ 2 / 2X Z , X Zn na as s = - + 2 2 2X Z X Z 2Z 2en n na a a s s s = + - - = = - = 22e 2 Znas= = 2Za 2t a s S . e :2 222 2 2222Ze Z e Z nn n eaaass s= = = 2Za 2ta s S . : n x xpn= . p :p pZp qn-= = :2 2 2p q p q p qp Z p Z , p Zn n nxpna a a - + =

! p p p q 12 . :2p qe Zna= :2 2p q p q: p Z , p Zn na a - + 2 2 2p q p q p q: p Z p Z 2Z 2en n na a a + - - = = 2p q2e 2Zna= = e :22 2 2222Z p qp q p qe Z e Z nn neaaa= = = : 11 . . %95 %5 ( )0.975Z 1.96 = ) 86 (1 ( 380 2 ( 385 3 ( 290 4 ( 400 : 4 .2 pqe Zna= p2e Znas= m

12 .( )20.025 0.9751 1pq 12 2e Z 0.05 2 n 20 n 400n n 0.051 p q , e 0.0521 0.95 0.05 , 0.025, Z Z 1.96 22a = = = = == = = a-a = a = = = ==

0.025Z 2 . . : (Null Hypothesis & Alternative Hypothesis) ( )0H : = ( )0H .( )1H : 0H 0H < > ( )1H . (Statistical Errors) 0H : ( ) a 0H 0H . :0H ) | 0) P(H = P( a = 0H ) ( 0H ( ) C 0H :0H) | 0H) P( = | 0P(H a = :) 0HP ( a =12 . ( ) ( )1 x x1PX x 1 ; x 0,1x- = = q -q = 5 . 01H :2q = 11H:2q . 0H 1 4 ) 88 (1 (382 (483 (584 (316 : : 1 . 5 p = q n 5 = .1n 5 ,2= q =( )x 5 x 55 51 1 1PX xx x 2 2 2- = = = ( ) ( )0 01 1H: H:2 2) P X 1 P X 4q= q== + 00 HH) P ( = | 0P(H a =( ) ( ) ( ) ( )1 1 1 12 2 2 2P X 0 P X 1 P X 4 P X 5q= q= q= q== = + = + = + = =( )5 5 5 25 5 5 51 1 1 1 1 12 31 5 5 10 1 4 5 2 2 2 2 32 32 8 + + + = + + + = = 13 . 10 1X ,..., X q 01H :2q = 12H :3q = . 10ii 1 X 8= ) 86 (1 ( 0.01 2 ( 0.05 3 ( 0.1 4 ( 0.15 : 2 . :1 ( ( ) a ) 00 HH) P (X = | 0P(H a =2 ( X q nii 1Y X== n q . : ( )10ii 1Y X Binn 10,== = q~( ) ( ) ( ) ( )0101 iH:i 12x10x 8x n x 10 10 10P X 810xPY 8 PY 8 PY 9 PY 1010 10 101 1 1 1 1 560.058 9 10 2 2 2 2 2 1024q===-a = = = = + = + = = = + + = = nii 1Y X~= = X ~

: ( ) b ) ( 0H 0H :0H ) | 0) P(H = P( b = 0H ) ( 0H )1H ( ( ) C 1H :1H) | 0H) P( = | 0P(H b = :) 1HP ( b =14 . . 01H :2q = 11H:4q = 0.2 a = ) ( . ) ( ) 85 (x 0 1 2 3 41 1 2 2 3 2P X x ;2 10 10 10 10 101 5 5 5 40 45P X x ;4 100 100 100 100 100 = q = = q =

1 ( {} 4 = 2 ( { } 3 = 3 ( {} 2 = 4 ( { } 1 = : 1 .1| )2q = 2) P(10 = 0H: P ( a = 1| )4q = ) P( b = 1H: P ( b = X 0.2 a = b :{ }1: X 1 P X 1 0.22 = a = = q = = 1 5 5 40 45P X 0, 2, 3, 4 | 0.954 100 100 100 100 b = = q = = + + + = : { }1: X 2 P X 2 0.22 = a = = q = = 1 5 5 40 45P X 0,1, 3, 4 | 0.954 100 100 100 100 b = = q = = + + + = { }1: X 4 P X 4 0.22 = a = = q = = 1 5 5 5 40P X 0,1, 2, 3| 0.554 100 100 100 100 b = = q = = + + + = P X 4 = 0.2 a = b . (Power Of a Test) ) a b ( . :0H ) | *0: 1 1 P(H b = -b = -

0H) | 0P(H =

: 1 - a ( ) b .2 - a b .0H) | 1 1H) P(H = | 1 1H) P(H = | *01 P(H b = -b = ( )*b 1-b . : ( ) b :*1 b = -b : : ( )*b :1H) | 1H) P( = | *0P(H b = :) 1*HP ( b = (Critical Rigion) : ( )0q 1q :(0 01 1H :H:q = q q = q ) k 0 k < :(( )( )10f xkf xqq> a :( )( )10f xP kf xqq a = > ( )0q :(0 01 0H :H:q q q < q :( )( )( )121 n2 11 nf x, xk ;f x, xqq> q > q.. T : T ) ( : : T k > T ) ( : : T k < 15 . X ( )xf x e ; x 0 , 0-q= q > q > . n 0 0H : q q 1 0H: q > q :) 85 (1 (iX K 2 (iX K 3 (( )2iX X K - 4 (( )2iX X K - : : 2 . :( )( )( )2 ii 1 22 i2 2 1 01 i1 1 inXX n 2X n1 ni 1 2 2n X n1 n 1 X11i 1ef X, Xeef X, Xeel=-ll -l=l l l= = = ll l.. 2 1l > l iT X =) ( iX k < .16 . ( )xf x e ; x 0 , 0-q= q > q > . 1X 2X . 0H : 1 q = 1H: 2 q = ) 88 (1 ( ( ){ } 1 2 1 2X, X X X K = + > 2 ( ( ){ } 1 2 1 2X, X X X K = + < 3 ( ( ){ } 1 2 1 2X, X X X K = + < 4 ( ( ){ } 1 2 1 2X, X X X K = + > : 2 . 01H : 1H: 2 q = q = :( )( )10f xKf xqq> 2 iX 1 2X X + . X q 1 2X X + ( ) 2, q . : n 1X,..., X l nii 1 X= ( ) n, l .( )( )n n 1 x1f x x e x 0 , 0n- -l= l > l >G 1 2X X T + = ) ( ( ) 2, q :( )( )221 t 2 tT1f t t e te t 0 , 02- -q -q= q = q > q >G ( )( )( )( )102 2t2 t t2 1t1Kf Tf T2 te KK k K 4e K ef T f T 41te-qq= - --q q=> > > > > 1 2K Kt ln K t K X X K - > < - + < : 17 . X ( )( )( )22xf x ; x 11q-q= q 3 ( 0.02 X 0.98 < < 4 ( X 0.98 > X 0.02 < : 2 . ) ( :( )( )102232x3 4; x 143 3 31 9x ; x 1f x4 4 4k k k X k1 1 1 f x2x x ; x 11 4 4 4; x 14114qq - < - - < > > > > - - < < -

1 12c c12 x1 1 40.05 P X c | 0.05 f x | dx 0.05 dx4 4114 - = > q= = q = = - 1 2 12cc32 1 1 x 1 1 c c0.05 x dx 0.45 32 x 0.45 329 4 2 4 2 4 2 4 = - = - = - - - 22 21 c c0.45 32 0.45 8 16c 8c 16c 8c 7.55 04 2 4 = - + = - + - - = c 0.98 =c 0.49 = - c 14 X 0.98 > .18 . ( )xf X e ; 0, X 0-q= q q > > H: q q 1H: q > q

. n ) : 86 (1 ( X c < 2 ( X c >3 (1 2c X c < < 4 (2X c > 1X c < : 1 . : : ( )( )( )2 ii 1 22 i2 2 1 01 i1 1 inXX n 2X n1 ni 1 2 2n X n1 n 1 X11i 1ef X, Xeef X, Xeel=-ll -l=l l l= = = ll l.. 2 1l > l iT X =) ( iX k m 0m .2 - ) ( n 2 1X, ... , X, X n X Z t . ) ( :1 ( 2s X n )n ( :0XXXZ ;n- m s= s =s 2 ( 2s X n : ( n 30 > X :0XXX SZ ; SS n- m= = : ( n 30 X t n 1 - :0n 1 XXXSt ; SS n-- m= =3 ( ) ( 2s ( ) n 30 > X :0XXXZ ;n- ms= s =s 19 .n 2 1X ,..., X , X n m 2S ) : 84 (1 (X mS-2 (X mS/ n 1-+3 (X mS/ n 1--4 (X mS/ n- : 4 . .X m: ZSn-= m2SX ~ N ,n m :( )22n 2nS: c =s 2s m( )( )22n 1 2n 1S:--c =s 2s m20 .n 1X ,..., X n 36 = l . 0H : 3 l = 1H: 4 l = %5 a = iX 144 = ): 87 (1 ( 3 2 ( 12 3 ( 2 4 ( 10 : : 2 . : n 1X ,..., X n 30 > iX nm2ns . n 1X ,..., X n 36 30 = > :( )( ) ( )2i i iX~ Nn , n X ~ Nn , n X~ N36 3 , 36 3 m s l l l .2m = s = li22iX n144 36 3 36 6Z 2 3 1236 3 6 3 3nX 144, n 36 , 3 - m- = = = = = = s= = m = s = l = : 0H 0H 0H 3 l = ) . 1H 4 l = (.21 . 1, 2, 3, 4, 5 ( )2N , m s . 0H : 2.5 m = 1H: 2.5 m ) 89 (1 (10102 ( 10 3 (224 ( 2 : 3 . ( )2s n 30 0 0H : m = m t :( ) ( ) ( ) ( ) ( ) ( )0i2 2 2 2 2 2i21x 3 2.5 1 22tS2 10 2 2n2 25x1 2 3 4 5X 3n 5x x 1 3 2 3 3 3 4 3 5 310 10S Sn 1 5 1 4 2 -m -= = = = =+ + + + += = =- - + - + - + - + -= = = =- -