996,,,,,,, 997 7, 40 ; 998 4,,, 6, 8, 3, 5, ( ),, 3,,, ;, ;,,,,,,,,,

,, AB,, ( CIP) /, 000 ( /, ) ISBN 704009448 F47 CIP ( 000) 86786 55 00009 0064054588 ht tp www hep edu cn ht tp www hep com cn 006404048 787960/ 6 05 370 000 730,,

996,,,,,,, 997 7, 40 ; 998 4,,, 6, 8, 3, 5, ( ),, 3,,, ;, ;,,,,,,,,,

3,,,,, 4,,, A, B, A, B ( ), 999, ( ),,,,,, 000 3

8 3 4 7 5 4 ( A) 9 ( B) 3 33 33 4 3 50 4 56 5 6 ( A) 66 ( B) 69 70 3 70 3 78 33 86 34 94 35 03 ( A) 09 ( B) 3 5 4 5 4 9 43 44 3 ( A) 36 ( B) 38 39

5 39 5 43 53 48 54 50 55 56 ( A) 63 ( B) 65 66 6 66 6 7 63 78 64 90 65 94 ( A) 05 ( B) 07 0 7 0 7 73 8 ( A) 8 ( B) 9 30 8 30 8 37 83 45 84 49 85 54 ( A) 60 ( B) 6 63 9 63 9 7 77 79 95 95 98 3 300

3 4F 30 5t 3 33

,,,,,, h ( ), h g, g = 98 ( / ),,,,,,,???,,,,,

,,,, ; ;, ( ) ; ( ), ; ( 3),,,,,,, ( ) 05,, 05, ( ) ( r n) De Mor gan 048 06 058 Buffon 4 040 048 05069 Pearson K 000 6 09 0506 Pearson K 4 000 0 05005,,,,,,,,, = {, } =, =, = {, },, 6,, 6 = {,,, 6 },,, 6,, = {,,, 6} 3, r n

3, = 0,,,, = = {0,,, } {0,,, } 4, (! ) t, 0t<, = {t 0 t < + }, = {t 0 t < } = [ 0, + ),,,,,,, 7,,,,, 7,,,,,,, A, B, 5, 6 A 5 B 5 C A, B, C, A,,,, 6 ( 6 ), 6,,, B C, C 4,,,

4,,,, ( ),,,, A, B,,, A A 6 5, = {,, 3, 4, 5, 6}, B C B = {,, 3, 4} C = {, 4},,,,,,,,,,, A B, A B, B A, A B, A B B A A B, A B A, A A B, B A, A B ( ), A = B,, A = B A B, 3 ( ) A B A B ( ),

5 A B, A + B, A B A B, 7, A B= {,, 3, 4, 5} 4 ( ) A = B = 5 A B A B ( ), AB ( AB), AB A B, 7, 5 A B A B = {, 3} A B A B, A- B 7, A B 6 A - B = {5} A B,, AB, AB=, A B,, 3 4 7 A, A, A,, A = - A, A = = A,, A, A, A, A, A A, A A =, A A =, A, A 8 A, A,, A, A, A,, A A, A,, A, A A A, A = A, A,, A A, A,, A, A A A, A =

6 A, A,, A,, A, A,, A,, A, A,, A,,, A, A, = A,, A, A, A,, A,, A = 9 A, A,, A,, ( ) A A j =, j,, j =,, ( ) A = A, A,, A,, A A ) 0 ( ) ( 8, A, B, C, D, P, F ( ) A B ( [ 90, 00] ), D ( [ 60, 70) ), ( [ 80, 90) ), P ( [ 60, 00] ), C ( [ 70, 80) ), F ( [ 0, 60) ), A, B, C, D, F ; P F, P = F ; ABCD P, P = A BCD 9, A, B, C, ( ) A ; ( ) AB ; ( 3) ABC ; ( 4) AB C + A BC + A B C; ( 5) A + B+ C; ( 6) A + B + C ; ( 7) ABC + AB C+ A BC; ( 8) AB + AC+ BC; ( 9) A B C ; ( 0) AB C + A BC + A B C+ A B C ; ( ) ABC;, ( 9 ( 6) ( ) ),

7,, 9,,, ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( ) ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( )

8 3 ( ) A ( BC) = ( AB) ( AC) ( ) ( ) A ( BC) = ( AB) ( AC) ( ) 4 ( A = ) = A( ) 5 ( ) A B= A B ( ) ( ) A B= A B ( ),,,,,,, A?,,,,,?, A, A,,,,,,,,,,,, A ( ), A, P ( A), A, A

9,,,,,,,,, A, r ( A ), f ( A) r ( A) f ( A ) = F, F A F, f ( A ) F A,, f ( A) ( ) f ( ) = ; ( ) A, f ( A ) 0; ( 3) A, A,, A,, f = A = = f ( A ),, f ( A),, f ( ) = 0, f ( A),, A P ( A), A, P ( A), F,,, ( ) P ( ) = ; ( ) A, P ( A ) 0; ( 3) A, A,, A,, P = A = = P ( A ),,,,,

0,,,,,, 933,,,, A A, F P ( A) F, A P ( A ) P ( A), A P ( ), ( ),, P ( ), F,, P ( ) P ( ) = ; A, P ( A ) 0; 3 A, A,, A,, P = A = = P ( A ), A, P ( A ) A, P ( ), (, P ),,,,, P ( ) = 0 F,, (, F, P ),,, F

, A i=, =,,,, A = = 3 P ( ) = P = A = P ( ), P ( ) = 0 = P ( A ) = P ( ) = ( ) A, A,, A, P = A = = P ( A ) A =, = +, +,, 3 3P ( A ) = - P ( A ) 4P ( A- B) = P ( A) - P ( A B) A B, ( ) P ( A - B) = P ( A) - P ( B) ( ) P ( A ) P ( B) 50P ( A) 6P ( AB) = P ( A) + P ( B) - P ( AB) 6,, P ( A BC) = P ( A ) + P ( B) + P ( C) - P ( AB) - P ( AC) - P ( BC) + P ( ABC) 0 5, A, P ( A ) = P ( A 0), =,, 3, 4, 5 ( ) 5 ; ( ) ; ( 3) A 0, A,, A 5 A =, = 0 5 = P ( ) = P = 0 A = 5 = 0 P ( A ) = 5 5 P ( A 0) + P ( A 0) = 6P ( A 0), = P ( A 0) = 6, P ( A ) =, ( =,, 3, 4, 5) 6 ( ), ( ), ( 3) A, B, C, ( ) P ( A) = P ( A 0) = 6,

5 5 ( ) P ( B) = P ( A ) = - P ( A 0) = = 6, ( 3) P ( C) = P ( 3 A ) = P ( A 0) + P ( A ) + P ( A ) + P ( A 3) = 7 = 0 6 P ( A ) = 05, P ( A B) = 0, P ( B) = 04, ( ) P ( AB) ; ( ) P ( A- B) ; ( 3) P ( A B) ; ( 4) P ( A B ) ( ) AB+ A B= B, AB AB, P ( AB) + P ( A B) = P ( B) P ( AB) = P ( B) - P ( AB) = 04-0 = 0 ( ) P ( A ) = - P ( A ) = - 05= 05, P ( A - B) = P ( A) - P ( AB) = 05-0= 03, ( 3) P ( A B) = P ( A) + P ( B) - P ( AB) = 05+ 04-0= 07, ( 4) P ( A B ) = P ( AB) = - P ( AB) = - 07= 03 3,,,,,,,,, ( ) ; ( ) ( ), = {,, }; ( ), P { }= P { }= = P { } = P ( ) = P ( ), = { } = = P { } =, ( =,,, ) P { },

3 3,, A, P ( A) A, A m,,,, m, A = {,,, m } = m 3 P ( A) = { j = j } m P { j } = m j =, P ( A ) = A A A =, 0, 3, 7, ( ), ; ( ), ( ) 0, C 0= 0, 0 3, C 3= 3,, A P ( A) = C 3 C 0 = 3 0 ( ) 0 C 0,, C 3C 7, C 3, B, C P ( B) = C 3C 7 C 0 P ( C) = C 3 C 0 = = 45 = 7 5, 3 45 = 5 3,, 3, 4, ( ),, 3, 4 ; ( ) ; ( 3) ; ( 4) ( ) 4 4! = 4, 4

4 ( ), ( ), ( 3), ( 4) ABCD ( ) A, P ( A) = 4 = ( ) B ( 3! ) =, P ( B) = 4 = ( 3),, 3,,!, C 3=, P ( C) = 4 = ( 4),,,, P ( D ) = 4 N ( N ),, ( ) ; ( ) ; ( 3) ; ( 4) k( k) N, N, ( ), ( ), ( 3), ( 4) A, B, C, D ( ),,!, P ( A) =! N ( ),, C N (! ) ( P N ), P ( B) = P N N ( 3) C C, N -, ( N - ), P ( C ) = ( N - ) N, P ( C) = - P ( C ) = N - ( N - ) N ( 4) k ( C k ), - k

3 5 N - ( ( N - ) - k ), k C k ( N - ) - k, P ( D) = Ck ( N - ) - k N 5 a + b, a, b, ( ), ( ) ; ( ) ; ( 3),, ( a + b) ( a + b)!, ( ), ( ), ( 3) A, B, C ( ) a,, a + b-, ( a + b- )!, A a [ ( a + b- )! ], P ( A) = a [ ( a + b - )! ] ( a + b)! = a a + b ( ) a, - b - ( P - b ) a + b- ( ( a + b- )! ), B a P - b [ ( a + b- )! ], P ( B) = a P - b [ ( a + b - )! ] ( a + b)! ( 3) C C P b [ ( a + b- )! ] P ( C ) = P b [ ( a + b - )! ] ( a + b)! = a P - b P a + b C,, P ( C) = - P ( C ) = - = P b P a + b = C b C a + b C b C a + b 5 ( a + b) ( a + b) ( a + b),, 5, ( ), a a + b,, a, a + b,,,

6, ( ),,, ( 3) ( 3),,,,,,,,,,, t= S ( ) ( ) A A S ( A ), A A, P ( A ) = ts ( A) t P ( ) = ts( ) = S ( ) P ( A) = S( A ) S( ) ( ) ( ),, ( ) ;, 6,,,,, ( ) 0, 0, 60, ( 0, 60), 0 A, = ( 0, 60), A = ( 50, 60),

4 7 P ( A) = 0 60 = 6 7 ( ) 7 8, 0,, 7 0,, x, y, = {( x, y ) 0 x 60, 0 y 60} A, A = {( x, y ) ( x, y ), x - ( 3) y 0},, P ( A) = S( A ) S( ) = 60-40 = 5 60 9 3 4,, A, B, 8, 475 644 9 5 56 8 500 700 00, 8 00 = 683%,?, 500, 500 5

8,,, 5 500 = 5% A, B P ( B), A, B, A, B, P ( B A), P ( B A ) = 5 5/ 00 = 500 500/ 00 = P ( A B) P ( A),,, P ( A) > 0, P ( B A) = P ( AB) P ( A) ( ), ( 4), A, B P ( B A) = S ( AB) S ( A) 4 P ( B A) = P ( AB) P ( A) = S ( AB) / S ( ) S ( A) / S ( ) = P ( AB) P ( A),,, 3 (, P ), A, B, P ( A) > 0, P ( B A) = P ( AB) ( ) P ( A ) A, B B F, P ( B A) F, A, A, P ( A) > 0, P ( A) ( ) P ( A) =

4 9 ( ) B, P ( B A) 0 ( 3) A, A,, A,, P ( = A A) = = P ( A A ), (, P ) A, P ( A) > 0, P ( A ), A=, P ( A ) P ( ), P ( A),, ( ) 9 0, 3, 7, ( ) ( ), ; ( ), A ( =, ) ( ) A,, 7 9,, 9, P ( A A ) = 9 ( ) A,,, ( ) ( ) P ( A A ) ( ) P ( A A ) = P 3 = P 0 5, P ( A ) = 3 0 P ( A A ) = P ( A A ) P ( A ) = 9 9,,,,, a b? 9, 0,, 7, 9

0, ( ) ( ), P ( B) > 0, P ( A B) = P ( A) P ( B A) ( P ( A ) > 0) ( 3) P ( A B) = P ( AB) P ( B) ( 4) P ( AB) = P ( B) P ( A B) ( P ( B) > 0) ( 5) ( 3) ( 5), P ( B A ) P ( A B) P ( A B) 0, 60%, 0%,, A, B,, P ( A) = 60% P ( B A) = 0% P ( A B) = P ( A) P ( B A) = 60%0% = 6% 0, 3 7, ( ),,,,,, A ( =, ), A A, P ( A ) = 3 0 P ( A A ) = 9 P ( A A ) = P ( A ) P ( A A ) = 3 0 9 = 5,, ( 3) ( 5) P ( A A A ) = P ( A ) P ( A A ) P ( A 3 A A ) P ( A A A A - ) ( 6)

4,,,, 0, 3, 7, ( ),,,, A ( = ), P ( A ) = P ( A A ) + P ( A A ) = P ( A ) P ( A A ) + P ( A ) P ( A A ) P ( A ) = 3 0, P ( A ) = 7 0, P ( A A ) = 9, P ( A A ) = 3 9, P ( A ) = 3 0 9 + 7 0 3 9 = 3 0 3,, 60%, 40%,,, 80%,, 40%, A, A, B P ( A ) = 60%, P ( A ) = 40%, P ( B A ) = 80%, P ( B A ) = 40%, P ( B) = P ( AB) + P ( A B) = P ( A ) P ( B A ) + P ( A ) P ( B A ) = 60%80% + 40%40% = 64%, B

5, A ( A A ) B AB A B, ( A A ) B, P ( B A ) P ( B A ) 5 5 ( ) {A }, A =, B, P ( B) = P ( B) = P ( B ) = P [ B ( A ) ] = P [ ( B A ) ] = = P ( A ) P ( B A ) P ( A ) P ( B A ) ( 7) P ( B A ) 3,,,,, 4 ( ), 9 ( ),,,,,,,,

4 3 A, B, P ( A B), P ( A B) = P ( AB) P ( B) P ( AB) = P ( A) P ( B A) P ( B) = P ( A) P ( B A) + P ( A ) P ( B A ) P ( A B) = P ( A) P ( B A) P ( A ) P ( B A ) + P ( A ) P ( B A ) P ( A) = 3 0, P ( B A ) = 9 P ( A ) = 7 0, P ( B A ) = 3 9 P ( A B) = 3 0 9 3 0 9 + 7 0 3 9 = 9 ( 8) ( 8),,, ( Bayes T homas) {A }, A = =, B, P ( B) > 0, P ( A B) = P ( A B) P ( B) = P ( A ) P ( B A ) P ( A ) P ( B A j ) j,, ( 9) 5, 45%, 35%, 0%, 4%, %, 5%,, ( ) ; ( ), A ; A ; A 3; B

4 P ( A ) = 45%, P ( A ) = 35%, P ( A 3) = 0%, P ( B A ) = 4%, P ( B A ) = %, P ( B A 3) = 5% ( ) P ( B) = P ( A ) P ( B A ) = 3 = 45%4% + 35%% + 0%5% = 35% ( ) ( ) P ( A B) = P ( A B) P ( B) = P ( A ) P ( B A ) P ( B) = 45%4% 35% = 54% 5,,,,,, 5, B A, P ( B) > 0, A B, P ( A) > 0, B A P ( A B) = P ( A ) ( 0) P ( B A) = P ( B) ( ) P ( A) > 0, P ( B) > 0, ( 0) ( ) P ( AB) = P ( A ) P ( B) ( ),, A B B A, A B P ( A ) = 0, P ( B) = 0 ( ),, 4 (, P ), A, B, P ( AB) = P ( A ) P ( B) A B, A B 6 ( ) A 5, B, P ( A) = 4 6, P ( B) = 3 6, P ( AB) = 6 = 3

5 5 P ( AB) = P ( A ) P ( B) = 3 A B ( ) A 4, B ( ), P ( A) = 3 6, P ( B) = 3 6, P ( AB) = 6 = 3 A B P ( AB) P ( A ) P ( B) 5 ( ) A, A,, A, < j, A, A, P ( A A j ) = P ( A ) P ( A j ) ( 3), A,,, A, A, A 3,,, P ( A A j ) = P ( A ), j,, j =,, 3,,, P ( A A A 3) P ( A ), 6 A, A,, A ( ) k ( k ) A, A,, A k ( < < < k) P ( A A A k ) = P ( A ) P ( A ) P ( A k ) ( 4) A, A,, A, =,, >,,,, 5,, k ( k)

6,, A, A,, A, m ( m ), A, A,, A, P ( A ) = - P ( A ) = - ( - P ( A ) ) ( 5) = = = m=,, 7,, 05, 06, 08, ( ) ( ) A ( =,, 3), A A A 3+ A A A 3+ A A A 3, A + A + A 3 ( ) P ( A A A 3+ A A A 3+ A A A 3) = P ( A A A 3) + P ( A A A 3) + P ( A A A 3) = P ( A ) P ( A ) P ( A 3) + P ( A ) P ( A ) P ( A 3) + P ( A ) P ( A ) P ( A 3) = 05( - 06)( - 08) + ( - 05)06( - 08) + ( - 05) ( - 06)08 = 06 ( ) P ( A + A + A 3) = - P ( A ) P ( A ) P ( A 3) = - 05040 = 096 ( Bernoulli),, 7,,,,,

5 7, A, A, A,, 0,, 000, 000, 8,,, P ( A) = p, P ( A ) = q, p > 0, q> 0, p + q=,,, A p, A 3 ( ), A p ( 0< p < ),, A k ( b ( k;, p ) ) q= - p b( k;, p ) = C k p k q - k A A, =,,,, A k ( Bk) C k A A A k A j A j A j - k,,, k,, k ( C k ), j, j, j - k,,, k - k,, k, P ( A ) = p, P ( A A A k A j A j A j - k ) = P ( A ) P ( A ) P ( A k ) P ( A j ) P ( A j - k ) = p k q - k b( k;, p ) = P ( B k) = C k p k q - k 4, A p, A k ( k) g ( k, p ) = q k- p A k k k A k- k A, 3 4

8 8 0, 3 7,, ( ) 0, 0 0 3 ( ),, 3 3 A P ( A ) = 3, =,,, 0 ( ) B 0, Bk 0 k k= 0,,, 0 P ( B) = - P ( B ) = - P ( B0) = - 3 7 3 7 P ( B3) = C 3 0 0 0 ( ) C 3, D 3 4 P ( C) = 7 0 3 0, P ( D ) = P ( A A ) = P ( A ) P ( A ) = 9 5 9, 9 ( ),, j, j A k k k=,,, 5, 5, B, C j, BC j 7 0 7 0 0, P ( B) = - 8 9 5, P ( C) = - 8 9 5 B ( B ), 8, j 8, P ( C B ) = - P ( C B ) P ( C), B C, B C 7 8 5

9 ( A) ( ), ; ( ) ( ) ; ( 3), ; ( 4) 0, 3 7,,,,,? 3,, A, B 5, C 5,, A, B, C, A+ B, A- B, B - A, AB, AC, A + B 4 9 ( 5), ( 8), ( 0), ( ) 5? ( ) x - a < x - a ; ( ) x > 0 x 0; ( 3) x > 0 x < 8; ( 4) x > 0 x ; ( 5) 0 0 ; ( 6) 0 0 ; ( 7) 0 0 6 5, A =,, 3, 4, 5 B 5 = 0,,, ( 3, 4, 5),, ( ) ( 3) ( 5) 5 A 5 = A 5 = = 3 B 5 = 0 = B ; ( ) A; ( 4) 5 = A B = ; B B ; = = 3 B = 3 ; ( 6) A A A 3A 4A 5 B 3; ( 7) A B 5; ( 8) 5 A B 5 = 7 AB= A( B - A) = ( A- B) ( B - A) ( AB) 8 3 6 6 5 5

30 9 P ( A) = 04, P ( B) = 05, P ( A- B) = 05, P ( AB), P ( AB), P ( B - A), P ( A B ) 0 P ( A) = 04, P ( BA ) = 0, P ( CA B ) = 0, P ( A+ B+ C) N N, ( ), N ( ) k ( kkn ) ; ( ) ; ( 3) N,, 30, 6, ( ) ; ( ), 3 ( 365), ( 365 ) 4 5, 9, 9,, ( ) ; ( ) j ; ( 3) j ; ( 4) 3 ; 5 4, 6 5, k 7, 0, 4,,, ( ) 3 ; ( ) 3 ; ( 3) 3 8 0, 3 9 5, 3,, 3,, 0 3 AAACE H I I MMN T T, MAT H E MAT I CI AN 0,, 3,,,, 3 a, b ( ) ( ), ; ( ), ; ( 3),

3 4 A (, P ), P ( A) > 0, ( ) BA, P ( B A) = ; ( ) B, B (, P ), B B =, P ( B + B A) = P ( B A) + P ( B A) 5, ( ), ; ( ), 6,, 4 70 4 3,,, ( ) ; ( ) ; ( 3) 8 3, 0, 7 A, 3 B; 5 ; 8,, A, ; B,,, 9 30, 0, 00, 006, 0 005 ( ),, ; ( ), 30 9, 3, 3,, 3 ( ) ; ( ) 3, 3 3 ELISA ( Enzyme Linked Imriunosorbent Assay ) EL ISA 95%, % 000, ( ),?, 00,,? 3,,,, 80%, 08? 33,,,, AB C,, ABC?? 34 P ( A) = 0, A B

3 35 P ( A) > 0, P ( B) > 0, AB, A B?? 36ABC, A, B, C 37 38, 09, 095, 08,, 39,,, 00, 0, 03,, 00 03, 40 ( ),, 0;, 06; ( ) ; ( ), ( B), 0, 0,,,, A, B, AB r?, A B, AB r 3( ), N,,, k? 4 k,,,, m 5 l, 6, p, p,,,? 7 0 5 5, 3 7 5, ( ) p ( ), q 8, AAAA, BBBB, CCCC,, 06, 0, ( ) ABCA ; ( ) ABCA, AAAA

,,,,,,,,,,,,,,,,,,, 3, 4, 5, 6, ;,, [ 0, + ),,,,,, -,,, X = X ( ),, y= f ( x ), x, y x, x X = X ( ),,

34 ( ), X,, (, P ), X = X ( ),, (, P ) X, X a, { X ( ) = a}, I R, X I X,,,, X,, {X = a }, {a< X b}, X, = {,, 3, 4, 5, 6} X X ( ) =,,,, X, X = {, } X ( ) =, =, -, =,, X (, P ), X, X x, x, X, x, {X = x },, X x 3 X, {x, =,, }, p ( x ) = P {X = x } =,, ( ) {p ( x ), =,, } X p ( x ) p, X

35 X x x x P p p p, {p ( x ) } ( ) p ( x ) 0, =,,, ( ) ( ) p ( x ) = ( 3), X, X ( ) =, =, 0, = P {X = }= P { }= P {X = 0}= P { }= X X 0 P X =, ; 0, X ( ) P {X = }= a, =,, 3; 3 ( ) P {X = }= a 3, =,, a ( ), a = 7 38 ( ) 3 3 = P {X = } = = = = P {X = } = = = a 3 a 3 = a 38 7, 3 = a - 3 = a,, a = X {p ( x ) }, X

36, P {a X b} = P ( {X = x }) = a x b P {X = x }, I, a x b = p ( x ) ( a b) ( 4) a x b 3 X ( ), P {X < }= 0, P {X }= P {X = }= 7 38 3 = 9 9, P {X < }= P {X = }= 9 9, P {X I } = p ( x ) ( 5) x I P {X < 5}= P {X = }+ P {X = }= 9 9 + 7 38 4 9 = 5 9, P {X 3}= P {X = }+ P {X = }+ P {X = 3}=, P {X 4}= P {X = }+ P {X = }+ P {X = 3}=,,,,,, ;,, 0,, ( ),,, 4 [ a, b], X ( ) X = [ a, b],, c[ a, b], B= ( c, d ] [ a, b], X ( ) =, [ a, b], ( 6) P {X = c}= P { = c}= 0 P {c< X d }= P { B }= d - c b- a ( 7)

37, P {c< X d }= P {X d }- P {X c}, ( 8) X, X ( -, x ] P {X x } F ( x ) = P {X x } x ( -, + ), F ( x ) ( -, + ), F ( x ), ( 8),, 4 X, F ( x ) = P {X x } x ( -, + ) ( 9) X, X F ( x ) 5 4 X x < a, {X x },, F ( x ) = P {X x }= 0 a x < b, {X x }= {ax x }, [ a, x ] [ a, b],, x b, F ( x ) = P {X x }= P {a X x }= x - a b- a {X x } = {a X b}, F ( x ) = P {X x }= P {a X b}= b- a b- a =, X F ( x ) = 0, x < a ; x - a, a x < b; b- a, x b 4, ( ) x x, F ( x ) F ( x ) ; ( ) F ( - ) = lim x - F ( x ) = 0; F ( + ) = lim F ( x ) = ; x + ( 3) F ( x + ) = F ( x ),,,

38, F ( x ),, X,,, 6 X, X, P {X = 0}= P {X = }=, x < 0, 0x <, F ( x ) = P {X x }= 0 F ( x ) = P {X x }= P {X = 0}= x, F ( x ) = P {X x }= P {X = 0}+ P {X = }= + =, X F ( x ) = 0, x < 0,, 0x <,, x, X, F ( x ), X 0,,,,, X F ( x ), X, F ( x ) F ( x ) 6 X, X 7 X

39 F ( x ) = 0, x <, 9, x <, 9 5, x < 3, 9, x 3 F ( x ), X, F ( x ),, 9 3, 9, 6 9, 4 9, X P {X = }= 9 9, P {X = }= 6 9, P {X = 3}= 4 9 4, [ a, b] 7 X,, X [ a, b], X [ a, b], [ a, b], [ a, b] b- a, X [ a, b], X [ a, b], X [ a, b] b- a, [ c, d ] [ a, b], P {c X d } = d c b - a dx = d - c b - a [ a, b] X, 0, f ( x ) = a x b; b- a 0, ( 0) f ( x ) X, X F ( x ) = x - f ( t) dt, 5 X, f ( x ), F ( x ) = P {X x } = x - f ( x ) X, f ( t) dt ( )

40 5, ( ) f ( x ) 0 x ( -, + ) ; ( ) + - f ( x ) dx =,,, X,, 5,,, X 3) P {a < X b} = F ( b) - F ( a ) = b a f ( x ) dx ( ), F ( x ) P {a< X b} ( 3 x, ( ), P {X = x }= 0 ( 3), ( ), f ( x ), F ( x ) = f ( x ) ( 4) 8 X [ a, b], 5, X, 5, F ( x ) = x = a x = b, f ( x ) = F ( x ) = 0, x < a, x - a, a x < b, b- a, x b 0, x < a, b- a, a < x < b, 0, x > b ( 5) F ( x ), f ( x ) =, x = a x = b ( 6) b- a

4 (, f ( a) f ( b), ), ( 0) X [ a, b], b- a [ a, b] 0, [ a, b], X [ a, b], 4 X,,,,,,,, 0 ( x ) 0 3 4 5 6 7 8 9 0 ( ) 3 3 ( f ) 0 0 0 0 3 0 3 0 0 0 0 0 0, x, 0 x 0 x = 0 = = x f = 0 = 0 0 + 0 + 0 + + 9 0 + 0 0 = 5,,,,, x, X ( ) x P {X = }= p, = 0,,,, 0,

4 0 x p = 0 6 X x ( =,, ), P {X = x }= p, =,,, = x p < ( 7), X, EX, EX = = x p ( 8) = =, EX = x p, x p, = x p <,?,,,,, =, x p,, x p = 9 5,, 3, 3 X, E X X 0,,,,, P {X = 0}= C3 = C 3 5 0, P {X = }= C 3C = 6 C 3 5 0, P {X = }= C 3C = 3 C 3 5 0 E X = 0 0 + 6 0 + 3 0 =, ( ),

43, ( ),, [ a, b],,, X, f ( x ),, f ( x ) [ a, b], x / [ a, b], f ( x ) = 0 a = x 0< x < < x + = b, X x = ( x, x + ) x + P {X x } = f ( x ) dx ( 9) x x,, P {X x }f ( x ) x ( 0) x x 0 x x p f ( x 0 ) x 0 f ( x ) x f ( x ) x X, x f ( x ) x ( ) = 0 X,,, ( ) b a x f ( x ) dx = + - x f ( x ) dx, X X,,, 7 X, f ( x ), + - X, EX, EX = + - 0 X f ( x ) = x f ( x ) dx <, x f ( x ) dx ( ) cos x, x ; 0,

44 E X f ( x ) -, 0,, E X, E X = + x f ( x ) dx = - -, E X = 0, E X x cos x dx, X f ( x ) = E X, x, 0x <, - x, x, 0, + EX = x f ( x ) dx = x dx + - 0 x ( - x ) dx = x 3 3 + x - 0 3 x 3 =,, X, g ( x ), X g ( x ) Y= g ( X ) Y, g ( x ), X Y, X, Y g ( x ), Y X,, Y X, X Y, 4,, Y= g ( X ),,, Y, X Y= g ( X ), X, g ( x ) ( ) X, P {X = x }= p =,,

45 = g ( x ) p <, Eg ( X ), Eg ( X ) = g ( x ) p ( 3) = + - ( ) X, f ( x ), g ( x ) f ( x ) d x <, Eg ( X ), Eg ( X ) = + - g ( x ) f ( x ) dx ( 4) (,, ) Y = g ( X ), Y, y j, j =,, P {Y = y j } = P ( g ( x ) = y j {X = x }) = P {X = x } g ( x ) = y j E g ( X ) = y j P {Y = y j } j = = y j j = = j = g ( x ) = y j P {X = x } g ( x ) = y j g ( x ) P {X = x } = g ( x ) P {X = x } = X 9, X X 0, E ( X - EX ) 9 ( ) P E X =, E ( X - EX ) = ( 0 - ) 0 + ( - ) 6 0 + ( - ) 3 0 = 44 0 + 004 6 0 + 064 3 0 = 036 3 X, 0 6 0 3 0

46 E X - E X f ( x ) = EX =, x, 0x <, - x, x, 0, ( ) + E X - E X = E X - = x - f ( x ) dx - = x - x dx + x - ( - x ) dx 0 = ( - x ) x dx + 0 = ( - x ) x dx + 0 ( x - ) dx ( x - ) ( - x ) dx = 3,,,,, ( ) a, Ea = a ( ),, g ( x ), g ( x ), Eg ( X ), E g ( X ), E [ g ( X ) + g ( X ) ] = E g ( X ) + Eg ( X ) ( 5) ( 3) E X, a, ) E ( X + a ) = E X + a ( 6) (,, ( ) g ( x ) a, ( ), g ( x ) + g ( x ) g ( x ) + g ( x ) g ( x ) + g ( x ) p g ( x ) p + g ( x ) E g ( X ), Eg ( X ), E [ g ( X ) + g ( X ) ] g ( x ) = g ( x ) + g ( x ), ( 3) ( 5) ( 3) ( ) g ( x ) = x, g ( x ) a, ( 3) p

47 4 E X, E X, E( X - EX ) = E X - ( E X ) ( 7) ( X - EX ) = X - X E X + ( E X ), ( 5) E ( X - E X ) = E [ X - X E X + ( E X ) ] = E X - EX E X + ( E X ) = E X - ( E X ),,,, ( ),,,,,,,, 8 X, E X, X - E X X,, E ( X - E X ), DX VarX,, E ( X - EX ) D X X X X, X - EX,, X, E ( X - EX ) = 0, X E X,, ( X - EX ) X E X, DX = E ( X - E X ) X E X, X - E X, E X - E X, ( ), ( 3) ( 4), g ( x ) = ( x - E X ), X, P {X = x }= p, =,,, DX = E ( X - E X ) = ( x - E X ) p ( 8) X, f ( x ),, DX = E ( X - E X ) = + - 4, ( 7), ( x - E X ) f ( x ) dx ( 9) DX = E X - ( E X ) ( 30), X DX, a, ( ) D a = 0; ( 3) ( ) D ( X + a ) = DX ; ( 3)

48 ( 3) D ( a X ) = a DX ( 33) 5 X, X DX = E ( X - EX ) = 036, EX = 9 ( 30) DX EX = 0 0 + 6 0 + 3 0 = 8, D X = EX - ( E X ) = 8- ( ) = 036 6 X,, E X = + = x 4 4 f ( x ) = x, 0x <, - x, x, 0, EX =, - x f ( x ) dx = 0 x 3 dx + x ( - x ) dx + 0 3 x 3 - x 4 4 = 7 6, DX = E X - ( E X ) = 7 6 - = 6 7X,, l ( C) = E ( X - C) ( 34) C= EX, l( C), DX l( C) = E ( X - C) = E [ ( X - E X ) + ( E X - C) ] = E [ ( X - E X ) + ( X - E X ) ( EX - C) + ( EX - C) ] = E ( X - EX ) + ( EX - C) E ( X - EX ) = D X, C= EX,, l( C) C= E X, DX 7 X, C X, X C X - C, E ( X - C) ( ),, C E ( X - C), 6 C= E X X, DX

49, E X k 9X, k> 0 EX k ( E X k < ), X k, E X k X k X t, s ( 0< s< t) X ( ), f ( x ), E X s x = x s f ( x ) dx s x + x s f ( x ) dx s > f ( x ) dx s x + x t f ( x ) dx s x > P { X s } + E X t < k, E X k, E ( X + C) k,, E ( X - E X ) k E X ) k 0X, k, EX k, E ( X - X k, E X - E X k X k E X X, D X = E ( X - E X ) X, EX <, X, 3 h( x ) x, X, E h( X ), > 0, P {h( X ) } E h( X ) ( 35), X f ( x ), Eh ( X ) = + - h( x ) f ( x ) dx = h ( x ) f ( x ) dx h + h( x ) f ( x ) dx h ( x ) ( x ) < h ( x ) f ( x ) dx h ( x ) h ( x ) f ( x ) dx = P {h( x ) } k ( ) X k, E X <,

50 > 0 = k E X P { X } ( 36) k ( ) X, > 0 P { X - E X } DX ( 37) 3 X 0 a, P {X = a }, { X - EX > 0}= { X - EX > = } P { X - E X > 0} = P { X - EX > = P { X - E X > = } DX = 0,,, P { X - E X > }DX / = 0 P { X - EX > 0}= 0 P { X - EX = 0}=, 3, a EX 3, X, P {X = a }= X a 3 3, X DX = 0 X a, E X = a,

3 5, X x, x, X x =, P {X = x }= p, P {X = x }= - p ( 0< p < ) ( 38) p EX = p x + ( - p ) x ( 39) D X = p ( - p ) ( x - x ) ( 40) x = 0 p, P {X = }= p, P {X = 0}= - p ( 0< p < ) ( 4) X p X p 0, X p EX = p, DX = p ( - p ) ( 4), A, P ( A) = p, P ( A ) = - p, x, x X ( ) = x, A A ( ) x, A ( A ) ( 43) p, A, A X, X, 0, X p 0- n,,, P {X = x }=, =,,,, ( 44) X {x, x,, x } E X = x = x ( 45) = DX = ( x - x ) ( 46) =,,, = {,, }, P { }=

5, =,,, X, X X,, = {,,, 6}, X ( ) =, X {,,, 6}, A p ( 0< p < ), X A, X 0,,,, k, 0k, {X = k} A k,, P {X = k}= C k p k ( - p ) - k k= 0,,,, ( 47), X ( 47), X, p, X b(, p ), b( k;, p ) = C k p k ( - p ) - k b( k;, p ), k= 0,,,, ( p + q) ( q= - p ),, =, b(, p ) p 0 8 N, N, N ( N + N = N ),,,, X,,, p = N N, X b, N N k N P {X = k}= C k N, N N - k 0k ( 48), EX = kb( k;, p ) = k= 0 k= = pc k - - p k- ( - ) - ( k - ) q k= - = pc k - p k ( - ) - k q k= 0 - = pb( k; -, p ) k= 0 k C k p k q - = p ( 49) k

3 53, b( k; -, p ), ( 0k- ) - p EX = k= 0 k b( k;, p ) = [ k( k- ) + k] b( k;, p ) k= 0 = k ( k- ) b( k;, p ) + k= k= 0 kb ( k;, p ) = ( - ) p b ( k- ; -, p ) + p k= - = ( - ) p b( k; -, p ) + p k= 0 = ( - ) p + p = p + p q ( 50) DX = E X - ( EX ) = p + p q- p = p q ( 5), A p, X A, X, 4 g ( k, p ) = q k- P {X = k}= q k- p g ( k, p ) k ( 5) p ( ), ( 5) p,, E X E X = p q - = = p = p q - = q + = p p ( 53) ( 54) DX = EX - ( EX ) = q p ( 55) 9 X, m,, P {X > m+ X > m}= P {X > } ( 56) P {X > m+ } P {X > m+ X > m}=, ( 5) P {X > m} P {X > m} = k= m + q k - p = q m j = q j - p = q m

54, P {X > m+ }= q m+, P {X > }= q + P {X > m+ X > m}= qm = q = P {X > } q m ( 56), m,, ( 56),, N, N, N ( N = N + N ),, X, X ( ) N P {X = k}= Ck N C - C N k 0k ( 57) a < b, C b a = 0 ( 57) 8,,, N, N N,,, ( 8), C k N C - C N N k C k N N k N N - k ( 58) N, N, N, N N p, N N - p, k,, lim N C k N C - C N N k = C k p k ( - p) - k ( 59), ( Poisson) E X = N N, ( 60) D ( X ) = N N N N N - N - ( 6) X

3 55 P {X = k}= k ( k! e- k0) ( 6) > 0, X, X P ( ), ( 6) k k! e ( ), K = 0 k k! e- = e e - = ( 6) EX = K = 0 k k k! e- = K = k - ( k - )! e- E X = + = K = 0 k k! e- = ( 63) D X = EX - ( EX ) = ( 64),,,,, 0 = 0, 95%, ( )? X, a, X a,, a P {X a }095 X = 0, a k= 0 0 k k! e- 0 095 4 k= 0 5 k= 0 0 k k! e- 0 0966 < 095 0 k k! e- 0 0953 > 095, 5 95% 4( ), A p ( ),, p ( > 0

56 ), k, lim b( k,, p ) = k k! e- ( 65), b(, p ), p, = p, b( k;, p ) ( p ) k e - p ( 66) k! 800, 0005 ( ), X 800, X b ( 800, 0005), = 8000005= 4, P {0 X } = P {X = k} = b( k; 800, 500) K = 0 K = 0 K = 0 4 k k! e- 4 = e - 4 + 4 + 6! 038 P {X > }= - P {0X }- 038= 0769 4,,,,, [ a, b],,, 5 8,, X, X X, f ( x ) =, a x b b- a 0, ( 67)

4 57 X [ a, b], X U[ a, b] [ a, b] [ a, b] 0, [ a, b] + - b- a f ( x ) dx = b a f ( x ) dx = ( 67) F ( x ) = 0, x < a, x - a, a x b b- a, x b, ( 68) EX = a + b, ( 69) DX = ( b- a ) ( 70),,, ( ),,,, A,, A A ( ), X, f ( x ) = x e- x 0 0, x < 0 > 0, X, X e( ) ( 7) X F ( x ) = - x e- x 0, 0, x < 0 ( 7) ( 7) E X =, ( 73)

58 DX = ( 74),, 5 X r s, P {X > r + s X > s}= P {X > r } ( 75),, X, 000, 3 000,, E X = 000, X F ( x ) = = EX = 000 - e- x 000, x 0, 0, x < 0 P {X > 000}= - P {X 000}= - F ( 000) = e - 000, 3 000 e - 3, - e - 3 X, ( x ) = ( x - ) e- ( - < x < ) ( 76),, > 0 X, X N (, ), + + - e - x dx = ( 77) ( x ) dx =, N (, ) ( ), - E X =, DX = ( 78), ( x ) 4,

4 59 ( x ), x ; x =, x = = 0, =, X N ( 0, ), X, 0( x ), 0( x ) = x e- ( 79),,,, 4,,,,,, ( ),,,, ( x ), ( ( x ) ) x ( x ) = - ( t) dt = x - e - ( t - ) dt ( 80) x,, x,, ( 0 ( x ),,,, x 0 0( x ) 0( x ), (, 0( x ) x = 0! ) 0( x ) 0( x ) x < 0 0( x ), 0( - x ) = 0( x ), x < 0, 0( x ) = 0( - x ), 0( - x ) 0( x ) 0( x ), 0( x ) x = 0, - x 0( - x ) = - 0( t) dt = + x 0( t) dt = - 0( x ) ( 8)

60 0( x ) + 0( - x ) = ( 8), 0( x ) = 05, x < 0, 0( x ) = - 0( - x ), 0( - x ), 0( x ) 3 X N ( 0, ), ( ) P {X 96}, P {X - 96}, P { X 96 }, P {- < X } ( ) P {X a }= 0709, P { x < b}= 094, P {X c}= 098, a, b, c ( ) 0( x ), P {X 96}= 0( 96) = 0975, P {X - 96}= 0( - 96) = - 0( 96) = - 0975= 005, P { X 96}= P {- 96X 96}= 0( 96) - 0( - 96) = 0( 96) - = 0975- = 095, P {- < X }= 0( ) - 0( - ) = 0( ) - [ - 0( ) ] = 0( ) + 0( ) - = 09775+ 0843- = 08855, ( ) a = 053, b78 P { X < b}= 0( b) - = 094, 0( b) = P {X c}= 098< 05, - c= 053, c= - 053 ( + 094) = 096 c< 0,, 0( - c) = - 0( c) = 0709, 3 6 X N (, ), Y= a x + b, ab, a 0, Y N ( a + b, a ) Y F Y ( x ), f Y ( x ), X ( x ), ( x ), a > 0, F Y( x ) = P {Yx }= P {a X + bx } ( 83) F Y( x ) = P X x - b a f Y ( x ) = F Y ( x ) = a x - b a = x - b a = a x - b a ( 84) ( 85)

4 6 a < 0,, f Y ( x ) = F Y ( x ) = P f Y ( x ) = F Y ( x ) = - a x - b a = a YN ( a + b, a ) X x - b a = [ x - ( a + b) ] e- a x - b a a e- = - x - b a = - x- b - a X N (, ), = X - N ( 0, ) X a x - b a ( 86) ( 87) a ( 88) X N (, ) N ( 0, ), X = + 3 X N (, ), 0( x ), ( x ), 0( x ), 0( x ), x - ( x ) = 0 ( x ) = x - 0 ( 89) ( 90) 3, 6 ( 84) ( 85) 4, 6,,,,, 4 X N ( 8, 05 ), ( ) ( 9), ( 7) ; ( ) P {75X 0}; ( 3) P { X - 8 }; ( 4) P { X - 9 < 05} ( ) ( 9) = P {X 9}= P X - 8 05 9-8 05 = P X - 8 05 = 0( ) = 09775; ( 7) = P {X 7}= P X - 8 05 7-8 = P X - 8 05 05 - = 0( - ) = - 0( ) = 0075;

6 ( ) P {75X 0}= P 75-8 05 X - 8 05 0-8 = P - X - 8 05 05 4 = 0( 4) - 0( - ) = 0( 4) + 0( ) - = 09 4 6833+ 0843-0843; X - 8 ( 3) P { X - 8 }= P < = 0( ) - = 09775-05 = 09545; ( 4) P { X - 9 < 05}= P {85< X < 95}= P < X - 08 < 3 05 = 0( 3) - 0( ) = 09 8650-08430573 5 X N (, ), 50 350 936%, - x + x 09, x? P {X > 50}= P {X < 350}, x =, = 350+ 50 = 300, P {X < 350}= P X - 300 350-300 < = 50 0 = 0936 50 43, 35 X N ( 300, 35 ), X - P { - x < X < + x }= P < x = x 0-09, 0 x 9 x = 095, 645, x 64535= 5758 5,, N (, ) X, Y= ax + b, a 0, Y X, X, Y X,, Y, Y, g ( x ), X, Y Y= g ( X ) ( 9) Y X,,,,,,

5 63 X Y= g ( X )?, ( ) B, C= {x g ( x ) B}, {YB}= {g ( X ) B}= {X C} ( 9) P {YB}= P {g ( X ) B}= P {X C} ( 93) ( 93), X Y 6 X, Y= X, x 0, P {Yx }= P {X x }= P {- x X x } ( 94) P {Y= x }= P {X = x }= P ( {x = x }{x = - x }) ( 95) X Y, X Y, ( 93) X Y= g ( X ),, X Y Y= X 7 X P {X = - }= 4, P {X = 0}=, P {X = }= 4 Y 0,, ( 9), P {Y= 0}= P {X = 0}= P {X = 0}= P {Y= }= P {X = }= P ( {X = }{X = - }) = P {X = }+ P {X = - } = 4 + 4 =, X Y, Y y, =,, C= {x j g ( x j ) = y }, ( 9) ( 93), {Y= y }= {g ( X ) = y }= {X C} ( 96) P {Y = y } = P {X C} = x j C P {X = x j } ( 97) Y, Y X

64,,,,,,,, N (, ) X Y= a X + b, 6 Y, a + b a,, X F X ( x ) f X ( x ), Y= g ( X ), ( 93), Cx= {t g ( t) x } F Y( x ) = P {Yx }= P {g ( X ) x }= P {X Cx} ( 98) P {X Cx } X F X ( x ) f X ( x ), Y, F Y ( x ) P {X Cx } = Cx f X ( t) dt ( 99) 8 X, F ( x ), F ( X ) [ 0, ] F ( x ),, F - ( x ) Y= F ( X ) F Y( x ), F Y ( x ) = P {Yx }= P {F ( X ) x } F ( x ) [ 0, ], x < 0, F Y ( x ) = P {F ( X ) x } = 0, x >, F Y ( x ) = P {F ( X ) x }=, 0x, F ( x ), F Y( x ) = P {F ( X ) x }= P {X F - ( x ) }= F ( F - ( x ) ) = x, F Y( x ) = 0 x < 0 x 0x x > Y= F ( X ) [ 0, ] 9 F ( x ),,

5 65 9( ( ) ) X N ( 0, ), Y= X Y F Y ( x ), F Y( x ) = P {Yx }= P {X x }, x < 0, x 0, F Y( x ) = P {X x }= 0 F Y( x ) = P {X x }= P {- x < X < x }= 0( x ) - Y= X F Y ( x ) = 0( x ) - x 0 0 x < 0 f Y ( x ) = F Y ( x ) = x 0( x ) x 0 = 0 x < 0 x x e- x 0 ( 00) 0 x < 0 ( 00) ( ), ( ) = ( ) 30 ( ) X,, Y= lnx N (, ) Y= ln X N (, ), X = e Y, YN (, ),, x > 0, F X ( x ) = P {X x }= P {e Y x }= P {Yln x }= ( ln x ) x 0, F X ( x ) = 0 X f X ( x ) = F X ( x ) = x ( ln x ) x > 0 0 x 0 x e- ( ln x - ) = x > 0 x 0 0,,, Black_ Sch oles, P 0,,, P, r,

66 P = P 0e r r = ln P = lnp - lnp 0 P 0 P 0,, P r, 30 ( ) X Y= lnx N (, ), X = e Y, + E X k = E e ky = e ky ( y - ) - e- dy = + - exp - ( y - ) - k ( y - ) + k 4 +, k=,, k+ k dy = e k + k E X = e + DX = E X - ( EX ) = e + - e + = e + ( e - ) ( A) 30 0,,, 0 0, 0 0,,, X ( ) P {X= } = a, =,,, 00; ( ) P {X= } = a, =,,, ( ) ( ) a 3 X F ( x ) = 0, x < - 5 5, - 5x < - 3 0, - x< 0, 0x <, x

67 X 4 0, 7, 3,,, ( ) ; ( ) 5 X 3, P {X > - 3 }, P { X < 3 }, P { X + > } 6 X A P {05< X08} 7 F ( x) = F ( x ) = 0, x 0 Ax, 0< x, x > + x, ( ) - < x< + ; ( ) 0< x < +, ; ( 3) - < x< 0, 8X,,, 0, x < 0 0, x< 0 ( ) F ( x ) = x, 0x <, ( ) F ( x ) = x, 0x <, x, x 9 X x, 0x < ( ) f ( x ) = ae - x, ( ) f ( x) = a F ( x ), P - < X 0, P X x = - x, x < a 0, F ( + x) + F ( - x ) =, - < x < + ( ) 3 X EX R, X, P {X> }, R % % 3% 4% 5% 6% p 0 0 0 03 0 0 0,?? 3 (, ) 00,,, K %, K f ( x ) = 5, 0x5 0,

68 4 ( 5) 5, ( 3) ( 33) 6 3, ( 37) 7 07, 03, 08, 0, ( ) ( ) ( ) ; ( ) ; ( 3) ; ( 4) 8 P,,, 9, P { = }= P { = }, E, D, E, P { = 3} 0,, 3, ( ) [ 0, 5], 3,,, 500, 0 000 005, 800, 3 5 43, 3,, 000, 000 5 N (, ) 6 X N ( 0, 0 ), 00, 96, 7 X N ( 0, 5 ), X 00V, 00V< X < 40V X > 40V, 0, 00, 0, X; ( ), 00V40V 8 N ( 70, 00), 5%,? 9 R, R 0 3 p 0 04 03 0 30 X [ a, b], X+ ( > 0) [ a +, b + ] 3 X [ -, ], X

69 DX= 4 3X, Y= X + ( > 0) 33, [ a, b], 34 6 [ 500, 500] E L 35X, Y= - e - X 36 X, 0, E X = 5, 37, f ( x ), = f ( y) = yf ( y ), y > 0 0 y 0 38 X f X ( x), Y= a X+ b, ( a 0) Y f Y( y) = f X a y- b a = a ( B) + - F ( x ), a> 0, [ F ( x + a) - F ( x ) ] dx, E = 000 ( ), ; ( ), X ; ( 3), Y 3 X N (, ), = 40 ( ), 30( ) 50( ) 05 70 00, 4 m,,, E = m - - m 5 b a f ( x ) dx ( b - a ) b a f ( x ) dx, f ( x ) d [ a, b]

3,,,, ( ),, 3 3 X, X,, X (, ), ( X, X,, X ) (, ) n,,, n 3 ( X, X,, X ) (, ), F ( x, x,, x ) = P {X x, X x,, X x } ( 3) ( X, X,, X ) X, X,, X,, ( X, Y), ( x, y ), F ( x, y ) = P {X x, Yy } ( X, Y) {( t, s) tx, sy }( 3 ), ( X, Y) 3 P {x < X x, y < Yy }, P {x < X x, y < Y y } = P {( X, Y) } = P {( X, Y) } ( ) } - P {( X, Y) ( ) }

3 7 - P {( X, Y) ( ) } + P {( X, Y) } = F ( x, y ) - F ( x, y ) - F ( x, y ) + F ( x, y ) ( 3) 3 3, ( ) ( ) 0F ( x, y) ; ( ) F ( x, y ) x y ; ( 3) F ( -, y) = lim x - F ( x, y ) = 0, F ( x, - ) = F ( -, - ) = F ( +, + ) = lim F ( x, y) = 0, y - lim F ( x, y) = 0, ( x, y) ( -, - ) lim F ( x, y) = ( x, y) ( +, + ), ( X, Y) F ( x, y), F ( x, y ) X Y F X ( x ) F Y( y ), F X ( x ) = P {X x }= P {X x, Y< + }= F ( x, + ) ( 33) F Y( y ) = P {X y }= F ( +, y) ( 34) F X ( x ) F Y( y ) ( ) F ( x, y ), ( X, X,, X ) F ( x, x, x ), F ( x ) = F ( +,, +, x, +,, + ) ( 35) F ( x, x,, x ) =,,, 33 ( X, Y), ( X, Y)

7 3,,, ( X, Y), X, Y,, 34 ( X, Y) ( x, y j ),, j =,, P {X = x, Y= y j }= p j,, j =,,, ( 36) ( 36) ( X, Y), X Y p j ( ) p j 0,, j =,,, ( ) p j = j, ( 3), 3 X Y y y y j P {X = x } x p p p j p j j x p p p j p j j x p p p j p j j P {Y= y j } p p p j X Y, X, Y p X, =,, ; p Y j, j =,,,, p X = P {X = x } = P { j {X = x, Y = y j }) = j P {X = x, Y = y j } = j p j, =,, ( 37) p Y j = P {Y = y j } = p j j =,, ( 38) ( 37), ( 38) P {X = x, Y= y j }= p j,, j =,,,,, 3 3 3, X, Y,

3 73, X Y X, Y 0,,,, ( X, Y) (, ), (, ), (, ), 0, ( X, Y),, P {X = 0, Y= 0}= 3 = 9, P {X = 0, Y= }= 3 = 9, P {X =, Y= }= 3 = 9, X, Y ( ) X Y 0 p X 0 9 9 9 4 9 9 9 0 4 9 9 0 0 9 p Y j 4 9 4 9 9,, ( X, Y) D,, P {( X, Y) D } = p j ( 39) ( x, y ) D j, F {( x, y) } = P {X x, Y y} = 3 ( X, Y), P {X 0, Y = 0}, P {X 0, Y 0 }, P {X Y= 0}, P {X = Y}, P { X = Y } X Y p j ( 30) x x, y y j - 0 0 0 0 0 03 005 0 05 0 0 P {X 0, Y= 0}= P {X =, Y= 0}+ P {X =, Y= 0} = 005+ 0= 005,

74 3 P {X 0, Y0}= P {X = 0, Y= - }+ P {X = 0, Y= 0} = 0+ 0= 03, P {X Y= 0}= P ( {X = 0}{Y= 0}) = P {X = 0}+ P {X 0, Y= 0} = 0+ 0+ 0+ 005= 035 P {X = Y}= P {X = 0, Y= 0}+ P {X =, Y= } = 0+ 0= 03, P { X = Y }= P {X = 0, Y= 0}+ P {X =, Y= - }+ P {X =, Y= } = 0+ 03+ 0= 06 35 ( X, Y), F ( X, Y), f ( x, y ), ( x, y ), F ( x, y) = x y - - f ( s, t) dsdt, ( 3) ( X, Y), f ( x, y ) ( X, Y) ( ), X Y 35, f ( x, y), ( ) f ( x, y) 0, ( ) + + - - f ( x, y ) dxdy =, f ( x, y),,,,, ( 3) D, P {( X, Y) D } = D f ( x, y ) dx dy ( 3), ( 3), F X ( x ) F X ( x ) = P {X x } = P {X x, Y < + } = x - = x - + - f ( s, t) dsdt + f ( s, t) dt ds ( 33) - ( 33), X, f X ( x ) = + - f ( x, y) dy ( 34)

3 75, Y, f Y( y) = + - f ( x, y ) dx ( 35) ( 34), ( 35) f X ( x ) f Y( y ) ( X, Y) f ( x, y ) 33 ( ) G, S ( G), ( X, Y) G, G, G, ( X, Y) f ( x, y) = C=, S ( G), + + - - f ( x, y ) = C, ( x, y ) G, 0, f ( x, y ) dxdy =,, ( x, y) G, S( G) 0, ( 36) ( X, Y) ( 36), ( X, Y) G ( 3), ( X, Y) G, D, P {( X, Y) D }= D f ( x, y ) dx dy = S( G) dxdy = S ( D G) S ( G) S ( D G) D G D G ( 37) 34 ( X, Y) f ( x, y ), ( X, Y ) g ( x, y ) f ( x, y) = g ( x, y ) = k xy, 0x, 0y, 0, ; k xy, 0x y, 0, ( ) k ( X, Y ) ; ( ) k ( X, Y ) ( ), + + - - f ( x, y ) dx dy = 0k x ydx dy =, 0

76 3 k = 4, f ( x, y ) = 4xy, 0x, 0y, 0, ( X, Y) f X ( x ), f Y ( y ), 0x, x < 0 x <,, f X ( x ) = + f X ( x ) = f X ( x ) = f Y ( y ) = - 0 f ( x, y) dy, 4x ydy = x f X ( x ) = 0 x, 0x, 0, y, ( ), + + - -, k = 8, g ( x, y) dx dy= g ( x, y ) = 0y, 0, 0x y = 8 k = kx ydx dy = 0dx x 8xy, 0x y, 0, kx ydy ( X, Y) g X ( x ), g Y ( y ), 0x, x < 0 x >, g X ( x ) = + - g ( x, y ) dy g X ( x ) = x 8xy dy = 4x ( - x ), g X ( x ) = 0, g X ( x ) = 4x ( - x ), 0x, 0,,

3 77 g Y ( y) = 4y 3, 0y, 0,,,, ( X, Y) ( x, y ) = - e- ( - ),,,,, > 0, > 0, <, ( X, Y) (, ;, ; ), ( X, Y) N (, ;, ; ) 33, (, ) ( x - ) ( x - ) ( y - ) ( y - ) - + ( 38), 33,,, ( X, Y) N (, ;, ; ), X ( x ) Y( y ), u= x -, t= X N YN X ( x ) = + = + = = y-, - - ( x, y ) dy - e - e - ( x - e- ) + - ( x - ), (, ), (, ) Y( y) = ( - [ u - u t+ t ] ) e - - ( y- ), dt e- ( t - u) ( - ) dt ( x, y) X ( x ), Y ( y ),, = 0, ( x, y),

78 3 ( x, y) = X ( x ) Y( y ) ( 39) ( ), 4 ( ),,,,,, = 0, ( 39), ( 3) (, 5 6 ) 3 X,, F X ( x ) = P {X x }, - < x < + A, A {X x } x, P {X x A }F ( x A ), F ( x A ), - < x < + A, X 35 X [ 0, ], X >, X, F ( x X > ) = P {X x, X > } P {X > } X [ 0, ], P {X > }= x, P {X x, X > }= 0 x >, P {X x, X > }= F ( x ) - F ( ) = F ( x ) - F ( x ) X,,,

3 79 F ( x ) = 0, x < 0, x, 0x,, x >, x > P {X x, X > }= x - < x, x > F ( x X > ) = 0 x x - < x x > A Y A= {Yy }, P {Yy}> 0, F ( x Yy) = P {X x, Yy} = F ( x, y) P {Yy } F Y( y) ( 30), X Y, ( 30) X Y, x y, {X x } {Yy }, P {X x, Yy }= P {X x }P {Yy} F ( x, y) = F X ( x ) F Y( y) ( 3) F ( x Yy ) = F X ( x ) x, y, {X x } {Yy }, X Y, ( 3) 36 X, Y F ( x, y ), F X ( x ), F Y ( y ), x y, F ( x, y) = F X ( x ) F Y( y) X Y,, X Y X Y,, 3 X Y X

80 3 Y,, A B, P {X A, YB}= P {X A}P {YB} ( 3) 3,,, A = ( x, x ], B= ( y, y ] 3 X Y, g ( x ), g ( y), g ( X ) g ( Y) = g ( X ), = g ( Y), x, y, D x= {t g ( t) x }, D y = {t g ( t) y}, 3, P { x, y }= P {g ( X ) x, g ( Y) y} 36 = P {X D x, Y D y } = P {X D x }P {Y D y } = P { x }P { y} 37 X, X,, X, F ( x, x,, x ), F ( x ), =,,,, x, x,, x F ( x, x,, x ) = F ( x ) F ( x ) F ( x ) X, X,, X,, ( ), ( X, Y), P {X = x, Y= y j }= p j, (, j =,, ), P {Y= y j }> 0, P {X = x Y= y j }= P {X = x, Y= y j } = p j P {X = y j } p Y j ( 33) P {X = x Y= y j } Y= y j, X = x, p j j, P {X = x Y= y j }= p j, =,,, Y= y j, A, B Borel, {X A}, {X B }

3 8, X x ( =,, ),, p j, =,, ( ) p j 0 ( ) p j =, P {X = x Y= y j }= p j, =,, Y= y j, X,, P {X = x }> 0, X = x, Y P {Y= y j X = x }= p j = p j, p X j =,, ( 34) ( 33), ( 34),, j =,,, p j = p X p j = p Y j p j ( 35) 36 X Y 3, Y= 0, X X = 0, Y P {Y= 0} = 0+ 005+ 0= 05, Y= 0, X P {X = 0 Y= 0}= P {X = Y= 0}= P {X = Y= 0}= P {X = 0, Y= 0} = 0 P {Y= 0} 05 = 08 P {X =, Y= 0} = 005 P {Y= 0} 05 = 0 P {X =, Y= 0} = P {Y= 0} 0 05 = 0 P {X = 0}= 0+ 0+ 0= 03, X = 0, Y P {Y= - X = 0}= P {Y= 0 X = 0}= P {Y= X = 0}= P {Y= -, X = 0} = 0 P {X = 0} 03 = 3 P {Y= 0, X = 0} = 0 P {X = 0} 03 = 3 P {Y=, X = 0} = 0 P {X = 0} 03 = 0 3, X Y,, j, P {X = x, Y= y j }= P {X = x }P {Y= y j } ( 36),, j, ( 36), X Y, ( 36), x y, P {X x, Y = y j }= P ( {X = x, Y = y j }) x x

8 3, x, y, = P {X = x, Y = y j } x x = P {X = x }P {Y = y j } x x = P ( {X = x }) P {Y = y j } x x = P {X x }P {Y = y j }, P {X x, Y y}= P ( {X x, Y = y j }) y y j, 33X, = P {X x, Y = y j } y y j = P {X x }P {Y = y j } y y j = P {X x }P ( {Y = y j }) y y j = P {X x }P {Y y } Y, P {X = x, Y= y j } = p j,, j =,,, p X p Y j,, j =,,, X Y p j = p X p Y j,, j =,, ( 37) 37 X Y 3, X Y? P {X = 0}= 0+ 0= 03, P {Y= - }= 0+ 03+ 05= 055 P {X = 0, Y= - }= 0,, P {X = 0, Y= - }P {X = 0}P {Y= - }, X Y,,, 3, X Y,, p j = p X p Y j, X

3 83 Y P {X = 0}= P {X = }= P {X = }= 3 P {Y= 0}= P {Y= }= P {Y= }= 3 X Y, X Y ( 37), 3 3 X 0 Y 0 p X 9 0 p Y j 9 3 9 9 0 3 0 9 9 3 3 3 3 Y X 0 p Y j 0 p X 9 9 9 3 9 9 9 3 9 9 9 3 3 3 3 ( X, Y), F ( x, y ) f ( x, y ) Y= y, X {Y= y }, P {X x Y= y}= P {X x, Y= y } P {Y= y} ( 38) 0, X, ( 38) 0 0, P {X x Y = y}= y0 lim P {X x y - y < Y y} P {X x, y - y < Y y } = y0 lim P {y - y < Y y} y y - = y0 lim x - = y x - y y - f ( u, y ) du f Y( y) f ( u, t) du dt f Y ( t) dt y ( 39), y, f Y ( y ) > 0, P {X x Y= y }Y= y X, F X Y ( x y ) ( 39)

84 3 F X Y ( x y ) = x f ( u, y) du ( 330) - f Y ( y ), f ( x, y) f Y( y) Y= y, X, f X Y( x y ), x, f X ( x ) > 0, X = x, Y F Y X ( y x ) = y - f Y X ( y x ) = f ( x, y) f x ( x ) f ( x, t) dt ( 33) f X ( x ) ( 33),,, f ( x, y) = f X ( x ) f Y X ( y x ) = f Y ( y ) f X Y( x y ) ( 333) 38 ( X, Y) D = {( x, y ) x + y } f Y X ( y x ) f X Y( x y ) ( X, Y) f ( x, y ) = f X ( x ) f X ( x ) = +, f Y ( Y) -, x x <, f ( x, y ) dy = f Y( y ) = f Y X ( y x ) = f ( x, y) f X ( x ) =, y y <, f X Y ( x y ) =, x + y 0, - x, x, - y, y, 0, 0, - x, y - x 0,, x - y - y 0, 39 ( X, Y) N (, ;, ; ), f X Y ( x y ) f Y X ( y x ) 3, X N (, ), YN (, )

3 85 f X Y ( x y ) = f ( x, y ) f Y ( y ) = - e- ( - ) ( x - ) e - ( y- ) - ( x - ) ( y - ) + ( y - ) = - e- ( - ) x - y- -, ) ) = - e- ( - ) x- - ( y- ) Y= y, X N ( + ( y- ), ( - ) ), X = x, Y N ( + ( x - ), ( -, 34 ( X, Y) f ( x, y ), f X ( x ) f Y( y ), X Y f ( x, y ) = f X ( x ) f Y( y ) ( 334) X Y x, y, F ( x, y ) = F X ( x ) F y ( y ) = x f - X ( u) du y f Y ( t) dt - = x - y - f X ( u) f Y( t) dudt, f X ( x ) f Y( y ) ( X, Y), f ( x, y) = f X ( x ) f Y( y) f ( x, y) = f X ( x ) f Y ( y ), F ( x, y) = x - y - f X ( u) f Y ( t) dudt X Y = x - f X ( u) du y - = F X ( x ) F Y( y ) f Y ( t) dt 30 34 X Y, X Y 34, X Y

86 3 X f ( x, y) = Y 34, X Y 4xy, f X ( x ) = f Y ( y ) = 0x, 0y 0, x, 0x 0, y, 0y 0, f ( x, y ) = f X ( x ) f Y ( y ) g ( x, y ) = g X ( x ) = g Y ( y ) = 8xy, 0x y 0, 4x ( - x ), 0x 0, 4y 3, 0y 0, > x 0> y 0> 0 ( x 0, y 0), g ( x 0, y 0) = 0, g X ( x 0) g Y ( y 0) = 6x 0( - x 0) y 3 0 0, X Y g ( x 0, y 0) g X ( x 0) g Y ( y 0) 3 ( X, Y) N (, ;, ; ), X Y = 0 3 X N (, ), YN (, ), N (, ;, ; ) N (, ), N (, ) ( x, y ), X ( x ), Y ( y ), = 0, ( x, y) = X ( x ) Y( y ) 33 ( X, Y), g ( x, y ), g ( X, Y) ( X, Y) ( X, Y)

33 87 P {X = x, Y= y j }= p j,, j =,, zk, k=,,, Z= g ( X, Y), ( 39), Z P {Z = zk} = P {g ( X, Y) = zk} = g( x, y j ) = z k P {X = x, Y = y j } ( 335) k=,, 3 ( X, Y) 3, = X + Y = X Y = X + Y -, 0,,, 3, 4, = X Y -, -, 0, 4 ( 335), P {= - }= P {X + Y = - } = P ( X = 0, Y = - } = 0, P {= 0}= P {X + Y = 0} = P {X = 0, Y = 0} + P {X =, Y = - } = 0 + 03 = 05, P {= }= P {X + Y = } = P {X =, Y = 0} + P {X =, Y = - } = 005 + 05 = 0, P {= }= P {X + Y = } = P {X = 0, Y = } + P {X =, Y = 0} = 0 + 0 = 0, P {= 3}= P {X + Y = 3} = P {X =, Y = } = 0, P {= 4}= P {X + Y = 4} = P {X =, Y = } = 0 P {= - }= P {X =, Y = - } = 05, P {= - }= P {X = Y = - } = 03, P {= 0}= P {X = 0, Y = - } + P {X = 0, Y = 0} + P {X = 0, Y = } + P {X =, Y = 0} + P {X =, Y = 0}, = 0 + 0 + 0 + 005 + 0 = 035, P {= }= P {X =, Y = } = 0, P {= 4}= P {X =, Y = } = 0 ( X, Y) X Y - 0 0 0 0 0 03 005 0 05 0 0

88 3, ( x, y j ) g ( x, y j ) ( X + Y, X Y ) Y X 0-0 0 0 0-0 Y X 0-0 0 0 0 0 0 0 03 0 005 0 3 03-005 0 0 05 0 0 4 05-0 0 0 4 g ( x, y ), g ( X, Y),, P { = - }= 0, P { = 0}= 05, P { = }= 05+ 005= 0, P { = }= 0, P { = 3}= 0, P { = 4}= 0, P { = - }= 05, P { = - }= 03, P { = 0}= 0+ 0+ 005= 035, P { = }= 0, P { = 4}= 0 33 X, Y,, = X + Y P { = k}= P {X + Y= k} k = P {X =, Y = k - } = 0 k = P {X = }P {Y = k - } = 0 k = = 0 e- = e - ( + = ( + ) k k k - ( k - ) e- ) = 0 ( + ), k e- k= 0,,, k- ( k- ) = X + Y + ( X, Y), f ( x, y), g ( x, y ), g ( X, Y) ( X, Y)

33 89 Z= g ( X, Y) F Z( z ) F Z( z ) = P {Z z} = P {g ( X, Y) z } = P {( X, Y) D z } = D z f ( x, y) dx dy ( 336) D z = {( x, y) g ( x, y ) z }, f Z ( z ), z, f Z( z) = F Z ( z ) ( 337) 34 ( ) ( X, Y) f ( x, y), X + Y 34, z, D z = {( x, y ) x + y z },,, F Z( z ) = P {Z z } = P {X + Y z } = f ( x, y) dx dy D z = + = + = z - dy z - y - f ( x, y ) dx - dy ẕ f ( u - y, y ) du - + - f ( u - y, y) dy du ( 338) f Z( z ) = + -, f Z( z) = + - 34D z= {( x, y) x + y z } f ( z - y, y ) dy ( 339) f ( x, z - x ) dx ( 340), X Y, ( 339) ( 340) f Z( z) = + - = + - f X ( x ) f Y ( z - x ) dx f X ( z - y) f Y( y ) dy ( 34) ( 34) f X ( x ) f Y( y), f X f Y( z ), ( 34) f Z ( z) = f X f Y ( z ) = f Yf X ( z ) ( 34) ( 34) ( 34),

90 3 35( ) X N (, ), YN (, ), X Y, X + YN ( +, + ) f X + Y ( z ) = + f - X ( x ) f Y ( z - x ) dx = + - = + - = + - e - e - e - ( x - ) e - ( z - x - ) dx ( x - ) + ( z - x - ) dx + x - - ( z - - ) + ( z - - ) - ( + ) dx ( z - - = e - ( + + ) ( 343) X + YN ( +, + ) ) 35, 35 X, Y N (, ) N (, ),, a, a X + byn ( a + b, a + b ) ( 344) b 0,, 36( ) ( X, Y) f ( x, 35D z= {( x, y ) = x y z } f Z( z ) = F Z ( z) = + 0 y ), y f ( zy, y ) dy - 0 Z= X Y z, D z = {( x, y) x y z }( 35) F Z ( z) = P X Y z = D z f ( x, y ) dx dy = + z y 0 + 0 - - + z y f ( x, y ) dx dy f ( x, y ) dx dy Z - yf ( zy, y) dy

33 9 = + - y f ( zy, y ) dy ( 345) 37( ) X, Y F ( x ), G( x ), f ( x ), g ( x ), X Y, M= max {X, Y}, N = min {X, Y}M N F M ( z ) = P {Mz }= P {X z, Yz }= P {X z }P {Yz} = F ( z ) G( z) ( 346) M f M ( z) = F M ( z) = F ( z ) G( z ) + F ( z) G ( z ) = f ( z ) G( z) + F ( z) g ( z) N ( 347) F N ( z ) = P {N z } = P ( {X z } {Y z }) = - P {X > z, Y > z } = - P {X > z}p {Y > z } = - [ - F ( z ) ] [ - G( z ) ] ( 348) N f N ( z ) = F N ( z ) = f ( z ) [ - G( z) ] + g ( z ) [ - F ( z ) ] ( 349) 38 ( X, Y) G= {( x, y) 0x, 0y}, X Y S f ( s) ( X, Y) f ( x, y ) = F ( s) S,, ( x, y ) G, 0, ( x, y ) G F ( s) = P {S s} = f ( x, y ) dx dy, x ys s0, F ( s) = 0, s F ( s) =, 0< s<, 36, f ( x, y ) dx dy = - xy s s dx s dy x = s ( + ln - ln s) F ( s) = s 0, s0, ( + ln - ln s), 0< s<, s 36

9 3 f ( s) = F ( s) = ( ln - ln s), 0< s<, 0, ( X, Y) Z= g ( X, Y), EZ,, Z (, Z ), Z= g ( X, Y) ( X, Y), P {X = x, Y= y j } = p j,, j =,,, EZ = Eg ( X, Y) = g ( x, y j ) p j ( 350), j ( X, Y), f ( x, y),, EZ = Eg ( X, Y) = + g ( X, Y) = X Y, EX Y = 39 ( X, E X Y E X Y = + + + - - + - - g ( x, y ) f ( x, y ) dx dy ( 35) x y j p j, ( X, Y),, j x yf ( x, y) dx dy, ( 35) ( X, Y) Y) 34 g ( x, y ), g ( x, y ) = + - - 8xy, x yg ( x, y) dx dy = dx x y 8xy dy 0 x = 4 9 0 x y, 0, 30 ( X, Y) 3, E X Y E X Y = 0( - )0+ 000+ 00 + ( - )03+ 005+ 0 + ( - )05+ 00+ 0 = 0 3, X Y, [ 0, 0], 000,,

33 93, 500, Z, Z= g ( X, Y) = ( X, Y) E Z= 0 f ( x, y) = 0 0 0 = 0 dy 0 0 y 000Y, YX, 000X + 500( Y- X ) = 500( X + Y), Y> X, 0x 0, 0y 0, 00 0, g ( x, y) f ( x, y ) dx dy 000y 00 dx + 0 = 0 0 y( 0 - y ) dy + 5 0 0 = 0000 3 + 5500 4 6667( ) 0 0 dy y 0 500( x + y ) 00 dx 3 y - 0y - 50 dy,,,, ( ) X, Y,, E ( X + Y) E ( X + Y) = E X + EY; ( 353) ( ) X, Y,, E XY, EX Y= E X EY ( 354),,,, ( X, Y) f ( x, y ), F X ( x ) F Y ( y ), ( 35),,,, E ( X + Y ) = E X + E Y, E X + Y E ( X + Y ), E ( X + Y) E X Y = E ( X Y ) = E X E Y E X Y

94 3 + - - = + + - - = + = + ( ) E ( X + Y) = + - x + - = E X + E Y ( x + y ) f ( x, Y) dx dy x f ( x, y) dx dy + + + - - - f ( x, y) dy dx + + x f X ( x ) dx + + - yf Y( y) dy y f ( x, y ) dx dy y - + - ( ) X, Y, f ( x, y) = f X ( x ) f Y ( y), E X Y= + + - - = + - = E X EY x yf X ( x ) f Y( y ) dx dy xf x ( x ) dx + - yf Y ( y ) dy f ( x, y) dx dy ( ) X, X,, X ;, E ( X + X + + X ), E ( X + X + + X ) = EX + E X + + EX ( 355) ( ) X, X,, X,, E ( X X X ), E ( X X X ) = E X EX EX ( 356) 34,,,, 37 ( X, Y), E X, EY, E [ ( X - E X ) ( Y- E Y) ], X Y, cov( X, Y) cov( X, Y) = E[ ( X - E X ) ( Y- EY) ] ( 357) ( 350) ( 35), ( X, Y),

34 95 P ( X = x, Y= y j }= p j,, j =,, X Y cov( X, Y) =, j ( x - EX ) ( y j - E Y) p j ( 358) ( X, Y), f ( x, y ), X Y cov( X, Y) = + + - -, ( 357) ( x - E X ) ( y - EY) f ( x, y ) dx dy ( 359) coo( X, Y) = EX Y- E X EY ( 360),, 35 ( ) cov( X, X ) = DX ; ( ) cov( X, Y) = cov( Y, X ) ; ( 3) cov( a X, by) = a bcov( X, Y), a, b ; ( 4) cov( C, X ) = 0, C ; ( 5) cov( X + X, Y) = cov( X, Y) + cov( X, Y) ( 6) X Y, cov( X, Y) = 0 X, Y,, X + Y,, X Y, D ( X + Y) = DX + DY+ cov( X, Y) ( 36) ( ), X, D ( X + Y) = DX + DY ( 36) Y) Y, cov( X, cov( X, Y) E ( X - E X ) ( Y- EY) D X DY ( 363) ( ) ( 5) D( X + Y) = cov( X + Y, X + Y) = cov( X, X ) + cov( X, Y) + cov( Y, Y) = D X + D Y + cov( X, Y) X Y, ( 6) cov( X, Y) = 0, D( X + Y) = DX + D Y 3 ( X, Y) 3,

96 3 cov( X, Y) X P {X = 0}= 03, P {X = }= 045, P {X = }= 05; Y 30, P {Y= - }= 055, P {Y= 0}= 05, P {Y= }= 0, E X = 003+ 045+ 05= 095, EY= ( - )055+ 005+ 0= - 05 E X Y= 0, cov( X, Y) = EX Y- E X EY= 09505= 045 33 ( X, Y) 34 g ( x, y ), cov( X, Y) D ( X + Y) ( X, Y) g ( x, y ) = 8xy, 0x y 0, 34,, E X = + E Y= + g X ( x ) = g Y( y) = 39, 4x ( - x ), 0x 0, 4y 3, 0y 0, - x g X ( x ) dx = 0 x 4x ( - x ) dx = 8 - yg Y( y) dy = 0 y 4y 3 dy = 4 5 E X Y= 4 9 cov( X, Y) = EX Y- E X EY= 4 9-4 5 8 5 = 4 5 EX = + 5 - x g X ( x ) dx = 0 x 4x ( - x ) dx = 3

34 97, E Y = + - y g Y( y ) dy = 0 y 4y 3 dy = 3 DX = E X - ( EX ) = 3-8 5 DY= E Y - ( E Y) = 3-4 5 = 5 = 75 D ( X + Y) = DX + DY + cov( X, Y) = 5 + 75 + 8 5 = 9 ( 36) ( 36) 36 ( X, X,, X ), X, =,,,, (,,, D = X = = DX + < j, X, X,, X, D = X = ), = X, j cov( X, X j ) ( 364) DX ( 365) = ( 36) ( 36), 38 ( X, X,, X ), X DX, =,,,, j = cov ( X, Yj ) (, j ), j =,,, ( j ) ( X, X,, X ), X= ( X, X,, X ), D X ( 364) = (,,, ), ( 366), D ( X) = DX ( 366) = (,,, ), DX = D ( X) 0 D X,,

98 3,, X Y K, X = K X, Y= K Y, X Y X Y, K, cov( X, Y) = K cov( X, Y),, X = X - EX DX, Y = Y- EY cov( X, Y ), cov( X, Y ) X Y,, cov( X, Y ) = DY cov( X, Y) D X DY ( 367) 39 ( X, Y), X Y,, X, Y= X Y cov( X, Y) D X ( 363), X, Y DY ( 368) 34 X, Y, Y= a X + b, ( a 0, b ), D X, X, Y cov( X, Y) = cov( X, ax + b) = a cov( X, X ) = ad X, X, Y = D Y= D( ax + b) = a DX, cov( X, Y) DX DY = a DX DX a DX = a a ( 369), a > 0,, X, Y= ; a < 0, X, Y = - 34, X Y, X, Y 34, P {Y= ax + b}= ( 370), X Y, X, Y =, ( 370), X Y, X Y X, Y = X Y?

34 99 37 ( X, Y), D X, DY, X, Y = X Y,, a 0 b,, a > 0, X Y = ; P {Y= a x + b}= a < 0 XY = -, X, Y = X, Y= 0, X Y, DX, DY, 35( 6) cov( X, Y) = 0, X, Y= 0 X, Y= 0, X Y ( 35 ),, X, Y = 0 X Y,, X Y, DX, DY, X Y ( ) cov( X, Y) = 0 ( ) E X Y= E XE Y ( 3) D ( X + Y) = DX + DY 35 [ -, ], X = sin, Y= cos X Y, EX = D X = - sin d = 0, E Y = - cos d= 0, - sin d =, D Y = E Y = EX Y = sin cos d = 0 - cos d =, E X Y= E XE Y, X Y X Y X Y X + Y = X, Y = X, Y= 0 X, Y <,, X Y,,, X Y - Y= a X + b+ ( 37) a 0, b, X, Y X, Y X, Y X, X, Y X a X

00 3 + b, Y X ( ),, X Y ( 37), X, Y 0< < X, Y,, X, Y 0,,,,, ( ) 30 ( ) ( X, X Y= y j X Y= y j Y), Y= y j, X P {X = x Y= y j }= p j =,,, x p j <, E [ X Y = y j ] = x p j ( 37) ( ) ( X, Y), Y= y, X f X Y ( x y ), + - x f X Y( x y) dx < X Y= y, E [ X Y = y ] = + - x f X Y ( x y ) dx ( 373) X Y= y 36 ( X, Y) 3, Y= 0, X 36, Y= 0, X P {X = 0 Y= 0}= 08P {X = Y= 0}= 0P {X = Y= 0}= 0 E [ X Y= 0] = 008+ 0+ 0= 0 37 ( X, Y) 38, E [ X Y= y ] ( y < )

34 0 38, y <, y <, f X Y ( x y ) = E [ X Y = y ] = + - - y, x - y ; 0, x f ( x y ) dx = - y x - - y - y dx = 0 ( ) C, E [ C Y= y ] = C ( 374) ( ) k, k, E [ X Y= y ], =,,, E [ k X + k X Y= y ] = ke [ X Y= y ] + k E [ X Y= y] ( 375) ( 3) X Y, E [ X Y= y] = E X ( 376) E [ X Y= y] Y, Y y, E [ X Y= y ], E [ X Y], Y,, E [ X Y= y ], E [ X Y] ( ) C, E [ C Y] = C ( 377) ( ) k, k, E [ k X + k X Y] = ke [ X Y] + ke [ X Y], ( 378) ( 3) X Y, E [ X Y] = E X ( 379), ( 4) g ( x ), E [ g ( Y) X Y] = g ( Y) E [ X Y],, E [ g ( Y) Y] = g ( Y) ( 380) ( 5) E ( E[ X Y] ) = E X ( 38) ( 4) Y= y, g ( Y) g ( y) ( ), E [ g ( Y) X Y= y ] = E [ g ( y ) X Y= y] = g ( y) E[ X Y= y ] E [ g ( Y) X Y] = g ( Y) E [ X Y] ( 5), g ( y ) = E [ X Y= y], g ( Y) = E[ X Y], E ( E [ X Y] ) = E ( g ( Y) ) = + = + = + - g ( y) f Y ( y) dy - E [ X Y = y] f Y ( y ) dy - + - x f X Y ( x y ) dx f Y ( y ) dy

0 3, = + + - - + - - = + = + x - + - = + - = E X [ x f X Y( x y) f Y( y ) ] dx dy [ x f ( x, y) ] dx dy xf X ( x ) dx f ( x, y ) dy dx 39 ( X, Y) N (,,,, ) E [ X Y] Y= y, X 330 X, N + ( y - ), ( - ) E [ X Y= y ] = + ( y - ) ( 38) E [ X Y] = + ( Y- ) ( 383) Y Y= a X + b+ ( 384) X, 0, E [ Y X ] E [ Y X ] = E ( ax + b+ X ] = E [ ax X ] + E [ b X ] + E[ ] = a X + b+ E = a X + b ( 385),,, E X X, E ( X - E X ) = mine ( X - C) ( 386) C, X, X Y, Y Y, X Y, X, Y, X Y g ( Y) ( ), g 0, g 0( Y) X, g 0( Y) X

3503 E [ X - g 0( Y) ] = mine [ X - g ( Y) ] ( 387) g, Y y, X Y, X, Y= y X,, X, E [ X Y= y ], E [ X Y],, ( 387),, E ( X - E ( X Y) ) = min g E ( x - g ( Y) ) ( 386) E ( X - g ( Y) ) = E ( ( X - E [ X Y] ) + ( E [ X Y] - g ( Y) ) ) = E ( X - E [ X Y] ) + E ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) + E ( E [ X Y] - g ( Y) ) ( 38) ( 380) E ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) = E ( E [ ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) Y] ) = E ( ( E [ X Y] - g ( Y) ) E[ ( X - E [ X Y] ) Y] ) = E ( ( E [ X Y] - g ( Y) ) ( E [ X Y] - E [ X Y] ) = 0 E( X - g ( Y) ) = E ( X - E [ X Y] ) + E ( E [ X Y] - g ( Y) ) E( X - E[ X Y] ) 35 a,, a lim a= a > 0, N, > N a- a <,,, X, X,, X,,, X, X,, X, X, > 0, N, > N,

04 3 X - X < X, X, X - X,,,,,, > N, X - X <,, X - X >, 33 X, X, X, 0, {X } X, X, X,, > limp { X - X > }= 0 ( 388) P X P - lim X = X,, A p, A ( ) p,, A, A,, p ( A ) P p,,, P 37 ( ) A A p ( 0< p < ), p P - lim = p lim P, E = p, D > 0 { - p < } = ( 389) = D ( ) = p q, ( q= - p ), > 0, P - p > p q

3505 = = = 0lim P lim P - p > lim p q = 0 - p > = 0 A =,,, 0 A, E = p, ( 305) lim P - = E < = ( =,, ) 0, = 38 ( ),,,,, E D,, C, DC=,,, > 0 limp - = =, = P D - = = = lim P E < = ( 390) D E = = D( ) C = = - = E < = C 0,,,,, E =, > 0, limp = = - < = ( 39) P,, 39 ( ),,,,

06 3,, E =, limp = - < = ( 39),, = E,,,?,,,,,, 3, 4,,,,,, =,,,,,,, ( ),,, = 30 ( ),,,,, E =, D ( ) = > 0, =,,, lim P = - x = x - e- t dt ( 393),, 30 N ( 0, ), =

, a N (, ), =, = - a N ( 0, ) ( 394) = a N,,,, > 0, P > 0,, = = = - > 0,, - >?, P = = a - N 0, - > = - P = - - = - P = / / - 0 / - = - 0 ( 395) / 0( x ) N ( 0, ), 0 ( 395) / P = - > 0,, - > = 33 00,, 00g 0g, 0kg 3507, =,,, 00, 0 0 =, E = 00, D ( ) = 0, E= i= 00E = 0000, D= 00,

08 3 P { > 000}= P - 0000 000-0000 > 00 00 = P - 0000 > 00 = - P - 0000 00-0( ) = - 09775= 0075, 3 X B(, p ), 0< p <, limp X - p P q x = x e - - t d t ( 396) 3,,, X B (, p ), P {a X < b} = C k p k q - k,, a k< b X - p,, 3, P q a N ( 0, ) X a N ( p, p q), P {a X < b}= P a - p p q X - p p q b- p 0 p q - a - p 0 p q, < b- p p q P {a X < b} 33 0 000, 07,, 6 800 7 00, B ( 0000, 07), E = 000007= 7000, D = 00000703= 00, P {6800< < 700}= P = P - 00 00 < - 7000 00 < 00 00-7000 < 436 4583 0( 436) - = 099999 333, [ 0, 000] ( ) 0%,, 00 095,? X, X, A = {X

09 00}, A =,,,, P {A }= 09, B(, 09), {X 00}= 64, 5 3 P {X 00}= P 08, 08 = P { > 08} = P = - 0 - - 09 090 > 3 = 0 08-09 090 3 095 ( A) F ( x, y) ( X, Y), x < x, y < y, F ( x, y ) - F ( x, y ) - F ( x, y ) + F ( x, y ) 0 5, 3,,, X, =, ( ) ( X, X ) ; ( ) P {X = 0, X 0}, P {X = X }, P {XY= 0} 3 ( X, Y) ( X, Y) f ( x, y) g( x y ) f ( x, y) = g( x, y ) = ( ) k, k, ( ) k e - 3x- 4y, x > 0, y > 0 0, k e - 3x- 4y, x > y> 0 0, 4 ( X, Y) G= {( x, y) 0x, 0y }, ( ) ( X, Y) ; ( ) X Y ; ( 3) P {Y< X } 5 ( X, Y) G= {( x, y ) > y > x > 0}, ( ) ( X, Y), ( ) X Y 6 X Y F x( x ), F Y( y) F X( x ) = 0, x< 0 x, 0x, F Y( y ) =, x> 0, y< y-, y, y> X Y, ( ) ( X, Y) ; ( ) = X, = Y, (, ) G( x y ) ; ( 3) P X<, y > 3 7 X Y, x, x, y, y ( x < x ; y < y ), {x < Xx }

0 3 {y < Yy } 8 X = X 9 ( X, Y), X P {X= x }= p X, =,,,, =,,, X= x, Y P {Y= y j X = x }= p j, j = ( ) ( X, Y) P {X= x, Y= y j}= p j,, j =,, ; ( ) Y P {Y= y j }= p Y j ; ( 3) Y= y j X P {X = x Y= y j }= p j, =,, 0 ( X, Y) X Y - 0 p X 0 p p p 3 0 p 0 p Y j 4 4 ( ) ( X, Y) p, p, p 3, p ; ( ) X Y X Y, ( X, Y), X x Y y y y 3 p X 8 x 8 p Y j 6 ( X, Y), P {X = x, Y= y }= p j, =,,, m, j =,,,, ( p j ) m X Y ( p j ) m 3, B, C, x + Bx + C= 0 p q 4 3 f X Y ( x y), f Y X ( y x ), g X Y ( x y), g Y X ( y x ) 5 D y = x -, y= x +, x = ( X, Y) D f Y X ( y x ) f X Y( x y) 6 D= {( x y) } a x b, ( x ) < y < ( x ) }, ( x ), ( x ), ( x ) < ( x ) [ a, b], ( X Y) D 7 ( X Y), X f X ( x ) x, X= x

Y f Y X ( y x), ( ) f ( x, y) ; ( ) Y f Y( y ) ; ( 3) f X Y( x y) 8 3 X Y, X Y 9 5 X Y 0 X [ 0, ], YN ( 0, ), X Y, ( X, Y) ( ) XN (, ), YN (, ) ( X, Y)? ( X, Y) f ( x, y) = x + y e- ( + sin xsin y) 3 ( X, Y) 3 ( ) = X+ Y, = X - Y, ( ) (, ) 4 ( B, C) 3, = B+ C 5 X P ( ) ( ), =,,,,, X P = = 6 X, X,, X + X 7 ( X, Y), X = X+ Y, X = X- Y, ( X, X ) ( x, x ) = X Y 3 e - ( x - 4) 3 + ( x - ) 8 ( 344) 9 ( X, Y) G= {( x, y) 0< x, 0< y}, = X Y 30 ( X, Y) X Y - 0 0 03 0 03 0 0 0 EX Y, E ( X+ Y) 3,,, [, ], 3 ( 353) ( 354) 33 60, 40, 0, 34 9, 35 33

3 36 X X cov( X, X ) 37 3 (, Y) X ( X, Y) cov( X, Y), cov( X, Y) 38 7 cov( X, Y), cov( X, X ) 39 ( X, Y) G= {( x, y) x + y }, X Y 40 33 X Y X, Y 4 A B, r A, r B, r A r B V V= 6 6 6 9 x, - x A B P, r P Dr B) ( ) r A r B ; ( ) Dr P ; ( 3) ( 0x ), x Dr P, D r P m in( Dr A, 4 A, B r A r B,, A, B, AB, A B P ( ) A B =,? AB,, A, B? 43 r r f, r r f r - 3% % % 3% 4% 5% 6% 7% r f % 005 005 0045 009 003 006 003 005 5% 005 005 0 05 0075 005 005 005 % 006 004 003 00 00 00 00 0 ( ) ; ( ) r f = 5%, 44 40 E [ Y X= x ] E [ Y X ] 45 ( X, Y) D= {( x, y) x + y x }, E [ X Y] E [ Y X] 46 0, 3,, 0 47 00, 07,, 5,, 95%? 48, ( ),, [ - 05, 05]

3 ( ) 300, 5 ; ( ), 0 09 ( B), 5, 4, 3,, 3, X j, =, ; Yj j, ( ) ( X, X ) ; ( ) X ; ( 3) X = j ( j = 0,,, 3), X ; ( 4) Y ; ( 5) ( X, Y) ( X, Y) 5 9, 9 ( ),, Y, =,,, X, 0 ( ) ( Y, Yj ) ( j ), YYj ; ( ) ( X, X j ) ( j ), X X j ; ( 3) cov ( X, X j ), X X j ; ( 4) X 3 Xb ( 5, p ), Yb ( 5- X, p ), ( ) X= k ( k= 0,,,, 5), Y ( ) ( X, Y) ( 3) E [ Y X ] E [ Y X ] ; ( 4) E Y, DY 4 ( X, Y) P {X= K, Y= S}= N! K! S! ( N - K - S! ) p K p S ( - p - p ) N - K - S ( K 0, S0, K + SN ) N, p, p ( 0, ), p + p < ( ) X b( N, p ), Yb( N, p ) ; ( ) Y= S, X b N - S,, X= - p K p Y b N - K, ; - p ( 3) Y Y= S X 5 Y X, X [, ] p

4 3 ( ) ( X, Y) ; ( ) Y ; ( 3) Y= y X ; ( 4) Y= X ; ( 5) P {YX} 6 ( X, Y) D= {( x, y ) 0y, y x 3- y} ( ) X, Y, XY ; ( ) ( X, Y), X Y ( 3) f Y X ( y x ) f X Y ( x y) ; ( 4) E [ Y X ] E [ X Y] 7 X p ( ), X p ( ) X + X = ( =,, b, + 8 Y =, ( ) ( X, X k= X ) ; ( ) cov ( X, X ), X, X 0, Yk, Y> k, ( k=, ) ), X j 9 ( X, Y) G= {( x, y) 0x, 0y }, U= ( ) ( U, V) ; 0 XY X> Y V= ( ) ( U, V) 0 X Y X > Y 0 ( X, Y) N (,,,, ), E [ X Y] = + ( Y- ), XY= D[ Y X] = E [ ( Y- E [ Y X ] ) X ], Y X D[ Y X ] = E [ Y X ] - ( E [ Y X ] ) X, Y, X, Y, X Y L ( X) = E Y+ X Y Y ( X - E X ) X 3 ( X, Y) D= {( x, y) 0x, 0y}, ( ) P {3X Y}; ( ) Z= min {X, Y}

4, ;,,,,, 43 F t,,, 4,,,,,,,,,,,,, X ( ),

6 4, X, X ( ), X, X, X, 4 ( ) X, ( ) X,, X,,, X,,,,,,, X 3 4,,,, X, ( ), ( ; p ),,, X ( x, x,, x ), x X,,, ( x, x,, x ),, ( X, X,, X ), 4 ( X, X,, X ) X, X, X,

4 7, X, X X X,,,,,,,,,,,,,,,,,, ( X, X,, X ),, ( x, x,, x ),, X, x, ( X, X,, X ) ( x, x,, x ),, R X F ( x ), 4, X ) F ( x, x,, x ) = ( X, X,, F ( x ) ( 4) =, X, f ( x ), f ( x, x,, x ) = f ( x ) ( 4) = f ( x ) f ( x, x,, x ) X, p ( x ) = P {X = x }, x X, p ( x, x,, x ) = P {X = x, X = x,, X = x } = = p ( x ),

8 4 p ( x ) p ( x, x,, x ) 4 X, X N (, ), f ( x, x,, x ) = = = exp - exp - x - = ( x - ) ( 43) 4 X, p ( 0< p < ), P {X = }= p, P {X = 0}= - p ( X, X,, X ) P {X =, X =,, X = }= p s ( - p ) s ( 44) k( k) 0, s= + + +, p ( ) ( ),,, X, X = ;, X = 0, X p p, 43 X, ( X, X,, X ), P {X =, X =,, X = }= P {X = k} k= = k= k k! e- = k( k), s= + + + s!!! e-, ( 45),,,, ;,,,, ( 43) ( 44) ( 45), ( x, x,, x ) (,,, ),, (, ; p

4 9 ),, ( 43) ( 45) 5, X ( X, X,, X ), X,,,,,,,,, 4, ( X, X,, X ),,,,,,, 43 ( X, X,, X ) X, 44 X, EX = 5, D X =, ( X, X,, X ) X, S = X + X + + X, X = S, S X ( X, X,, X ) U = ( X - 5), U, U,,,

0 4,, ( X, X,, X ) X, X, X = ( X + X + + X ) S 0 = ( X - X ), = S 0, S = - S 0 = - S = ( X - X ),, S,, S S 0 S 0,, S S 0 X, S E S =, ES 0, S,, 3, S, S = 4 - = ( X - X ) A k = X k, k, = A k k, 5

43 B k = ( X - X ) k, k, = B k k,,,,, 6 ( X, X,, X ) X, X ( ) X ( ) X ( ), ( X ( ), X ( ),, X ( ) ), X ( ), X ( ) X ( ), X ( ) - X ( ),,, ( X, X,, X ) X, ( ), U ( X, X,, X ; ),,,, 45 X N (, 0), 0,, ( X, X,, X ) X, U= ( X - ) 0 U,, U N ( 0, ) U, 43, X ( X, X,, X ), ( )

4 ( ), F t,,, 44 X F ( x ), ( 0< < ), F P {X > F }= ( 46) - F ( F ) = F ( F ) = - ( 47) F X ( ) F ( x ), F ( x ), F F = F - ( - ) ( 48) X, f ( x ), F + f ( x ) dx = ( 49) F 4 ( 4),, x x = F, N ( 0, ) u, u - 0( u ) = ( 40) 0( u ) = - ( 4) 4,,,,,,, ( 4), u, (, y

43 3! ), 45 X, F ( x ),, T P { X > T }= ( 4) F ( T ) - F ( - T ) = - ( 43) T X,, ( ) F ( x ) 4 43, ( 43), F ( T ) = - - F ( T ) = ( 44) ( 45) T = F / ( 46) 43 46 = 005, 005 0( u 005) = - 005= 095, u 0 05= 645 005 u 00 5, 0( u 0 05) = - 005= 0975

4 4 u 0 05= 96, X N ( 0, ), X f ( x ) = x - e - x, x > 0 ( 47) 4 X, X,, X, X N ( 0, ), =,,,, ( a ) = + 0 ( x ; ) = X = X + X + + X x a - e - x dx ( a > 0) ( ) x - e - x, x > 0 ( 48) ( 47), =, ( 48) = k, ( 48), = X + X + + X k, = X k+, = X + Y ( 47), z > 0, f ( x ) = f ( y ) = k k x k - e - x, x > 0; y - e - y, y > 0,,, z f ( z ) = = 0 f ( z - y ) f ( y ) dy k+ k z 0 ( z - y) k - e - ( z - y) y - - e y dy = e - z k+ k z 0 ( z - y) k - y - dy = e - z z k+ - k+ k z 0 - y z k - y z - d y z

43 5 = e - z z k+ - k+ k 0 ( - t) k - t - dt t = y z = k, k+ k z k+ - e - z = k+ k + z k+ - e - z ( a, b) = x a - ( - x ) b- dx, 0, ( a, b) = ( a ) ( b) ( a + b), = k+, ( 48) 46, ( 48), X ( ) 3,, x = -,, x, x = -,,, =, x = 0, ; =, x = 0, y, x ) 44 =, 4, 0, 0, 46, 4 ( ) X ( m), Y ( ), X Y, ( X + Y) ( m+ ( ) X ( ), E [ X ] =, D [ X ] = X, X,, X m+, ( ) X ( m), 46 4, X X + X + + X m, Y X m+ + 44 X m + + + X m +, X Y

6 4, X + Y X + X + + X m +, X + Y ( m+ ) ( ) X, X,, X, X ( ) X X + X + + X, E X = EX = = E X = = D X = =, E [ X 4 ] = 3 ( ( A) 9 ), D[ X ] = E [ X 4 ] - ( E [ X ] ) = 3- = X, X,, X,, D [ X ] = D = X = = D[ X ] =, ( x ; ), 3 45 ( 50) ( ),, X ( ), P {X > ( ) }= P {X < - ( ) }= ( x ; ),,,,, P ( {X < - ( ) }{X > ( ) }) = P { - ( ) < X < ( ) }= - X ( 0), = 005, P {X < 3940}= P {X > 8307}= 005, P {347X 0483}= 095 ( > 45 > 50),, F, X ( m), Y ( ), X Y, Z= ( X / m) ( Y/ ) = X m Y, ( 49) Z X Y, X, X,, X m + F

43 7 43 Z ( 49), Z f ( x ; m, ) = m, m m x m - + m x - ( m+ ), x > 0 ( 40), X ( m), Y ( ), 46, X Y z > 0, f X ( x ) = m m x m - e - x, x > 0; f Y ( y ) = y - e - y, y > 0 Z 0= X Y, Z= m Z 0 X Y,, 3, Z 0 f Z 0 ( z ) = f X ( yz ) f Y( y) ydy 0 = m + m ( y z) m - e - yz y - e - y ydy 0 = z m - m + m y m + - e - y ( z + ) dy 0 = z m - ( + z) - m + m 0 t m+ - e - t dt t = y ( z + ) = m + m z m - ( + z ) - m + = m, Z= m f Z( z ) = m f Z 0 z m - ( + z) - m+ Z 0, z > 0, Z m z = m, m m z m - + m z - 47 X m, m+

8 4 F, X F ( m, ), ( 40), 43, X F ( m, ), X - F (, m) F, m3, x = m- m + x <,,,,, m, x, 45 F F,, 4 45F F F ( m, ), X F ( m, ), P {X > F ( m, ) }= P {X < F - ( m, ) }= X, f ( x ; m, ), F, P ( {X < F - P {F - ( m, ) }{X > F ( m, ) }= ( m, ) < X < F ( m, ) }= -, X F ( 5, 0), 4 YF ( 0, 5),, P {X > 333}= 005, P {X > 44}= 005 P {Y> 474}= 005, P {Y> 66}= 005 4 F, X F ( m, ),, ( 4), F 095( m, ) = F ( m, ) = X - F (, m) F - (, m), ( 4) F 005(, m), F 0 97 5( m, ) =, X F ( 5, 0), P X < 474 = 005, F 005(, m)

43 9 P X 44 = 095 66 t F, t, F,, t X N ( 0, ), Y ( ), X Y T = X Y/, ( 4) Y ( ), ( 4), T ( 4) T = X Y/ F (, ) t F ( 4) T 44 ( 4) T t( x ; ) =, + x - +, - < x < ( 43), T ( 4), X ; X,, T, f T ( t) f T ( t) T T, T,, t > 0, f T ( t) = f T ( t) ( ( B) 7 ) F = X Y/, F F (, ), 43, F f F ( x ) =, T = x - + x X Y/ = F - +, x > 0 ( A ) 37, t> 0,

30 4, t> 0, f T ( t) = tf F ( t ) = t, =, f T ( t) = f T ( t) =, t - + t + t + t, x > 0, ( 43) f T( t), x < 0, ( 43) 48 X t, X t ( ), ( 43) t, y, x = 0 46, = t 46t, - + - - + x 46 =, 5, 0,, t +, t, lim + x - + = e - x 46,, t x, t 5 t t ( ) t,, t( ) = - t - ( ), ( 44), X t( ), P {X > t( ) }= P {X < - t ( ) }= t,, P { X > t / ( ) }= t, X t( 8), = 005, t ( 8) = 860, t ( 8) = 306,

44 3 P {X > 860}= P {X < - 860}= P { X > 306}= 005,, t, t ( ) u, u 44,, ( ), ( ), ( ), ;,,,,,,, ;,,, ( F t ),, 4 X N (, ), ( X, X,, X ), X S, ( ) X N, ; - ( ) S ( - ) ; ( 3) X S, ( ), ( ) ( 3),,, ( B) 46,

3 4,, ( ),, ( 55 ) ( 6 ) ( ) 4 ( X, X,, X ) X N (, ), X S, ( ) U = X - N ( 0, ) ; / - ( ) S ( - ) ; ( 3) T = X - S/ t( - ) ( ) 4 ( ) ( ) 4 T = U - S / -, 4 U - S, ( ) ( ) 43, T ( - ), 4,,,,,, 4,, T ; -, S,, U,,, T U, ( ), ( 55 ) ( 63 )

44 33 43 X N (, ) YN (, ) ( X, X,, X ) X, X S ( Y, Y,, Y ) Y, Y S S S S, ( ) U = S = - + - S + ( X - Y) - ( - ) N ( 0, ) ; + S ( ) F = F ( -, - ) ; S ( 3) = =, T = - + - S ( X - Y) - ( - ) t( + - ) S + ( ) 4 ( ) X N,, YN, X Y, X Y, X - YN = - U= ( ) 4 ( ) -, +, 0= +, ( X - Y) - N ( 0, ) 0 - S ( - ), - ( - ) X Y,, 43 F = - S / ( - ) F ( -, - ) - S / ( - )

34 4 ( 3) = +, = =, 0= /, ( ), U = ( X - Y) - N ( 0, ) / ( ), 4, V= {( - ) S + ( - ) S } ( + - ) x, y, s> 0, s> 0, P {X x, Yy, S s, S s} = P {X x, S s}p {Yy, S s}( X Y ) = P {X x }P {S s}p {Yy}P {S s}( 4), X, Y, S S X Y U S S V, 44 T = U V + - ( X - Y) - t( + - ) V + - = S S/ = U V + - t( + - ) 43,,, = =, ( ), ( X - Y) - ( a) U = / N ( 0, ), = -, = + ( ) ( b) F = S F ( -, - ) S,, X Y, X Y,

44 35, 43,, = - r= /,, r,,, T ; r, F, r,,, U, T U 4, U T, F ( x ) X, F ( x ) X, C( F ) F ( x ), lim F ( x ) = F ( x ), x C( F ), X X, X d X F ( x ) d F ( x ) 45 X, X d X, Y P, X Y d X, ( ( B) 8 ), 44 ( X, X,, X ) X, X, E X =, D X = U = X - /, T = X - S /, X S, d ( ) F U ( x ) 0( x ), ( ) F T ( x ) d 0( x ),

36 4 F U ( x ), F T ( x ) ( x ) U T ( ) U = X - ( X + X + + X ) - = / X, 35 ( ), lim F U ( x ) = lim ( ) ( A) 6 P ( / S) T = P ( U x ) = 0( x ), x R X - S / = U ( ) 45, F T ( x ) d 0( x ) S 44,,, U T,, U ( 54 64 ) ; ( 64 ), T,,,,,, 64 ( A) X ( > 0), ( X, X,, X ), X [ 0, ], ( X, X,, X ), 3 X p ( 0< p < ), ( X, X,, X ), 4 ( x, x,, x ) ( u, u,, u ), u = x - a b ( b0, a )

37 u, x s u s x 5 ( X, X,, X ) X, Ak k X k k, A k P k ( ) 6 ( X, X,, X ) X, S X D[ X] = P ( ) S ( ) ; ( ) ( ) / S P ( ) 7 X F ( x), ( X,, X ), F ( x ) X () X ( ), X ( > 0), X ( ) X ( ) 8, ( ) x e - X dx ; 0 ( ) x 6 e - x dx ; 0 ( 3) x y 4 0 ( + y ) 3 dy 9 X N ( 0, ), E [ X ] = 35( - ) 0 m, ( X, X,, X ), X X 6), X, 6 ( X, X,, Y= ( X + X + X 3) + ( X 4+ X 5+ X 6) c, cy, X ( > 0), ( X, X,, X ), S = X + X + + X 3 u06, u 08, u 09u095 4 X N (, ), 997% X- < 0, = 05,? 5 095( 5), 005( 5), 099( 0) 00 ( 0) 6 F ( 4), F F 095( 4, 6), F 0975( 3, 7) F 099( 5, 5) 7 t t 005 ( 3), t 00 ( 5), t 00 ( 7) t 0005 ( 0)

38 4 ( B) X 04 06 09? X ( X, X,, X ), X X = 4, X 00, X- ( X ) 090 3 X, ( X, X,, X ), X, X - X X j - X ( j ) = - -, 4 X, X,, X, X N ( 0, ), =,, A, AA T = E, A T A, E X= ( X, X,, X ) T, Y= AX= ( Y, Y,, Y) T Y, Y,, Y, YN ( 0, ), =,,, 5 X + X + + X, X N ( 0, ), =,,, X N 0, X = ( X + X + + X ), S = - = ( X - X ), ( - ) S ( - ), X S 6 4 7 X f ( x ), F ( x) g ( x ) G( x) X ( ) F ( x ) + F ( - x) = ; G( x ) = F ( x) -, x 0 ( ) f ( x ) = g ( x ) 8 Y ; X X, X X Y d X ( )

5,,, 5,,, ( > 0),,, ( X,, X ) X, ( x,, x ),,,, h( X,, X ), h( X, X ) h ( x,, x ) h ( X, X ), ( X,, X ) ; h ( x,, x ), ( x,, x ),,,,,,,, r, (,, r), ( X,, X ),, =,,, r g ( ), g ( ) g ( ) g ( ), g ( )

40 5 5 X ( ) f ( x ; ) = ( x > 0), > 0 68, 30, 69, 43, 74, 98, 08,, 5, e- x, X, = E X, X = X x = ( 68+ 30+ + 5) = 77, 9 = x = 77 = X = X, = 68; X ( ) = min {X,, X }, X ( ) = max {x,, x }, = ( X ( ) + X ( ) ), = ( 08+ 5) = 80,,,, 5 = ( X,, X ), E =,, lim E =, 5 ( X,, X ) X, X, ( ) X ; ( ) S ; ( 3), ( X - X ) = ( ) E X = EX =, =,,, E X = E ( ) = = EX = E X =, = X = ( ) D X = DX =, =,,, D X = DX =,

5 4 E S = E [ - ( X - X ) ] = = E { - [ X - ( X ) ] } = = - [ E X - E ( X ) ] = = { - ( + ) - [ D X + ( E X ) ] } = = - ( - ) = = S ( 3) E [ ( X - X ) ] = E ( - = S ) = - E S = - lim E [, = ( X - X ) ] =,, g ( ), g ( ) g ( ) X N (, ), X, ( X ) > 0, E ( X ) E( X ) = D X + ( E X ) = +,,?,, 5, D < D, 53 X, EX =, ( X, X ) X

4 5 = X = X, D = D X < D = DX, 54 X N (, ),, > 0 ( X,, X ) X ( > ) = S = - ( X - X ), = E = E S =, E = E [ = = D X = = ( X - ) = = = ( X - ) = E ( X - EX ) ( - ) S ( - ), = ( X - ) ( ), ( - ) S D [ ] = ( - ), D [ = D = DS = ( - ) D[ ( X - ) ] =, ( - ) S ] = D = D [ ( X - ) ] = ( ) D [ = = 4, 4 -, = ( X - ) ] 3, 53 = ( X,, X ),, > 0, P { - < }= P { - }= 0, lim lim ( )

5 43 55 ( X,, X ) X, DX k, k=,,, X k EX k, k=,,, =, k, Y= X k, Y= X k, Y= P - lim Y= E Y= EX k X k E X k = Y = = X k =, X EX 56 X N (, ), ( X,, X ) S E S =, DS = 4 - ( 54),, > 0, 0P { S - E S }= P { S - } DS = 4 ( - ) 0, S 5?,, ( X,, X ) X, X,, X, P {X = x }= p ( x ; ), ( X,, X ) P {X = x,, X = x }= p ( x ; ),, = ( X,, X ) ( x,, x ) ;, ( x,, x )

44 5,, L ( ), L( ) = p ( x ; ), ( 4) = L ( ) ( x,, x ),, L ( ) X, f ( x ; ), ( X,, X ) f ( x ; ), ( X, X,, X ) ( x, x,, x ) =, ( X,, X ) ( x,, x ) ( x,, x ), L ( ) L ( ) = = f ( x ; ), ( 4), L( ), ( x,, x ), L( ) L ( ) 54 ( x,, x ) = ( x,, x ), L( ) = maxl( ), ( 43) ( x,, x ), ( x,, x ), MLE,,, r, L (,, r ) ( x,, x ) = ( x,, x ), =,,, r, L(,, MLE, =,,, r r ) = max (,, r ) L(,, r), ( 44), MLE, L(,, r) L = 0 lnl = 0, =,, r,, L

5 45 lnl, MLE 57 X N (, ),, - < < +, > 0 ( X,, X ) X, ( x,, x ) X f ( x ;, ) = L (, ) = lnl(, ) = - f ( x ;, ) = = ( ) - exp - exp - = ln ( ) - = ( x - ), ( x - ) ( x - ), lnl(, ) = 0 ( x - ) = 0 = lnl( ( ), ) = 0 - + ( x - ) = 0, 4 0= = A = B= lnl( x = x,, ) 0= ( 0, lnl ( ( ), ) C= lnl(, ) ( ) A < 0 AC- B = ( - 0, ) ( - 0 0 ) = - = = ( x - x ) 0 = 0 (, 0 0 ) ( 0, 0 ) = - ) > 0 4 0 4 0 = 0 = X ( X - X ) =, = x, = = ( x - x ) MLE, MLE MLE,,

46 5 58 X,, > 0 ( X,, X ) X, ( x,, x ) X L ( ) = x = x! e- P ( X = x ) = x x! e-, x = 0,, lnl( ) = - 0= d lnl ( d, x = 0,,, + ( = x ) ln - ln ( = ) = 0, - + = x = 0, x! ), x = x = = X, = x, u = g ( ) = h( u ) g ( ) g ( ) 59 5 5 P {X > 80} E X L( ) = = e- x, ( x > 0, =,, ) lnl ( ) = - ln - n x i= d d ln L = 0 - + x = 0, = 0= x, = X x = 77 P {X > 80}= e - 80 P ( X > 80) X, P ( X > 80) 7 7 = 0353 e - 8 0 e - 80 50 X [ 0, ],, > 0 ( X,,

5 47 X ) X, ( x,, x ), 0x,, x L ( ) = 0, L ( ) = 0 MLE MLE, L( ),, x, =,, max{x,, x }, L( ) = 0, 0 L ( ), = max {x,, x }, L ( ) = max {x,, x }, = max {X,, X } = max{x,, X }, X ( ) = max {X,, X }, X ( ) G( x ) g ( x ), G( x ) = P {X ( ) x } = P {max {X,, X } x } = P {X x,, X x } = P {X x } = = [ F ( x ) ] F ( x ) X 0, x < 0 F ( x ) = x, 0x, x > 0, x < 0 G( x ) = [ F ( x ) ] = ( x ), 0x, x > g ( x ) = x - 0< x < 0, E X ( ) = x g ( x ) dx = x dx = 0 0 +, = X ( )

48 5 + X ( ),,, 53, ;, k= E X k, EX ^ = X, D ^ X = = k= E ( X - EX ) k, A k= ( X - X ) X k, Bk= = = ( X - X ) k, k k ; k k, k= A k, k=,,, k= B k, k=, 3,,,, ME,,,,,,, = g (,, l ;,, s) A k, Bk g k, k = g (,, l;,, s) = g ( A,, A l; B,, B s) ME 5 X N (, ), ( X,, X ) X, = EX, = DX = E^ X = X = DX ^ = ( X - X ) =

53 49, N (, ) 5 50 E X =, = E X, = E ^ X = X, = x 53 X m, p, m, p ( X,, X ), ( x,, x ) ( ) p ; ( ) p q, q= - p ( ) X P ( X = x ) = C x mp x ( - p ) m- x, x = 0,,, m L( p ) = = Cx m p x ( - p ) m- x, x = 0,,, m, lnl( P ) = ( x ) lnp + = dlnl x dp = 0 = - p p 0= x = m = x m [ = ( m - x ) ] ln ( - p ) + ln ( ( m- x ) = = 0, - p, p p = X m E X = mp, p = EX m, E^ X = X, p = X m p ( ) g ( p ) = p q = p - p, p = X m, g ^ ( p ) = g ( p ) p x - p m = = X - m, DX = mp q, g ( p ) = p q = mp DX, X m- X = C x m ), g ^ ( p ) = m( X m ) = g ( p ) ( X - X ) = m ( X ) = ( X - X )

50 5 54 X x - e-, f ( x ;, ) = x 0, x <,, > 0 ( X,, X ) X, x - = t, E X = + D X = + - xf ( x ;, ) dx = + x - x e- dx + = ( t+ ) e - t dt= ( ) + ( ) = + 0 ( x - E X ) f ( x ;, - ) dx = + = ( 3) - ( ) + ( ) = E X = +, = DX D X = = EX - DX E^ X = X, DX ^ = ( X - X ), = ( x - - ) = = ( x - x ), e- x - dx = X - = ( X 0- X ) 54, X N (, ), X, x, = X,,,, ( N eymann ) 934,,, ( -, + )

54 5, > 0 -, P { - < < + }= -, ( -, + ) 55, ( X,, X ) X - ( 0< < ), = ( X,, X ) = ( X,, X ), P { < < }= -, ( 45) (, ) -, -, ( x,, x ), (, ) ( ( x,, x ), ( x,, x ) ) (, ),, 55 - (, ) -, 00 ( X,, X ) 00, (, ) 00, 00, 00( - ), 00 - = 095, 95, 5, (, ), 55 X N (, ),,, ( X,, X ) X - X, X U= X - /,,, U= X - / N ( 0, ) -,,, P < X - / < = -,, - u, u, u,

5 5 P - u < X - / < u = - P X - u < u< X + u = - X - u, X + u - - = 95%, 95% X - 96, X + 96, 95%, 95% 5N ( 0, ) X 96, u,, u= u ( X,, X ; ) u u, -,, P { < u< }= - P {u }= P {u }= (, ) 56 X f ( x ; ) = x e- x > 0 0 x 0 > 0, ( X,, X ) X ( ) X ( ) ; ( ) - ( ) Y= X, Y G( y ) g ( y ), G( y) = P {Y y } = P X y

54 53 F ( x ) = G( y ) = g ( y ) = = P X y = F y - e - x, x > 0 0, x 0 - e- y, y> 0 0, y 0 y e-, y > 0 0, y 0 Y ( ), X ( ), =,, = = X = X ( ) X = = X X ( ) ( ) X 5, X u= X, ( ) u, u = X ( ) -, P - ( ) < X < ( ) = - P X < ( ) < X ( ) = - - X ( ), X ( ) -,,, 57 X p, p, 0< p < ( X,, X ) p

54 5 p X X u = X - E X DX / = X - p 50 u N ( 0, ), p ( - p ) / -, P X - p p ( - p ) / < u - P {ap + bp + c< 0}-, a = + ( u ), b= - X - ( u ), c= ( X ) a > 0 ap + bp + c< 0 p < p < p, p = a ( - b- b - 4ac) p= a ( - b+ b - 4ac), -, p p P ( p < p < p ) -, ( p, p ) p - ( X - u X ( - X ) /, X + u X ( - X ) / ) X - p N ( 0, ), p ( - p ) / X - p X p, N ( 0, ) X ( - X ) /, P P {X - u X - p X ( - X ) / < u -, X ( - X ) / < p < X + u, X ( - X ) / }- 58 400, 08 p 95% 0, p, p p -

54 55 ( ( X - u X ( - X ) /, X + u X ( - X ) / ) ) = 400, X = 08 400 = 07, = 005, u 005 = 96, ( 07-96 07073/ 400, 07 + 96 07073/ 400 ) ( 03, 03) (, ) (, + ) ( -, ),,,,, 56 X, ( X,, X ) X -, = ( X,, X ) P ( < ) = -, (, + ) -, ; = ( X,, X ) P ( < ) = -, ( -, ) -, 59 ( 55) ( ), - U = X - / - P P > X - u = - - X - u P X - / > - u = - X - / < u = - 53N ( 0, )

56 5 P < X + u = - - X + u 50 ( mg ), 8 x = 06 X N (, 3), 095 X + u = 005, u 005 = 65, x = 06, = 3, = 8, 06+ 65 3 8 = 4 55,,, t F N ( 0, ) X N (, ), - < < +, > 0 ( X,, X ) X ( ) 55 - X - u, X + u, X u, -, > 0, > 0, + =, P u - < X - / < u = - X - u, X - u -, = = ( 54) 54 N ( 0, ) X u

5557 5 X N (, ),, = 450 36, x = 5440 95% = 36, = 450, 450 36 = 47 95% = 005, u 005 ( x - 47, x + 47) ( 593, 5587) = 96, u = 96, ( 593, 5587), 95%, 95%, 95% 5 X N (, ),, = 4 ( X,, X ) ( ) = 6, 09 095 ( ) 90%? ( 3) 95%? ( ), = X + u - X - u = u - = 90%, = 65 6 = 65, - = 95% = 96 6 = 96 ( ) u, ( u ), - = 90%, ( 65) 44, 44, 90% ( 3) - = 95%, 6,, - ;, -,, ;,,,,

58 5 ( ) S, T = X - S /, 4 T = X - t( - ) -, S/ P {- t ( - ) < X - S/ < t ( - }= -, P X - t ( - ) S < < X + t ( - ) S = -,, - X - t ( - ) S, X + t ( - ) S X t ( - ) S, ( - ) S S =, ( - ) S 4 = ( - ) -, P P - - ( - ) S ( - ) < < ( - ) = -, ( - ) S ( - ) S < < ( - ) ( - ) = -, - = ( - ) S ( - ), ( - ) S - ( - ) ( X - X ) ( - ), = ( X - X ) - ( - ), - ( - ) S ( - ), ( - ) S ( - ) -, ( X - ) =

5559 = ( X - ) = = - = ( X - ) ( ), X - ( ) ( X - ) = ( ) - 53, 5, x = 86, s= X N (, ), 90% - X t ( - ) s = 0, s=, = 5, t 0 ( 5- ) = 709 t ( - ) s = 709 5 = 406, 90% ( 86406) ( 889, 90) ( - ) S - ( - ), ( - ) S - ( - ) 0 0 ( 5- ) = 364, ( 5- ) = 385, - 4 364 = 974, 90% ( 974, 580) 4 385 = 580, X N (, ), YN (, ), - < < +, > 0, =, ( X,, X ) ( Y,, Y ) X Y, X = Y = = = X, S = Y, S = - - = - ( ), X Y, X - YN = -, + ( X - X ), ( Y- Y)

60 5 ( X - Y) - ( - ), + N ( 0, ) -, P ( X - Y) - ( - ) + - - ( X - Y) - u + < u = - ( 49), ( X - Y) + u ( ) = + = =, S = ( - ) S + ( - ) S, S ES = + - S, T = T t( m+ - ) -, ( X - Y) - ( - ), 43 S + P { T < t ( + - ) }= -, - - ( X - Y) - t ( + - ) ) S +, ( X - Y) + t ( + - ) ) S + ( 40) 54A, B 9, 9 A, 0 B, ( kg ) A 00, 05, 0, 5, 0, 98, 05, 6, ; B 0, 00, 05, 5,, 07, 06,, 0, 9 A X N (, ), B YN (, ),,, - 90% ( X - Y) t ( + - ) S + = 0, = 9, = 0 t 0 ( 7) = 7396, x = 09, y = 06, S = 550 8, S = 606 9,

556 S = ( - ) S + ( - ) S = 68, S= 846, + - ( 09-06) - 7396846 9 + 0 = - 359, ( 09-06) + 7396846 9 + 0 = 959-90% ( - 359, 959), S S, F = S / 43 F F ( -, - ) - S /, P ( F - ( -, - ) < F < F ( -, - ) ) = - P F ( -, - ) S S < < F - ( -, - ) S S = - / - F S ( -, - ) S, F - S ( -, - ) S, ( 4) F - ( -, - ) = F ( -, - ) 55, 3 5, S = 75 S = 00 (, ) / 95% X N (, ), YN / - ( F ( -, - ) ) - S F ( -, - ) S = 005, = 3, = 5, S F 005 ( 30, 4) =, F 005 ( 4, 30) = 4 75 = 034, 4 75 = 6, 00 00 ( 034, 6), 5 5 S

6 5 5 ( ) X - N ( 0, ) / X - t ( - ) S/ Xu Xt ( - ) S X - u X + u X - t ( - ) S X + t ( - ) S = ( X - ) X ( ) = ( X - ) = ( ), ( X - ) - ( ) ( X - ) / ( ) = ( X - ) / - ( ) = ( - ) S ( - ) ( - ) S ( - ), ( - ) S ( - ) S / ( - ) - ( - ) ( - ) S / - ( - ) 5 ( ), ( X - Y) - ( - ) / + / - N ( 0, ), ( X - Y) - ( - ) S / + / t( + - ) - S = = ( - ) S + ( - ) S + - ( X - Y) u / + / ( X - Y) t ( + - ) S / + / u u ( X- Y) - / + / ( X- Y) + / + / ( X- Y) - t ( + - ) S / + / ( X- Y) + t ( + - ) S / + /

63 ( ), S / / S / -, - ) F( F ( -, - ) S S F ( -, - ) S S, ( -, - ) S S, F F ( -, - ) S S ( A) X,, ( X,, X ) X = = ( X - ) X N (, ), ( X,, X ) X, C - C = ( X + - X ) 3 X p 0, ( X,, X ) X p = ( X ) -, B = - B p = ( X - X) 4 ( X, X ) N (, ) ( ) = 3 X + 3 X ; ( ) = 3 4 X + 4 X, 5 ( X,, X ) X ( ) X f ( x ; ) = ( ) X P {X= k}= p ( - p ) k- - x, 0x,, > 0; 0,, k=,,, p, 0< p < ; ( 3) X P {X= }= p, P {X= 0}= - p, p, 0< p < ; ( 4) X,, > 0 f ( x;, ) = ( ) x - e - x, x > 0, 0,

64 5 6 3,, 0, 5, 4, 3,, 0, 7,, 0,,, > 0 a, ( ) ; ( ) P ( X= 0) 7 X [ a, b], a, b, b> a, ( X,, X ) b 8 ( X,, X ) X, E X =, DX =,, 0, > 0, C= 9 X f ( x; ) = e - (x - ), x 0,,, - < < + ( X,, X ) X 0 X [ 0, U= X ( ) - ],, > 0 ( X,, X ), X ( ) = max {X,, X } p, 00, 0 p 95% X ( kg) N (, 0 ) 6, x = 8 95% 3 X N (, 9), ( X,, X ) -, ( ) = 0 ; ( ) = 00 4 0, x = 70, s= 90 X N (, ) 98% 5 X( ) N (, ) 5, x = 54, S = 06, 95% 6 XN (, ), 5, s= ( ) 95% 7, 0 8 x = 400 ( ) y = 50 ( ), S = 5 ( ) S = 64 ( ) N (, ) N (, ),,, - 95% 8 N (, ) N (, ),,,, / 95%

65 ( B) ( X,, X ) X, E X=, DX =, 0, > 0 = ( + ) kx k k= X [, + ], ( X,, X ) X, 3 X, f ( x; ) = e- x, x > 0 0, x0, > 0 ( X,, X ) X, = 50 x = 40 99%

6,,,,,, 44, 6 63 64,, 65,,,, 6,, 6, X F ( x ; ),,,,, X N (, ), =, ; =,

6 67,,,,, 0, H 0 0, X ( x, x, x ),,, ;,, H, = - 0,, H 0 0 H, ( ) 6, 00kg,,,, = 5k g,, ( kg) 993, 987, 005, 0, 983, 997, 995, 0, 005?, X N (, ), = ( 5), = R, H 0= 00 H 00 0, ;,,,?,,

68 6?,, ( X, X,, X ) A, H 0, P ( A H 0),, ( x, x, x ), A, H 0, ;, H 0, A H 0,, ( x, x,, x ) A,, H 0, H 0,,,,, ( X, X,, X ) A, H 0,? X N (, 0), 0 H 0= 0 H 0 0 ( X, X,, X ) X 44 4, U = X X - N ( 0, ), 0/ U,, X - P 0/ > u =, ( 6) u, x - 0 C= ( x, x,, x ) 0/ > u, ( 6)

6 69 x ( x, x,, x ), 0 0, C, A A = {( X, X, X ) C} ( 63),, P ( A H 0) = P {( X, X,, X } C H 0} X - 0 = P 0/ > u H 0 =,, H 0, ( 6) ( 63) A,, ( X, X,, X ) ( x, x,, x ) x - 0 > u 0, A, H 0, H ;, H 0 6, 6, 0= 5, 0= 00, = 9 = 005, u = 96, 6 x9998, u= x - 0 9998-00 = 0/ 5 9-005 u = 005< u / = 96, H 0, X F ( x ; ),, H 0 0 H ( 0 = ) ( 64) ( X, X,, X ), ( X, X,, X ) A, C(, C ( 6) ), A A = {( X, X,, X ) C} ( 65),, H 0, ( 65) A, P ( A H 0) = P {( X, X,, X }C H 0), ( 66),

70 6,, 00, 005 00 C ( 66),, ( x, x,, x ) C, H 0, H ; ( x, x,, x ) C, H 0 C H 0, ;, C C H 0,, C ( 66), U ( X, X,, X ; ) ( U = ( X - ) / 0), ( U N ( 0, ) ),, H 0 C ( ( 6) C, H 0 0 ), ( 64) X ( X, X,, X ), U ( X, X,, X ; ), ;,, H 0, C, P ( ( X, X,, X ) C H 0) ; 3 ( x, x,, x ), C C, H 0 ( H ) ; C, H 0, C,,, U ( X, X,, X ; ),,,,,,,,,,,,

6 7,,,, H 0, H 0,, ;, H 0,, H 0,, ( 66),, H 0,,,,,,,,,,,,,,,, ;, 6 X N (, ),,,,,

7 6 X N (, 0), 0, ( X, X,, X ),, U= X - ( 67) 0/ X, U, H 0= 0 H 0, 0,, C C= ( X, X,, X ) ; x - 0 0/ > u ( 68) H 0 0 H > 0, H 0 ( 67),,, X - P 0/ > u = ( 69) u, C= ( x, x,, x ) H 0 0 X - 0 0/ X - 0/ x - 0 0/ > u ( 60) X - 0 0/ > u X - 0/ > u ( 6)

P {( X, X,, X ) C H 0}= P P = X - 0 0/ > u H 0 X - 0/ > u H 0 6, X N (, 40 000),, 500,, 5, 575 500,, 5? H 0 500 H > 500, 0= 500, 0= 00, = 5 = 005, u = 645, x = 575, u= x - 0 575-500 = 5= 875 0/ 00 u= 875> u = 645, H 0, H, 3 H 0 0 H < 0,, x C= ( x, x,, x ) - 0 0/ < - u ( 6),,,,,, U,, U X N 6 73 (, ),, ( X, X,, X ) X,, ( 67)

74 6,, T = X - ( 63) S/, X, S ( ( 67) 0) 44 4, T - t t,,,,, 0 H 0= 0 H 0, C= ( x, x,, x ) x - 0 s/ > t ( - ) ( 64) t / ( - ) - t, x s H 0 0 H > 0, C= ( x, x,, x ) x - 0 s/ > t ( - ) ( 65) t ( - ) - t 3 H 0 0 H < 0, C= ( x, x,, x ) x - 0 s/ < - t ( - ) ( 66),, t, T,,, T 63 ( 6) 6, T 0= 00, = 9 005

5, t / ( - ) = t 00 5 ( 8) = 306 6 x = 9998, s = 9 8 ( x - x = s=, t= x - 0 = s, t = 0495< 306= t 0 05 ) = 9 8 = 9998-00 x - 9x = 469, 9 = - 0495 ( 8), H 0, 64 5 6, 9, 8,, 0, 6, 5,? ( = 005) H 05 H < 5 T 0= 5, = 6 = 005, 5 t ( - ) = t 00 5 ( 5) = 05 6 x = 0, s = 0 t= x - 0 0-5 = s/ 0, t= - 6-05= - t 00 5 6 = - 6 ( 5), H 0, X N (, ),,,, ( X, X,, X ), 6 75 W = - S ( 67)

76 6 S 44 4, W -,,, 0 H 0 = 0 H 0, ( 67) -, - P S < - ( - ) - S > ( - ) =, ( 68) - ( - ) ( - ) - - C= ( x, x,, x ) - s < - ( - ) - s > 0 ( - ) ( 69) 0 s S H 0 0 H > 0 H 0, P - S > ( - ) =, ( 60) ( - ) ( - ), C= ( x, x,, x ) H 0 0,, - S - 0 - s > ( - ) ( 6) 0 S - S > ( - ) - S > ( - ) ( 6) 0

P { ( X, X,, X ) C H 0} = P - S > ( - ) H 0 0 P - S > ( - ) H 0 = 3 H 0 0 H < 0 H 0, C= ( x, x,, x ) - s < - ( - ) ( 63) 0 - ( - ) ( - ) -,,,,,,, = 0, 0, ( 67) W?,, ( ( B), ) 65 X,, 7, ( ) 44, 405, 4357, 4394, 436, 487, 4683 0? ( = 005) H 0 = 0 H 0, = 005, 3, - ( - ) = 09 75 ( 6) = 37, ( - ) = 00 5 ( 6) = 4449 7 7 = 436, ( - ) s = x 6 77 ( x - x ) = 006 =

78 6, w = ( - ) s = 006 = 6789 0 0 w = 6789> 4449= 00 5 ( 6),, 66,, 64 (k g ) 64 0, ( ) 578, 57, 570, 568, 57, 570, 57, 596, 584, 570 x = 575, s = 7574,,,,,, H 0 64 H > 64, 0 = 64, = 0 = 005, 3, ( - ) = 0 05 ( 9) = 699, w = - s = 97574 = 065699= 00 5 ( 9) 0 64, H 0,,,, 6 63,, X N (, ), YN (, ), = -, r = /,, r, ( X, X,, X ) X, ( Y, Y,,

63 79 Y ) Y, X Y, S = - ( X - X ), S = = - ( Y- Y ) ( 64) =, S, 6- X N (, 0),, 0, 0 H 0 = 0 H 0 ( ) H 0 0 H > 0 U= U 0= X - 0 / X - 0 0 / H 0 0 H < 0 X N (, ),, 0 H 0 = 0 H 0 T = X - S/ T 0= X - 0 S/ - t

80 6 ( ) X N (, ),, 0 H 0 0 H > 0 T = X - S/ T 0 = X - 0 S/ - t H 0 0 H < 0 X N (, ), 0 ( 0 H 0 = 0 > 0) H 0 H 0 0 H > 0 W= - S W 0 = - 0 S - H 0 0 H < 0

63 8 S = + - ( - ) S + ( - ) S ( 65) ( x, x,, x ) ( y, y,, y ), x, y, s, s s,, r,,,,, 0= +, ( 66), U= ( X - Y ) - 0 ( 67) 44 43, U,, U U,, H 0 = 0 H 0 H 0 = H, U ( 0= 0 ), ( 68), x - y C= ( x, x,, x ; y, y,, y ) 0 > u

8 6 H 0 0 H > 0 H 0 H >, U ( 0= 0 ), ( 60), C= ( x, x,, x ; y, y,, y ) x - y > u 0 3 H 0 0 H < 0 H 0 H <, U ( 0= 0 ), ( 6), C= ( x, x,, x u ; y, y,, y ) x - y < - u 0 67 A X N (, 95 ), B YN (, 0 ), 00 75, 80 0 = 005,?, = 95, = 0, = 00, = 75, ( 66) 0= 68 x = 80, y = 0, u = x - y 80-0 = = - 38 0 68 = 005,, u = u 00 5= 96, u = 38> 96, H 0( H 0 ), 68, 50, 743, 50, 704,, 53 6? ( = 005) X Y, X N (, 53 ), YN (, 6 ), = 53, = 6, = = 50, ( 66) 0= 43 x =

63 83 743, y = 704, u = x - y 743-704 = = 34 0 43 = 005, u = u 0 05 = 645, u = 34> 645= u 0 05, H 0( H 0 ), = =,, = =,, ( 66) 0,, ( 67) U,, 0,, = =, ( 64) ( 65) S ( ( A ), ), T = ( X - Y) - ( 68) S + 44 43, T + - t, T t( + - ), T,, T ( 0= 0 ), ( 64), x - y C= ( x, x,, x ; y, y,, y ) s + > t ( + - ), T ( 0= 0 ), ( 65), C= ( x, x,, x ; y, y,, y ) x - y > t ( + - ) s + 3

84 6 3, T ( 0= 0 ), ( 66), x ( x, x,, x ; y, y,, y ) - y > - t ( + - ) C= s + 69 5 4948475354339575646444554440 46404754336433848544834 7? ( = 005) X N (, ) YN (, ),, = 5, =, = + = 7 x = 476, y = 44; s= 5 ( - ) s = ( x - x ) = 4696, ( - ) s = ( y - y ) = 4 = = + - {( - ) s + ( - ) s }= t= s x - y = + 476-44 = 566 594 5 + ( 4696+ 4) = 594 5 = 005, 5, t ( - ) = t 005 ( 5) = 060 t = 566060= t 0 0 5( 5), H 0( H 0 ), 60,,, 7 ( kg) 55, 47, 5, 60, 43, 53, 54, 8 4, 49, 46, 34, 38, 54, 38, 5 X N (, ) YN (, ),,,

63 85 = 7, = 8, = + = 5, x = 5, s = 0003; y = 44, s = 0005 ( 65) s= 00646, t= S x - y = + 5-44 00646 7 + 8 = 395 = 005, 5, t ( - ) = t 005( 3) = 77 t= 395> 77= t 0 05( 3), H 0( H 0 ), r r F,, r= / r, r,,,,, S S F = r S S ( 69) X Y ( ( 64) ) 44 43, -, - F, F F ( -, - ) r H 0 r = H r H 0 = H, ( 69) F F ( -, - ), F ( 43 ) P r S < F S - ( -, - ) r S > F S ( -, - ) =, ( 630)

86 6 F - ( -, - ) F ( -, - ) - - - F, C= ( x, y ) s < F s - ( -, - ) s s > F ( -, - ) ( 63) x = ( x, x,, x ), y= ( y, y,, y ) P {( X, X,, X ; Y, Y,, Y ) C H 0} = P S < F - S = ( -, - ) S S > F ( -, - ) H 0 ( ( 63) ) H 0 r H r > H 0 H >, H 0, F P r S > F ( -, - ) = ( 63) S C= ( x, x,, x ; y, y, y ) s > F ( -, - ) ( 633) s H 0 r r S S S, S S S > F ( -, - ) r P {( X, X,, X ; Y, Y,, Y}C H 0} = P S S P r = > F ( -, - ) H 0 S S > F ( -, - ) H 0 S > F ( -, - ), ( 634) S

63 87 3 H 0 r H r < F - H 0 H <, C= ( x, x,, x ; y, y,, y ) s < F s - ( -, - ) ( -, - ) - ( - ) ( - ) F,, F, F, r F 6( 60) 60, X Y, X N (, ), YN (, )?, 60, f = s = 0003 s 0005 = 06078, = 7, = 8 = 005 4, F ( -, - ) = F 0 05( 6, 7) = 5, F - F - ( -, - ) = ( -, - ) = F 005( 7, 6) = 570 = 0754 F ( -, - ) = 0754f = 060785= F ( -, - ), H 0( H 0 ), 6, 7 9, ( ) 5, 45, 55, 48, 5, 56, 47 5, 50, 48, 5, 50, 49, 5, 48, 53? ( = 005) X Y, X N (, ), YN (, )

88 6 = 7, = 9, x = 5057, s = 0745; y = 5033, s = 00438 f = s = 0745 s 00438 = 3984 = 005, 4 F ( -, - ) = F 005( 6, 8) = 358, f = 3984> 358= F 005( 6, 8) H 0( H 0 ),, 6, X Y, ( 69) F?,,, 6 ( ( B) ) 6- X N (, ), Y N (, ), H, H 0 = H 0 H > U= X - Y, 0 0 = + H 0 H <

63 89 X N (, ), YN (, ),,, H 0 = H H 0 H > T = S X - Y + S= ( - ) S + ( - ) S + -, H 0 H < X N (, ), Y N (, ),,,, H 0 = H H 0 H > F = S S, H 0 H <

90 64,,,, ( ),,, 4 4, p,,,, X p ( 0< p < ), P ( X = ) = p = - P ( X = 0) ( X, X,, X ) X, N = X + X + + X ( 635), N N p, P ( N = k) = C k p k ( - p ) - k = b( k;, p ), k= 0,,, ( 636) p,, p 0 ( 0< p 0< ) p H 0p = p 0 ( 636), p, p, H 0, N p 0,, k k, j k= max j b( ;, p 0) = 0 ( 637)

64 9, k = min j b( ;, p 0) = j ( 638) C= ( x, x,, x ) ( x + + x ) k ( x + + x ) k ( 639) P {( X, X,, X ) C H 0} = P {N k N k H 0} k = b (,, p 0) + = 0 = k b( ;, p 0) + = p, p H 0 p p 0 H 0, C = {( x, x,, x ) x + x + + x k }, k k = min j = j 3 H 0 p p 0 H 0, b( ;, p 0) ( 640) C = {( x, x,, x ) x + x + + x kk }, k j k = max j b( ;, p 0) ( 64) = 0 63 p 0= 007 0, H 0 p p 0 = 005 0, = 0 0 b( ; 0, 007) = 0047 = 005, = 4 ( 640) k = 4, C = {( x, x,, x ) x + x + + x 4} 0, 4, 007

9, ( 636) ( 638) ( 640) ( 64) k, k, k k,, X p ( 0< p < ), E [ X ] = p, D [ X ] = p ( - p ) ( X, X,, X ) X, U = X - p p ( - p ) = N - p p ( - p ) N = X + X + + X, X ( 64) 44 44,, U, 6 U p p,,, 63, p 0,, u u 6-3 p U H 0 H H 0 p = p 0 H p p 0 U= N - p 0 p 0 ( - p 0 ) U > u, H 0; - u Uu, H 0 H 0 p p 0 H p > p 0 U> u, H 0; Uu, H 0 H 0 p p 0 U < - u, H 0 ; H p < p 0 U- u, H 0, p 0 ( 64) U p,, U, 64, 00, 59, 005 50% H 0 p 05 = 00 U U

64 93 u = 59-50 000505 = 8> u 0 05= 64 H 0, 50% 6 ; p, p,,,, X, E [ X ] = X X,, D [ X ] = ( X, X,, X ) X, T = X - S /, ( 643) X S 44 44,, T, T,,, U, T 64, 0,, u u 6-4 U H 0 H H 0 H = 0 0 T = X - 0 S/ T > u, H 0; T u, H 0; H 0 H 0 > 0 T > u, H 0; T u, H 0; H 0 H 0 < 0 T < - u, H 0; T - u, H 0, 0 ( 643) T

94,, T, 65,,, ( Weibull),,,, 00, x = 784 ( kh ), s= 5 ( kh),? ( = 00) 365,, 3654= 750, 75kh H 0 75 H > 75 U 0= 75, = 00 t = x - 0 784-75 = 0= 56 s/ 5 = 00,, u = u 00 = 33 t = 56> u 00= 33, H 0, 65,,,, X r, X r,,, r,, p + p + + p r=, p = P {X = }, =,,, r ( 644) H 0 p = p, 0, =,,, r ( 645) p, 0, =,,, r, r, p, 0+ p, 0+ + p r, 0=

65 95, r =,, X 0,,, r3,,, H 0,, ( X, X,, X ) X 6,,,, N, N,, p r N + N + + N r= ( 646), ( N, N,, N r), p, p,, P {N =, N =,, N r = r}=!!! r p p p r r ( 647), =,,, r,, + + + r= r p, p,, p r, H 0 ( ( 645) ) r, r =,,, 64, ;,, r ),,, N ( =,,, ( N - p ) p ( - p ), =,,, r,, ( K P ear so), r = i= ( N - p ) ( 648) p

96, r, ;,,,, ( 648), 6 ( K Pearson ), ( 648) r-, [ 3], p 34 [ 4], p 697,, ( 646), r N - p, N - p,, N r - p r,, p r r r r ( N - p ) = N - = = = p = - = 0 r - 6 ( 645) H 0 ( ), 6 p, p,, H 0, 0 p p, 0, 0, r ( N - p, 0) 0= ( 649) = p, 0,, H 0, 6 0 r -, H 0 ( ) N p, H 0,,,,, P { 0> ( r- ) H 0} ( 650) ( r - ) r -

65 97,,, r N, N,, N r, 0 ( 649), C= {(,,, r) 0> ( r - ) } ( 65) P {( N, N,, N r ) C H 0}= P ( 0> ( r- ) H 0) A= {( N, N,, N r) C} H 0, ( 3), ( n, n,, n r) C ( 0> ( r- ) ), H 0 ( ( 645) ),, H 0;, H 0 66, 8 5, 6, 3, 6, ( = 005) X, X = ;, X = X = 3 p = def P ( X = ), =,, 3 H 0 p = 3, =,, 3 r = 3; p, 0= 3, =,, 3; = 6, = 3, 3= 6, = 5 3 ( - p, 0) 0= = 5 [ ( 6-5) + ( 3-5) + ( 6-5) ] = = p, 0 = 005, 3 ( r - ) = 00 5( ) = 599 0= < ( r - ) = 599,,, 636 / 3,

98,,,,, X F ( x ) ( X, X,, X ) X F 0( x ) H 0 F ( x ) = F 0( x ) ( 65) X,, r- a < a < < a r -, ( -, + ) r I = ( -, a], I = ( a, a ],, I r= ( a r-, ), r Y Y= XI, =,,, r X F ( x ) Y,,, p = P ( Y= ) = P ( X I ) = F ( a ), p = P ( Y= ) = P ( X I ) = F ( a) - F ( a - ), =,, r - p r = P ( Y= r) = P ( X I r) = - F ( ar - ) ( 653) F 0( x ), ( 653) r p, 0, =,,, r, H 0 Y p, 0= F 0( a ), p, 0= F 0( a ) - F 0( a - ), =,, r - p r, 0= - F 0( ar - ) ( 654), H 0 ( 65) ( 653) ( 654), Y H 0 p = p, 0, =,,, r ( 655)

65 99, Y,, r ( N - p, 0) 0= = p, 0 ( 656) N I i, r p, 0( =,, r ) ( 654) 0 ( ),, H 0 H 0 H 0 H 0 H 0 F ( a ) = F 0( a ), =,,, r, H 0 H 0 ( ( 65) ) H 0, H 0 ;,, H 0, H 0, ;, H 0, H 0, H 0,, ( ), [ 3], p 46 [ 5], p4 a r -, r a, a,,, ( 654) p, 0 p, 05 ( =,,, r ), X,,, 67,,,,,, 00,, 5

00 3 4 5 43 3 5 6 5 00 ( = 005) p X,,, X p, p = P {X = }= p ( - p ) -, =,,, p = H 0 p =, =,, p + = P {X > }= ( - p ), = 0,,,, 5, H 0 p =, =,, 3, 4; p 5 =, r = 5; p, 0=, = 4,, 3, 4, p 5, 0= ; = 43, = 3, 3= 5, 4= 6, 5= 5, = 00 r ( - p, 0) 0= = 3 = p, 0 = 005, 3 ( r- ) = 0 05( 4) = 9488 0= 3< 005( 4) = 9488,,, ( 68) F 0( x ),, F 0 4 H 0 F ( x ) = F 0( x ;,,, t ), ( 657), t,,, t, H 0

65 0 {N (, ) - < < +, > 0} H 0 ( 657),, ( 654) p, 0, p, 0(,,, t) = F 0( a ;,,, t) - F 0( a - ;,,, t) ( 658),,, t, F 0, p, 0 (,,, t), ( 649) 0,, ( ), =,,, t, ( 658) ( 657), ( 658), p, 0 p, 0= F 0( a ;,,, t ) - F 0( a- ;,,, t ), =,,, r ( 659) F 0( a 0;,,, t ) = 0, F 0( ar;,,, t) = ( ) 0 0= v = ( N - p, 0) p, 0 ( 660) ( R A F isher), H 0, F 0( x ;,,, t),, 0 r- t-, [ 8], p 98, p 30305 ( 3),,, r N, N,, N r,, 0 ( 660),,, C= {( n, n,, n r) 0> ( r- t - ) } ( r - t- ) r- t-,,, t+, r t+ 68 000 50,, x = 360 ( g), s 0= 4655 ( g ), 450g, 3700g 3 ( 450, 700], ( 700, 950], ( 950, 300], ( 300, 3450], ( 3450, 3700] 5 7 0

0 ( = 005) X H 0 X N (, ) = 50, r = 7; =, = 5, 3= 7, 4=, 5= 0, 6= 7= 3, = X, = S 0,, H 0 F 0( a ;, ai- 360 ) = 0 4655 ai= 450+ ( i- )50, i=,,, 6, i=,,, 6 p, 0= F 0( ai;, ) - F 0( ai- ;, ), i=,,, 7 F 0( a 0;, ) = 0, F 0( a 7,, ) =, 0, 6-5 0 3 4 5 6 7 p, 0 0063 0098 065 00 096 045 03 p, 0 35 490 85 050 980 75 65 5 7 0 3 ( - p, 0) 33 000 563 50 0040 4063 993 ( - p, 0) / p, 0 040 000 089 04 0004 940 63, 0= 7 = ( - p, 0) p, 0, 3 = 438 ( r - t- ) = 0 05( 4) = 9488( r = 7, t= ) 0= 438< 0 ( r - t- ) = 9488, H 0,

65 03, X, Y H 0 X Y ( 66),, ( ), X Y r q A, A,, A r B, B,, Bq p, j = P {X A, YBj }, =,,, r; j =,,, q; q p 0= P {X A }= p, j, =,,, r ; j = r p 0 j = P {YB j }= p, j, j =,,, q; =, X Y, X Y H 0 p j = p p 0j, =,,, r ; j =,,, q ( 66) ( r + q) p ( =,,, r ) p 0j ( j =,,, q) ( rq) p j ( =,,, r; j =,,, q), ( ) ( X, Y) ( X, Y), ( X, Y),, ( X, Y), N j AB j q r N = j, =, j =,, r; N j = r = q N = N j = = j = r = N j, j =,,, q q = j = N j, ( 3) ( r + q) p ( =,,, r) p j ( j =,,, q), 0 p p j, p p 0 j

04 6-6 B j q ( N 0) N N N q N, A N N N q N, r N r N r N rq N r, ( N 0j ) N N N q p = N, =,,, r ; p j = N j, j =,,, q, ( 660), H 0 ( 4) H 0 r q p = = j = p j =, r q 0= ( N j - N N j / ) = j = N N j 0 ( 663) ( r + q) ( r+ q- ) rq- ( r + q- ) - = ( r- ) ( q- ),, H 0,, ( 663) 0 ( r - ) ( q- ), 0, H 0, 69, 339 50 43 6 05 % 3 34 97% 56 83 339 65% ( = 00) A A ; B B, r = q=, = 43, = 6, = 3, =

05 ; = 05, = 34; = 56, = 83, = 339 ( 663) 0= 339 = j = ( j - j / 339) = 748 j = 00,, ( ( r- ) ( q- ) ) = 00( ) = 6635, 0= 748> ( ( r - ) ( q- ) ) = 6635 H 0, ( A) X ( > 0),, ( X, X,, X 0) H 0 = 0 H = 0, C= {( x, x,, x 0 ) x + x + + x 0 = 0} X N (, 9),, ( X, X,, X 5) H 0 = 0 H 0, C= {( x, x,, x 5) x - 0 c} c, 005 3, 536 ( ), 6, 0, 57, 578, 584, 593, 607, 73, 564, 589, 475, 495,,? ( = 005) 4,, 3h ( h) N ( 08, 8 ), ( h) 67, 0, 4, 0, 7, 50, 34? ( = 005) 5 0050cm,, 0, 0053, 0003, ( ) = 005, ( ) = 00, 6, 0 600 ( kg/ cm ),, 0, 0533, 064, 0688, 057, 0793, 079, 0600, 0683, 07, 0570 = 005,

06 7, N ( 80000, 4000 ) ( kg/ cm ), 00, 79600 ( ) = 005 8 X,, 7, 44, 405, 4357, 4394, 436, 487, 4683 0 ( = 005) 9,, 000g, 5g,, 0, ( ) 00, 030, 968, 994, 04, 998, 976, 98, 950, 048 ( = 005)? 0, 56, 74, 4653 / mm 3, 5480 / m m 3 ; 46 / m m 3, 490 / mm 3, ( = 00) ( = 00),,, = =,, ( 64) ( 65) S 3, 8h 90%, 00,, 60, 8h, ( = 00)? 4 7, 6 H, H / 6 ( = 005)? 5, 50min, 5, 0 3 4 5 9 68 8 0 ( = 00)? 6 0, 0,,,, 9,, 800, 0,,,, 9

07 0 3 4 5 6 7 8 9 74 9 83 79 80 73 77 75 76 9 0? ( = 005) 7 99, 8 30 4 78 8 90? ( = 005) 8 3 ABC 4, 00, A B C 4 3 59 4 4 0 48 7 8 3 38 7 9 9 55 9 64 44 00 ( = 00)? ( B) X N ( 0, ), 0, ( X, X,, X ) X, ( 67) W,, XN (, ) YN (, ),, r = / ( X, X,, X ) ( Y, Y,, Y )

08 X Y ( 69) F,, ( ) H 0 r = ; ( ) H 0 r; ( 3) H 0 r, 3,,,, 4000,, 4000 0,, 3 0 388 64 66 3 4 4000 ( ) ; ( ) H 0 = 0 H 00 4, a, b, c, A, B, C,, a ( b, c) A ( B, C) 50 0 4 4 3 50 ( = 005) X ) 5 X p ( 0< p < ), p ( X, X,, X H 0 p = p 0 H p p 0 ( p 0 )

09 Q= ( X- p ) p ( - p ) ( ),, ; ( ), Q ( r = ) ( 3) 4040, = 048 Q ( = 005) 6 A= x - 4 < x, 4 =,, 3; A4= x 3 4 < x <, 80, A ( =,, 3, 4) 6, 8, 0 36 0, X? p ( x ) = x, 0< x < ; 0,

7 t, ( A nalysis of V ariance, A NOV A ) ( R A Fish er ) 90 7 7 MBA, MBA 0 7 7- MBA ( ) A 96 83 5 33 8 3 0 46 4 0 938 A 78 36 79 4 05 87 6 9 90 A 3 3 4 6 83 08 63 97 3 7 89 995 A4 95 06 8 75 7 7 45 0 068 7,, A 4,, MBA?,,,

7,,,,,, ;,,, 7,,,,,,,, 7 7 MBA, MBA, MBM 7, MBA, A, =,, 3, 4,, B, j =,, 3, 4, 5,, MBA,,,,, ;,,, MBA,

7, 0, 5, 4, 0,,,,, ( B j, A ), j =,, 5, =,,, 4,, 7 7- MBA ( ) B B B 3 B 4 B 5 A 96 83 5 68 0 46 0 73 5 75 63 90 45 68 80 A 78 8 80 65 57 49 30 98 70 36 38 5 8 A3 3 40 6 4 95 80 8 73 50 74 60 85 63 6 0 A4 95 06 8 64 36 0 75 80 94 50 88 5 78 7 7 7 8 A r, A, A,, A r A X, A X ( =,,, r)

7 3, X N (, ) ( =,,, r), r,, r, r, A, A X ;, r, A X 73 A X X X 3 X 4 X k X A X X X 3 X 4 X k X Ar X r X r X r3 X r 4 X rk X r, H 0 = = = r ( 7), r k, X j, =,,, r ; j =,,, k, X j, j =,,, k A 73,, X j N (, ), j =,,, k, X j = + j, j =,,, k ( 7) A j j j, j N ( 0, ), ( 73) ( 7), ( 73), ( 7),, ( 7) = r r =, ( 74) = - =,, r, ( 75) X j = + + j =,,, r ; j =,,, k ( 7 ) r ( 7), ; A, A ( effect ), r = = 0 ( 76)

4 7, ( 7) ( 7 ), ( 73), ( 76) H 0 = 0, =,,, r ( 77) r ; H, r,0, ( 7 ), ( 73), ( 76),, r, X j N ( +, ), j =,,, k, f ( x j ) = ( ) - exp[ - ( x j - - ) ] ( 78) ( ; r; ) L ( ; r; ) - rk ) = (, exp [ - r k = j = ( x j - - ) ) ] ( 79), L r k V( ; r) = ( x j - - ) ( 70) = j =, V,, V V V = - k - = j = r k = j = ( x j - - ) = 0, ( X j - - ) = 0, =,,, r ( 7) ( 76) A r k = rk X j = X, ( 7) = j = = k k X j = X, ( 73) j = = - = X - X ( 74),,, ( =,,, r ) X, X A, = r k r k = j = ( X j - - ) = 7, A, A, A 3, A 4 r k rkj = = ( X j - X ) ( 75) 4

7 5, =,, 3, 4 = 938, = 9, 3 = 995, 4= 08, = 97075, = - 0375, = - 04875, 3= - 045, 4= 0575, A 4,,,, ( 7) ( 77) 7, X j, =,, r, j =,, k A, r k Q = ( X j - X ) ( 76) = j =, A, A A, A,, Q r k Q = [ ( X j - X ) + ( X - X ) ] r k = j = = ( X j - X ) + = j = r k = j = ( X j - X ) r ( X - X ) + k( X - X ) ( 77) = ( 7) ( 73) X X r k = j =, ( 77) Q = r k ( X j - X ) ( X - X ) = 0 ( 78) ( X j - X ) + = j = r = k( X - X ) = Q + Q ( 79) ( 79) Q,, A, ; Q,, Q A X, ( 7) ( 77), Q,

6 7 Q, X j, A ; Q, A, ( 7) ( 77) Q / Q > c ( c ) j = X j - N ( 0, ), Q ( rk- ), r k = k k j = Q = ( j - ) Q Q = j = r k = = j = ( j - ) k = k( - ) j = Q ( r ( k- ) ), j, = r r =, Q ( r - ), Q Q ( ),, 48, F =, H 0, F = Q / ( r - ) F ( r -, r ( k- ) ) ( 7) Q / r ( k- ) Q = Q = Q r k = j = r k = j = ( + j - ) ( j - ) r = k( + - ) = Q / ( r - ) Q / ( r - ) = F ( r - ; r ( k- ) ) ( 7) Q / r( k- ) Q / r ( k- ), C= {x F = Q / ( r - ) Q / r ( k- ) F ( r -, r( k- ) ) } ( 73) C= {x Q r- Q r ( k- ) F ( r -, r( k- ) ) } F ( r -, r( k- ) ) ( r - ) r( k- ) F

7 7, ( 7) ( 77) ;, ( 79) Q Q ; 3 ( 7) F ; 4, F F ( r-, r ( k- ) ), F, 74 74 F r Q= k( X - X) r- Q/ ( r - ) = Q = r k = j = Q= r k = j = ( X j - X ) r ( k- ) Q/ r( k- ) ( X j - X ) rk- Q / ( r - ) Q / r ( k- ), 74 75 8 Microsoft Excel 97 757 SS df MS F P Value F crit 73475 3 4385 005 088466 86665 46993 36 5834 4430775 39 75 74, SS df MS, F 74 F, 75 P Value F E xcel 97,, 999

8 7 F,, F > F P Valne< P Value 088466 3 36 F, 088466 005 005, P, 75, F c rit,, = 005; F Cr it = F 005 ( 3, 36) = 86665, F < F,,,,,,, 73 A B, A r, A, A, A r; B s, B, B, B s A, B, ( A, B j ) ( =,,, r ; j =,,, s) rs, ( A, B j ) X j,, X j N ( j, ) =,,, r ; j =,,, s rs, 7 7, A B X j, A B, A B X j,, = = s s j ( =,,, r ) ( 74) j = j = r r r s rs = = j = j ( j =,,, s) ( 75) j = r r = = s s j ( 76) j = A, r ( =,,, r ), A r,

73 9 A ;, j Bj, s j ( j =,,, s) B,, X = - ( =,,, r ) ( 77) j = j - ( j =,,, s) ( 78), A, j j,, r = = 0 ( 79) s j = 0 ( 730) j =, A H 0 = 0 ( =,,, r ) ( 73) B H 0 j = 0, ( j =,,, s) ( 73) A, B? r j = 0, j = j - - - j, ( =,,, r ; j =,,, s) ( 733) j = + + j, ( =,,, r; j =,,, s) ( 734), A B, B B j o, A A o o, A ox, B Bj ( j j o), A o A B 7,, r j ( =,,, r; j =,,, s) A A B B j r j = 0( =,,, r; j =,,, s),, ( 734) ;,, j r j 0; A, B X 7,,, 76 ( A, B j ) =,, rj =,, s,,

0 7 X j, X j = + + j + j, ( =,,, r ; j =,,, s) ; j N ( 0, ) ; r = s = j = j = 0 ( 735), A B X j ( 735) ( 73) ( 73),, ( 735),,, j,, j ( =,,, r ; j =,,, s) = rs r s = j = X j = def X, ( 736) 76 B A B B B s A X X X s X A X X X s X Ar X r X r X rs X r X X X s X = s s X def j = X, ( 737) j = j = r r = X j = def X j, ( 738) = X - X, ( 739) j = X j - X ( 740), ( 73) ( 73), Q r = r s Q = ( X j - X ) s = j = = j = [ ( X j - X - X j + X ) + ( X - X ) + ( X j - X ) ]

73 Q, Q, Q 3 r = r = s = j = ( X j - X - X j + X ) + s s ( X - X ) + j = r ( X j - X ) = Q + Q + Q 3, ( 74) Q A, A ; Q 3 B, B ; Q, A B, H 0( 73) H 0( 73) Q / Q c ( 74) Q 3/ Q c ( 743) CA, CB, c, c Q, ( ( r - ) ( s- ) ) ; H 0, Q / ( r - ) ; H 0, Q 3/ ( s- ), H 0 H 0, Q/ ( rs- ), H 0, H 0, Q, Q, Q 3, H 0, Q Q ; H 0, Q 3 Q, H 0 F A = Q / ( r- ) Q / ( r - ) ( s- ) = Q ( Q s- ) ( 744) r - ( r - ) ( s- ) F ; H 0, F B = s- ( r - ) ( s- ) F Q 3/ ( s- ) Q / ( r- ) ( s- ) = Q 3 ( r- ) ( 745) Q,, F A ( r -, ( r - ) ( s- ) ) F B ( s-, ( r - ) ( s- ) ), H 0 CA = {x F A = H 0 Q / ( r - ) Q/ ( r - ) ( s- ) F A ( r -, ( r - ) ( s- ) ) } CA = {x Q / Q s- F A ( r -, ( r- ) ( s- ) ) }

7 CB= {x F B = Q 3/ ( s- ) Q / ( r - ) ( s- ) F B ( s-, ( r- ) ( s- ) ) } CB = {x Q 3/ Q r - F B ( s-, ( r - ) ( s- ) ) }, 77 77 ( ) F A Q= r r- s( X - X ) = Q r - F A= Q/ ( r - ) Q/ ( r- ) ( s- ) B Q3= s s- r ( X j - X ) j = Q 3 s- F B= Q 3/ ( s- ) Q / ( r- ) ( s- ) Q= r s = j = ( X j - X ( r - ) ( s- ) Q ( r - ) ( s- ) - X j + X) r s Q= = j = ( X j - X) r s-, 7,,,, 76, 78 77 F A F B, F A F B, F A F A F B F B, Excel97, 78, 79, 005

73 3 7-8MBA ( ) B B B 3 B 4 B 5 A 96 68 75 45 788 A 78 8 57 98 38 704 A 3 3 4 8 74 63 748 A 4 95 64 75 5 8 955 6375 8 975 495 763 79 A, B (, 78 ) ; 75 798 SS df MS F P Value F Cr it 3366 3 034 0856 34903 6007 4 505 407 00346 3596 499 3577 0650 9 79, F, F P, 005, H 0, MBA, H 0,,, ( A, B j ) ( =,,, r ; j =,,, s),, ( A, Bj ), m ( m),, X j k, ( =,,, r ; j =,,, s; k=,,, m) AB X, ( 73) ( 73), X

4 7, H 03 r j = 0 ( =,,, r; j =,,, s ) ( 746) r j ( 733),,,,, j =,,, r j =,,, s X j = X X j = m X j k, m k = = s s j = X j, r r X j, = X = r r X = s = s j = X j = r s rs X j = = j = r rsm X j k = j = k= s m ( 747) A, B,, j, =,,, r ; j =,,, s = X -X, =,, r, j = X j - X, j =,, s, r j = X j - X -X j + X, =,, r, j =,, s ( 748) Q = r s m = j = k= r = s m = j = k= ( X j k - X ) [ ( X j k - X j ) + ( X j - X - X j + X ) + ( X - X ) + ( X j - X ) ] r = s m = j = k= r s ( X j k - X j ) + m ( X j - X - X j + X ) = j = r + sm ( X - X ) =

73 5 s + rm ( X j - X ) j = = Q + Q + Q3 + Q4, ( 749) Q QQ 3Q 4 Q 3Q 4 Q Q A B,,, H 0H 0H 0 3 W W W3, ccc3 Q 3/ Qc, Q 4/ Qc, Q / Qc3, Q ( r s ( m- ) ) ( 750) H 0 H 0 H 0 3 ( ( 73) ), ( ( 73) ), Q 3 ( r - ) ; ( 75) Q 4 ( s- ) ; ( ( 746) ), Q ( ( r - ) ( s- ) ) H 0 H 0H 03, Q ( rsm- ) Q Q Q 3Q 4, H 0, Q 3/ ( r - ) F A = Q / rs ( m- ) r - rs ( m- ) F,, F A ( r -, rs ( m- ) ), H 0 CA CA = {x F A = Q 3/ ( r- ) Q / rs( m- ) F A ( r -, rs( m- ) ) },

6 7 CA = {x Q 3/ Q r- rs( m- ) F A ( r -, rs( m- ) ) } H 0 F B, H 0, F B= Q 4/ ( s- ) Q / rs( m- ), s- rs( m- ) F,, H 0 CB CB= {x F B = Q 4/ ( s- ) F B ( s-, rs( m- ) ) } Q / rs( m- ) CB = {x Q 4/ Q s- rs( m- ) F B ( s-, r s( m- ) ) H 0 3, F A B= Q / ( r - ) ( s- ) Q/ rs( m- ) ( r - ) ( s- ) rs( m- ) F, F A B,, H 0 3 CA B CA B = {x F A B= Q / ( r - ) ( s- ) Q / r s( m- ) F A B ( ( r - ) ( s-, rs( m- ) ) } ( r- ) ( s- ) CA B = {x Q / Q F A B ( ( r - ) ( s- ), rs( m- ) ) } rs( m- ) H 0 3,, m( m), 70, F A F B F A B, F, Excel97 7 7, 005 7 70, A, B,,

73 7 70 ( m ) F A B Q3= r r- sm( X - X) = Q4= rm( X j - X) s- j = Q= A B r s m( X j - X ( r - ) ( s- ) = j = - X j + X) Q = r s m = j = l= ( X j k- X j ) rs( m- ) Q 3 r - Q 4 s- Q ( r - ) ( s- ) Q rs( m- ) F A= Q 3 / ( r- ) Q / rs( m- ) F B= Q 4 / ( s- ) Q / rs( m- ) F AB= Q / ( r - ) ( s- ) Q / rs( m- ) r s m Q= = j = k= ( X j k- X) rsm- 77 SS df MS F P Value F Crit 34067 3 40689 088793 0455956 838746 353573 4 3383933 636 00487 60597 5336 9633 074764 069756 00346 54333 40 8558 799473 59 7 F F P 005, 005 H 0H 03, MBA,, F B = 636 > F B= 60597, F B P 00487< 005,, 005 H 0, MBA ( 754)

8 7 = X - X = 950-836= 4 = X - X = 86-836= - 00 3= X 3- X = 938-836= 0 4= X 4- X = 94-836= 088 5= X 5- X = 55-836= - 85, B MBA,,, B MBA ( A) 6, ( cm ) 8 34 33 389 408 74 503 479 368 60 507 558 406 43 445 437 485 464, ( = 005), 5 500, A 9 0 9 B 9 0 3 C 4 6 3 5 7 ( = 005) 3 5, 35, 7, 80 8 86 30 3 7 30 4 76 5 5 85 0 005,?

9 4 A 5 5 7 9 9 6 6 8 A 7 7 7 5 5 5 9 A 3 5 7 6 8 7 6 8 8 8 A 4 8 0 5 6 7 7 7 7 = 005 ( )? ( )? ( 3)? ( B) 7 H 0 = = 3= 4 t t = 005, H 0? ( 74) =,, r A =,, r 3 ( 75) 4, A, F t

8,,,,,,,, 8,,, s v t s= vt, v t, s,, 8, 8 8 ( ) 3 4 5 6 7 8 9 0 35 60 5 30 35 40 5 0 50 45 440 50 380 475 385 55 450 365 540 50

8 3 X Y 8, X Y,,,,,,, 5 3 7,,, ( X, Y), =,,, 0, 8 8,,,,, 8,,, ( X, Y) ( X, Y), =,,, X Y, r = = ( X - X ) ( Y- Y) [ ( X - X ) = =, r ( Y- Y) ] / ( 8)? 8, y,, 0+ x,, y = 0+ x +

3 8, Y X Y= 0+ X + ( 8) Y, X,,, E = 0, X, E[ Y X ] = 0+ X ( 8) 0+ X 0 0+ X, ( 8) Y 8 X Y,,,, Y ( ) X,, X m,,,, f ( X,, X m) X,, X m Y X,, X m Y= f ( X, X,, X m) + ( 83),, 0, y = f ( x, x,, x m ), X, X,, X m,, ; Y,, ( ),, f,, f 0,,, K,,,,, f ( X ) = 0+ X ( 84), 0,

8 33,, X, X, 8 x X ( x, Y) =,,,, Y, Y,, Y,, Y Yj ( j, =,,, ) ( 84) Y= 0+ x +, =,,,, ( 85) ( =,,, ), E ( ) = 0 ( =,,, ) ( =,,, ) Y, x, E ( ) = D ( ) =, =,,, ( 86), ( H omoscedasticity), ( H ) x ; ( H ) x ; ( H 3) YY, Y ; ( H 4) ( 85), D ( ) = =,,, Y, Y, Y, D ( Y) =, =,,, ( 87) ( 85) 0, 0 Y = 0+ x ( 88) Y= 0+ x 8 8 ( x, Y) ( x, Y ), =,,,, ( 88)

34 8 Q( 0, ) = = a ( Y- 0- x ) ( 89) Q( 0, ) = minq( 0, ) ( 80), 0 Q( 0, ),, 0, Q 0 = 0 0, = = = = Q = 0 0, = ( Y- 0- x ) = 0; ( Y- 0- x ) x = 0, 0 Y = 0+ x ; x = = x, Y= Y = = x Y= x 0+ x = = 0, ( 8) ( 8) ( 83), 0= Y- x, ( 84) = = x Y- x Y x - x ( 85) =, o ( 89) Q 0 0 ( Least Squares Est imators, LSE) 0 ( 88) Y = Y+ ( x - x ) ( 86) ( x, Y) ( 0, 0), l xx = = ( x - x ) ( x - x ) = = l xy = ( x - x ) ( Y- Y) = = = x - x, x Y- x Y,

8 35 = l xy l x x ( 87) x ( =, ), Y = 0+ x ( 88) Y, e= Y- Y, =,,,, ( 89) e ( =,,, ) a b,, = ( 8) e= 0 e ( ax + b) = 0 ( 80) = 8, ( 84) ( 85) 0 0= 309576, = 4067736, 0 Y = 309576+ 4067736x 0,? ( 85) ( x - x ) = Y, ( 8) = Y, Y,, Y, E (, ) = = ( x - x ) E ( Y) l xx l x x ( x - x ) = ( = l 0+ x ) xx = 0 = ( x - x ) + l x x = ( x - x ) x =, ( 8) Y, Y,, Y, D ( ) = = ( x - x ) l x x l xx D ( Y) = ( 83) l x x

36 8, ( 84) ( 8) 0 Y, Y,, Y 0= - ( x - x ) x l x x =, 0 0 D( 0) = 0 Y ( 84) E ( 0) = 0 ( 85) + x l x x ( 86) Y, Y,, Y,, 0 ( H ) ( H 4), ( 85) 0 0 ( ( 84) ( 85) ) ( Best Linear U nbiased Estimat ors, BLU E) 0 0, ( 88) Y E ( Y x ), E ( Y ) = E( 0) + E ( ) x = 0+ x = E ( Y x ), Y E ( Y x ) cov( 0, ) = cov( Y- x, ) = cov( Y, ) - cov( x, ) = cov = = = = = - =, Y x l xx ( x - x ) l x x Y, = ( x - x ) l x x ( x - x ) Y - x cov(, ) l x x D ( Y) - x D ( ) - x l x x, ( 87) D ( Y ) = D ( 0) + D ( ) x + cov( 0, ) x = + ( x - x ) l x x ( 88)

837 8,, x Y, ( 84) ( 85), 0 0 9,?,, x Y,, E ( Y x ) x, x Y, E ( Y x ) x, = 0;, = 0,, x Y x Y H 0 = 0 ( 89), ( 85) ( H 5), N ( 0, ), =,,,, ( H 5), Y, Y,, Y, YN ( 0 + x, ), =,,,, 0, Y, 0N ( 0, N ( + x l x x ), ( 830), ) ( 83) l x x Y N ( 0+ x, + ( x - x ) l x x ) ( 83) ( H ) ( H 5), x Y, Y Y,, x Y ( 89)

38 8, SST = S ST = [ ( Y- Y ) + ( Y - Y) ] = = ( Y- Y = ) + = ( Y- Y) = l Y Y ( 833) = - Y) + ( Y ( 80), ( Y- Y ) ( Y - Y), = = = = = 0, S ST = SS E = = ( Y- Y = e ( x - x ) = e( x - x ) ( Y- Y = ) + = = ( Y - Y) ( Y- Y ) ( Y - Y) = SSE + S SR ( 834) ) = =, SSR = e ( Y - Y) = ( x - x ) = l x x,, E ( SSR) = E ( l x x ) = l x xe ( ) = l x x[ D ( ) + ( E ( ) ) ] = l x x + l x x = + l x x ( 835), H 0, E ( SS R) =, SSR ; H 0, 0, SS E, SSE = = ( Y- Y ) E( SSR) >, x =

= = ( 0+ x + - 0- x ) = [ ( 0-0) + ( - ) x + ] = = [ ( 0-0) + x ( - ) + + ( 0-0) + = x( - ) + x ( 0-0) ( - ) ] = ( 0-0) + ( - + x ( - =, ( 84) ( 8), ) = ) E [ ( 0-0) ] = - E ( 0 ) 839 = - = x + + = - ( x j - x ) x l x x = - - ( x - x ) x l x x E [ x ( - ) ] = - x E( ) = + = ( 0-0) x ( 0-0) ( - ), ; R( Yj ) ( x j - x ) = - x E ( Yj ) j = l x x = - ( x - x ) x l x x E ( SSE ) = D ( - 0) + D ( = + = = + x - ) = ( x - x ) x l x x x cov( 0, ) l x x - = x + = D ( ) - = x + + - + l x x = ( x - x ) x l xx - = = - + ( x + x + = x x l x x = ( x - x ) x l xx ( x - x ) x l x x ( x - x ) x =

- = ( x - x ) x - = x x ) l x x = ( - ) ( 836), SSE, E SSE - =, ( 837) SSE - S SR S SE, H 0 ( 89) S SR S SE c ( 838), c 0, SSR N 40 8 YN ( 0+ x, ( 0, ), =,,, N ) ( =,,, ), H 0 Y S ST ( - ) ( 839),, H 0 N l xx S SR, e= Y- Y ( =,, 0,, l xx = l xx ( ) ( 840), ) Y, Y,, Y e ( =,,, ), SSE, ( 8), SSE -,, H 0, S SR SS E, F H 0 -F F = SSR S SE / ( - ) ( 84),, F (, - ), F (, - ), H 0 C= {( x, y) F F (, - ) } ( 84)

84 C= {( x, y) S SR (, - ) F } ( 843) S SE - H 0, 8 8 F SSR = = ( Y - Y) MSR= SSR SSE = ( Y- Y ) = - MSE = SSE - SST = ( Y- Y) - = F = MSR MSE, 8 ( 8), F, 8 83 8 Excel97, 005 838 df S S M S F Significance F 30643 30643 9347 000708793 8 435357 79496 9 37560 83 Excel97, 83 Significance F F P, 000708793 005, 005 H 0 = 0,,, 8 F 005 F 00 5(, 8) = 53, F, F = 9347> F 005 (, 8),

4 8 ( 834) SSR SSE,, S SR S ST, SS E, x Y, R x Y R R, R = S SR SS T = - SS E SST ( 844) R ( 845),, R,, x Y,, R =, SSE = 0,,, x Y SS R= l x x ( 87), R = l xx l x Y = l YY l X X l Y Y = = = ( x - x ) ( Y- Y) ( x - x ) = ( Y- Y) ( 846) R x Y Y( x ), R = r ( 847), r R,,, x Y, x Y, H 0 = 0 r d ( 848), d, r = R = SSR S ST = + SSE SSR = + - F = F F + ( - ) ( 849)

843 > F, F F (, - ), r > F ( 848) d d = F (, - ) (, - ) + ( - ) F F (, - ) (, - ) + ( - ) ( 950) r d H 0, d r, r ( - ) = d,, 8, r = 0786034, = 005, r 00 5 ( 8) = 063, r r 005 ( 8), H 0 = 0, F t 0, 0 0, N - / l xx,, l xx N ( 0, ) ( 85) SSE ( - ), S SE SSR= l x x, SSE, t, - SSE / l ( - ) xx = - t( - ) ( 85)

44 8 = SSE - l x x ( 853), t l xx, H 0 = 0 t = t( - ) ( 854),, t ( - ) - t, C= {( x, y) t > t ( - ) } ( 855) H 0 = 0, 0-0 t( - ) ( 856) 0 = SSE 0 - + x l x x ( 857) 0, 0 + x l xx, H 0 0= 0, t 0= 0 t( - ) ( 858) 0 t 0 > t ( - ) ( 859), H 0 0= 0 8, 0= 309576, = 4067736, ( 857) ( 853) = 43404, = 3054 t t 0 = 736 t = 35964, 0 8 t 005 306, t 0 t, 005 H 0 0= 0 H 0 = 0 Excel97,, t 0 t P, t 0 t P 989

83 45 0-5 000708793, P 005,, H 0 = 0 F t F t P 000708793,,, 83,, x x 0, x 0 Y0 8, 0, 55,, Y0, 0N ( 0, ) Y0= 0+ x 0+ 0 ( 860) 0,,, Y0, Y0N ( 0+ x 0, ), Y 0,, E ( Y0 x 0) = 0+ x 0, E ( [ Y0- E( Y0 x 0) ] ) = min c E ( [ Y 0- c] ) ( 86),, 0,,, Y = 0+ x E ( Y0 x 0) = 0+ x 0 ; Y0 E ( Y0 x 0) Y 0= 0+ x 0, ( 83) Y 0N 0+ x 0, + ( x 0- x ) l x x ( 86)

46 8, Y 0 E ( Y0 x 0) E ( Y0 x 0), ( 86), Y 0- ( 0+ x 0) Y Y 0 0, Y 0 Y = 0 + ( x 0- x ) N ( 0, ), ( 863) l x x ( 864) SSE 0 SSE - ( - ) SSE Y 0, Y 0- ( 0+ x 0) t( - ) ( 865) Y 0 Y = SSE 0 - + ( x 0- x ) l x x ( 866) Y 0 ( 865), - t t ( - ), E ( Y0 x 0) = 0+ x 0 - Y 0- Y t ( - ) E ( Y0 x 0) Y 0+ Y t ( - ) ( 867) 0 0, Y t ( - ), x 0, 0, x 0= x, x 0, Y0, Y0, N ( 0+ x 0, ) ; Y 0= + x 0, ( 86) Y 0 Y0?, Y P ( Y0- Y 0 < ) = -, 0, ( Y 0-, Y 0+ ) Y0 - Y0 Y, Y,, Y, Y0 Y 0, Y0- Y 0, E( Y0- Y 0) = 0; D( Y0- Y 0) = D ( Y0) + D ( Y 0) = + + ( x 0- x ) l xx = Y - Y 0 0

83 47 Y0- Y 0 Y0 - Y 0 N ( 0, ) ( 868), SS E Y0 Y 0 Y0- Y 0 t( - ), ( 869) Y - Y 0 Y - Y 0 0 Y - Y = + 0 0 + ( x 0- x ) 0 l x x SS E - ( 870) ( 869) P { Y0- Y 0 Y - Y t ( - ) }= - ( 87) 0 Y0 - ( Y 0- Y - Y 0 t ( - ), Y 0+ Y - Y 0 t ( - ) ), ( 87) = Y - 0 t ( - ) 0 Y 0 ( 870),, x 0, ( x 0), x 0 x,,, x 0= x, x, 83, x,, x l x x,,,, x 0 x, SS E Y - Y 0 0 -, - t N ( 0, ) 83Y0, u t ( - ),, Y 0-0 ( Y 0- u, Y 0+ u ) ( 873) SSE = - 8 55

48 8 x 0= 55, Y0 Y0 Y 0= 309576+ 4067736= 53353, E ( Y0) 53353, Y0 E ( Y 0 ) 95% ( 53353-57966, 53353 + 59966 ) = ( 475963, 5905447), Y0 ( 55) = 3396, 95% ( 40034, 646496), x 0= 365= x, 95% ( 458-04449, 458+ 04449) = ( 355555, 5604449), 95% ( 873), ( 458-4358, 458+ 4358) = ( 37384, 5476) u 0 05 8, 95% 380,, ( 0< < ), x, - Y ( y L, y U ) 84, Y ( 0, y L ) ( 0, y U ) x, Y - Y + L U, L Q x L x U, 84, x x L x U, Y - ( y L, y U ),, x L x U x, ( y L, y U ), ( x L), x = x L Y -,, ( y L, y U ), 84, x L x U x Y - ( y L, y U ),

8449 x,, ( 873) x L x 0 x, Y ( x L ) - u = y L Y ( x U ) + u = y U y L+ u - 0 x L =, ( 874) x U = y U + u - 0, ( 875) 95% 380, ( 874) x L= 380+ 4358-309576 385, 4067736 385 84, Y x ( ( 84) ),, Y x,, 9 8, 84 85 Y x,, x Y,,, Y x, Y -

50 8 8-4 3 4 5 6 7 8 x ( ) 4 6 0 5 7 8 0 Y ( ) 6 7 9 0 7 4 3 6 lny 79 95 0 30 83 38 34 36 Y = 0e x ( 876) Y = 0+ x ( 877) 86 lny x,, lny x, ( 876) 85 86, 86 ( 876), lny x ( 876) Y x, 86 85 ( ), ( x, Y), =,,, ) Z = lny, =,,, ; ( 878) Z x Z = 0+ x, ( 879) 86, ( 876) Y x Y = e Z = e 0 e x ( 880) 0= e 0, = 84, lny x Z = 339 ( 434) + 0003 ( 4934) x t Y x

Y = 3708e 0 003 x ( 88) ( 877),, w = x, ( 877) Y w Y = 0+ w ( 88), w = x ( =,,, ), ( w, Y) ( =,,, ) ( 88) 0 ( 877) Y x 94 Y = 443 + 0057 x ( 883) ( 4547) ( 3630),,, ( 876) ( 877), Y = a + b x Y= x a x + b Z= Y, w = x, Z= a + bw 845, ( x, Y), x Y ( =,,, ), Y Y x x, w = lnx, 3 Y= a + blnx, Y= a + bw Y= a x b,

5 8 Z= lny, = lna, w = lnx, 4S Z= Y, w = e- x Z= + bw Y= a + be - x, z = a + bw,, 87, 8, Y x,,? R R R = - = ( Y- Y ) ( Y- ( 884) Y) =,,,,, R, R S S S = ( Y- Y ) = - ( 885), S ( 884) ( 885), R S, 8 ( 88) ( 883) R S,

8453 87 85, R S 85,,,,

54 8 8-5 R S Y = - 0+ 37x 09637 376 Y = 3708e 0003x 09804 75 Y = 443+ 00574x 0984 5746 85,,,,, Y p, x, x, x p Y,, Y x, x,, x p, Y= 0+ x + x + + p x p + ( 886) f ( x, x,, x p) = 0+ x + x + + p x p ( 887) p, p ( x, x, x p ; Y) ( =,,, ) ( 886) Y= 0+ x + x + + p x p +, =,,, ( 888), E ( ) = 0, D ( ) =, =,,,, ( 889) Y, Y,, Y,,,, ( 890), ( =,,, ),, ( 889) ( 890) N ( 0, ) cov(, j ) = 0, ( j ;, j =,,, ) ( 89)

85 55, ( 888) ( 89) 0, p, ( 888) ( 89) X= Y= X +, N ( O, I ) Y= ( Y, Y, Y), = (,, ), x x x p x x x p x x x p = ( 0,,, p ), I, O, ( 89) 0,,, p Q, Q( 0,,, ( Y- 0- x - - p x p ) ( 893) = p ) = Q( 0,,, p) = min Q( 0,,, p ),,, 0 p Q ( 0,, p ), 0 p p + - ( Y- 0- x - - p x p) = 0 = - ( Y- 0- x - - p x p) x = 0 = - ( Y- 0- x - - p x p) x p = 0 j =

56 8 0+ x + + x p = = x 0+ = x = + + = p = Y = x x p p = = x p 0+ = x p x + + = x p x Y = x py = p = ( 894), ( X X) = X Y ( 895) = ( 0,,, p), X X, = ( X X) - X Y ( 896), ( 894) 0= Y- x - x - - x p p, ( 897) Y = = Y, x k= x k ( k=,,, p ), p = l + l + + l p p= l Y l + l + + l p p= l Y ( 898) l p + l p + + l p p p = l p Y, l kj = ( x k- x k) ( x j - x j ), k, j =,,, p ; = l ky = ( x k- x k) ( Y- Y), k=,,, p = L ( 898), L= ( l kj ) pp, l Y ( 898), l Y= ( l Y, l Y,, l p Y), ( 898) L = l Y, ( 899) (,,, p ) L, = L - l Y ( 800) ( 897) O 0,,, p ( 896), Q, 0,,, p 0,,, p > p, X X X

85 57 Y = 0+ x + + p x p ( 80), ( 89), 0,,, p 0,,, p ( BLUE ) x, x,, x p Y, H 0 = = = p = 0 ( 80) H 0, Y, Y = 0+ x + + p x p, ( =,,, ), 8, S ST = = = = ( Y- Y ) ( Y- Y def = SSE + SSR, ) ( Y - Y ) = ( Y- Y ) ( Y - Y) = 0, = ( 894) S SRS SE, 9,,, H 0, F = SS R/ p SS E/ ( - p - ) p -p - F ( 803), F ( p, - p - ) p -p - F, F > F ( p, - p - ) H 0 86

58 8, R = SSR SST = - SSE SST 8-6 F SSR= ( Y - Y ) P MSR= SSR = P F = MSR MSE SSE = ( Y- Y ) - p - MSE = = SSE - p - SST = ( Y- Y ) - =, 0R, R,, R R R,, R Y x x,, x p, ( H 0 ( 80) ) p Y, Y,, x j Y, H 0 = 0 ( 804) H 0, x Y, x j j j, H 0 j > c, c, H 0 ( 804), j / l j F j = SSE / - p -, ( 805)

85 59 t j =, l j L - j, j / l j S SE / - p -, ( 806) F j F (, - p - ), tj t( - p - ),, F (, - p - ) F (, - p - ), t ( - p - ) t ( - p - ), H 0 F j > F (, - p - ) ( 807) t j > t ( - p - ) ( 808), H 0 ( 807) F, ( 808) t H 0 ( 804), F t,,, F j ( t j ), p -,, Y = 0+ x + + p x p, ( x 0, x 0, x 0p ), Y0, E ( Y0) Yo= 0+ x 0 + + p x 0p + 0, Y 0= 0+ x 0 + + p x 0p ( 809) 0< <, E ( Y0) - Y 0-0 t ( - p - ), Y 0+ 0 t ( - p - ) ) ( 80) o= + l kj ( x ok- x k) ( x oj - x j ), k= j = = SSE - p -,

60 8 l kj L - k j, k j =,,, p, 0< <, Y0 -, Y0- Y 0 t( - p - ), ( 8) = + + l kj ( x ok- x k) ( x oj - x j ), k= j = Y0 - ( Y 0- t ( - p - ), Y 0+ t ( - p - ) ), ( 8) E ( Y0) -, ( x 0, x 0,, x 0p ), ( x, x,, x p ), Y0- ( Y 0- u, Y 0+ u ) ( 83) u ( A) 9, ( ) 3 4 5 6 7 8 9 0 4 56 8 94 9 7 34 495 87 9 90 4 50 60 8 6 0, ( ) ; ( ) Y x, ; ( 3), 300 95%,,, ( = 005),?

6 3 4 5 6 7 76 69 8 9 88 70 84 80 63 84 90 9 76 7 3, Y ( % ) x x 789 00 000 55 00 00 809 00 000 574 00 00 853 08 000 607 08 00 ( ) ; ( ) ( = 005) ; ( 3) ( = 005), 4 0 Y x,, x, Y x x Y x x ( ) ( ) ( / ) 0 0 50 5 0 50 3 4 0 50 4 30 03 60 5 3 03 60 6 40 04 70 7 8 03 50 8 50 05 75 9 40 04 70 0 50 05 74 5 x y 8

6 8 x 3 5 9 3 6 7 30 y 0 97 0 8 7 3 36 450, ( ) ; ( ) y x, ( B) Y= 0+ x + N ( 0, ), =,,, ; ( H ) ( H 5), 0, 3 P ( Y0Y 0- ) = -, ( Y 0-, + ) Y0 - ( ) ;, Y 0 4 ( 894) ( 898), X X L

9 ( ),,,, 9,,,,,,,,,, 9, 3,, 3, 3 3, 3,

64 9 9 K K K 3 K4 K 5 K 6 K 7 K 8 K 9 K 0 K K K 3 00 8 89 97 9 90 99 94 9 84 86 90 9 93 9069 36679 00 85 90 9 94 88 98 95 94 87 76 94 83 84 893 873 003 79 78 9 97 9 9 94 78 87 86 87 87 9 8769 6345 004 89 9 9 89 86 88 88 73 86 8 95 90 90 876 763 005 80 87 90 90 87 9 88 84 8 84 9 79 86 865 60 006 83 74 88 97 9 8 9 77 8 84 87 8 99 8600 9945 007 87 9 87 80 87 80 9 80 79 90 9 8 85 8554 373 008 9 84 8 88 85 8 94 83 84 87 89 84 78 8546 405 009 85 77 86 96 77 7 95 84 8 88 86 86 98 8538 4559 00 80 90 9 78 8 9 84 87 86 73 9 85 85 849 49 0 89 9 9 79 83 69 95 80 87 8 94 86 76 8485 3099 0 9 8 94 9 78 8 86 77 87 70 85 84 86 8408 4 03 85 89 83 85 84 79 89 73 77 90 9 84 8 8400 06399 04 85 90 88 93 88 85 95 64 77 60 9 75 96 8369 06367 05 80 78 9 85 84 90 86 80 7 89 77 86 87 8346 07 06 83 9 86 93 85 66 80 69 84 89 79 89 89 8338 08656 07 83 8 80 69 8 96 95 86 80 86 88 8 74 833 0545 08 80 86 85 96 8 80 80 76 8 74 8 86 9 89 06473 09 8 9 93 7 83 84 87 78 85 67 87 8 78 808 04798 00 78 95 8 80 8 9 80 74 68 75 96 78 78 83-0683 0 83 89 88 73 8 79 87 7 85 68 96 74 8 83-047 0 74 89 79 87 86 78 9 73 74 7 88 87 74 800-03004 03 69 89 75 96 90 48 95 70 68 90 90 84 84 806-0646 04 8 93 86 66 7 77 89 8 80 64 9 90 7 8008-0593 05 73 84 77 9 73 63 85 68 75 68 87 83 9 783-476 06 80 90 9 48 79 57 90 67 87 70 9 84 70 773-066 07 73 96 78 88 79 54 89 60 69 63 9 7 83 7654-4788 08 67 90 80 86 74 57 83 60 60 78 86 69 8 7469-3374 09 79 85 88 65 75 6 88 60 68 70 87 78 64 7446-75 030 68 8 70 73 67 57 8 6 50 60 87 78 63 6900-596 03 68 0 80 8 80 75 9 8 68 66 0 76 8 6538-30483 03 30 0 87 60 68 55 94 66 65 86 0 8 6 579-60493 ( ) K K7 7, K 8K3 6 ; ( ) K K ( 3) 3,

9 65,,,, 3,, ( ),,,,, H, 90 Karl Pearson, Hotelling, p p, X= ( X, X,, X p ), = ( j ) pp, cov( X, X j ) = j, (, j =,,, p ) X X, X,, X p,, Y= l X= l X + l X + + l p X p, Y= l X= l X + l X + + l p X p, Yp = lp X= l p X + l px + + l p p X p,, Y= ( Y, Y,, Yp ), L= ( l j ) pp = ( l, l,, l p ) ( 9) Y= L X ( 9) D ( Y) = l l, =,,, p ( 93) cov( Y, Yj ) = l l j,, j =,,, p ( 94), Y = L L ( 95)

66 9 Y = l X l l = D ( Y), Y, Y X Y= l X, l l = Y X Y, Y X YX, l l = - YY,, Y- X Y, Y, Y,, Yp X p, Y,, ;, p p p 0; ep 9 X p 0, e, e,,, X, Y= e X =,,,, p ( 96) = D ( Y) = e Xe=, =,,, p ( 97) p ( ) L p, L= ( e, e,, ep),, X X 9,,, tr ( ) = tr ( ), t r( ) = + + + p p = D( X ) + D ( X ) + + D ( X p ) = tr( ) = + + + p p = D( Y) + D ( Y) + + D ( Yp ) ( 98), p p X, X, X p,, p,

9 67 p p 9 / k Y k / k k= k= p, k= ( =,,, p ), 70% 90%, p 9 X, X, X 3 = 0 07 = 7985, e = ( 0034, 0946, 09804) ; = 463, e = ( - 0555, - 08, 0790) ; 3= 03384, e3 = ( 083, - 05499, 0085) ; 0 Y= 0034X + 0946X + 09804X 3; Y= - 0555X - 08X + 0790X 3; Y3= 083X - 05499X + 0085X 3; + + 3= tr ( ) = 0, Y 7985%, Y, Y 9666%, Y, Y X, X, X 3 Y,, Y X 3,, X 3,,,,,,,,, X, X,, X p Z = X - E( X ) S t d ( X ), =,,, p ( 99) S t d ( X ) = X, =,,, p

68 9 cov( Z, Z j ) = j X X j cov( X, X j ) X R = j,, j =,,, p ( 90) j j, Z= ( Z, Z,, Z p) R= p p p p ( 9), Z R p0 ee ep p Y= e Z = p ek Z k = k= p k= ek( X k - E ( X k) ), =,,, p ( 9) kk 9 9 X, X, X 3 R= 0707 0 0707 0673 0 0673 = 7559, e = ( 0664, 0707, 0500) ; = 0000, e = ( - 03536, 00000, 09354) ; 3= 044, e3 = ( 0664, - 0707, 0500) ; ( 9) Y= 0664( X - ) + 05( X - ) + 00945( X 3-3) ; Y= - 03536( X - ) + 03535( X 3-3) ; Y3= 0664( X - ) - 05( X - ) + 00945( X 3-3) ;,, 3 X, X, X 3, Z, Z, Z 3, Y / 3= 5853%, Y Y ( + ) / 3= 986% 9, X X,,,, p

9 69 X= ( X, X,, X p), Q= X X X p X X X p = X X X X p X X X k = ( X k, X k, X p k) k p, p X, X,, X p X X j j = S = X = - k= - k= X k ( 93) k= ( X k - X ), =,,, p ( 94) ( X k - X ) ( X j k - X j ),, j =,,, p ( 95) X = ( j ) pp = - ( X k - X) ( X k - X ) ( 96) X = ( X, X,, X p) X, k=, X X, X,, X p, y j = lj X = p l j X, j =,,, p ( 97) = p, y j y j y y j y j k= lj Xk, k=,,, ; j =,,, p ( 98) y j = y j k k= j = - ( y k - y ) ( y j k - y j ), y, y,, y p k = = l j X, j =,,, p ( 99) = - l j ( X k - X ) ( X k - X) l j k = = l j lj,, j =,,, p ( 90) = L L, ( 9)

70 9 L= ( l, l,, lp ) p ( 97), 9 y = l X X, l l = y ; y, y,, y - X -, y X, l l i= ; y y, y,, y - 0;,, y,, 0 9 X p 0, e, e,, e p, X y = e X, =,,, p ( 9), y, =,,, p, y y j 0 ( j ), tr ( p ) = = p = = = tr ( ),,, j ( ) j ( ), j = p j j = p j =, X, X,, X p,, Zk= X k- X, S k=,,, ; =,,, p ( 93), Z= ( Z, Z,, Z p ), Z = X - X, =,,, p ( 94) S Z X R, R Z X, R X y, y,, y p,

9 7 93 y = e X, =,,, p, k X k, y k= e Xk, =,,, p ; k=,,, y k= ( y k, y k,, y p k) ( k=,,, ), 93 9 3 ( k, k,, k3), 3, k k S = 487767, S = 497994, k 6 S 6= 467339, 9, 93 y 5768 039444 039444 y 300 07694 05738 y 3 5859 074 0693 y 4 09940 007646 076958 y 5 085855 006604 08356 y 6 067087 0056 08873 y 7 050 00400 0975 y 8 09858 0097 0950, 8 95%, 3,, 70%, = 5768 y = 0349z + 035z + 07z 3 + 0307z 4 + 0349z 5 + 038z6 + 034z 7+ 036z8+ 0360z9+ 094z 0+ 0343z+ 044z + 03009z 3 ( 95)

7 9, y k, k,, k 3, y 9,,,,,,, 3 9, ( ),,,,, 9,, 7 6?,,, ( ),,,,, H Hotelling ( 936),,, 0 x= ( x, x,, x p ), y= ( y, y,, y q), p q, E ( x ) = 0, E ( y j ) = 0, =,,, p ; j =,,, q,, X x x, Y yy, X Y ( )

9 73 xy ; Z= ( x, x,, x p; y, y,, y q), Z = xx yx x y yy y x= x y y x x y= 0, ( 96) x y,, x y0, u = l x = m y ( 97) l m u, ( u, ) a, b, c, d, a c> 0, ( au + b, c + d ) = ( u, ) ( 98) u, v,,, m, D ( u ) = l xx l =, D ( v ) = m y ym=, ( 99) ( u, v) = cov( u, v) = l x ym, ( 930) ( 99) ( 930) u, v,, x y, ( u, v) x y,, x y, x y, u, v,, u, v, u = l x, v = m y ( 93) u, v, D ( u ) = D ( v ) =, cov ( u, u ) = cov ( u, v ) = cov( v, u ) = cov ( v, v) = 0, ( u, v ) l m, u, v, x y,, p, l

74 9 94 ( 97) ( u, v ), x y ; k- ( u, v), ( u, v ), ( u k-, v k- ) k ( u k, vk) u k= l k x, vk= mk y ( i) D ( uk) = lk x xl k=, D ( v k) = mk ( ii) cov( u k, u j ) = 0, j =,,, k- ; cov( u k, v j ) = 0, j =,,, k- ; cov( v k, u j ) = 0, j =,,, k- ; cov( v k, u j ) = 0, j =,,, k- ; yymk= ; ( iii), ( u k, v k) = lk x ymk,, ( u, v ) ( 97), ( u, v ) l m, ( l, m) = l x ym - ( l xxl - ) - ( m yym- ) ( 93) l m, 0, xym - xxl = 0 yxl - yym= 0 ( 933) l, m, ( 99) = l x ym = m yxl = ( 934) xx yy, ( 933), = - x x x y - y y yx m= y - y y xl ( 935) xy - yy yxl- xxl = 0, l = l ( 936) ( 936), l - - - - yx - xx xx x x = x x x y yy yy T T

9 75 T = x x x y y, - - p 0 ( 937) ( 930) ( 934), u, v,, l m ( 935) l, m ( 97), u, v,,, 93x y k ( u k, v k) k k, u k l k, vk mk mk= - y y y xl k ( 938) k 93, ( < p ), x y,,, x k= ( x k, x k,, x p k), y k = ( y k, y k,, y qk), ( k=,,, ) ( 96), x x = - ( x k - x) ( x k - x) ; k= y y = - k= ( y k - y ) ( yk - y ) ; x y = - y x = x y k= ( x k - x) ( y k - y ) ;, x y x, y x x x y y x y y x x x y y x y y = - xx x y - y y yx ( 939) p 0 l, l,, l p, m = - y y yx l, =,,, p ( 940)

76 9 m, m, m p, u = l x, u = m y, =,,, p, ( 94) ( u, v ) x, y,, =,,, p, ( 93), R xx x y = yx yy, ( 94) 94, 9, 7 6, x= ( k, k,, k 7) y= ( k8, k9,, k 3) 93 R, x y ( ) R x y,, k k 93 k8 k9 k0 k k k 3 k 048 064 0087 078 0754 04587 k 0003 0300-009 09756 036 05 k3 04663 079 098 0089 04006 0909 k4 09 067 099 046 0349 085 k5 04456 05355 04563 03378 0730 06080 k6 07584 05476 090 0407 0348 0400 k7 0350 0576 0947-0337 0535 00740 R 94,, 0985366,, l m,, u k 09648, v k 09794,,, k k

77 94 l m 0985366 l = ( 00863, 09648, - 00073, 0077, - 0095, 0094, 036) m = ( 0047, 00396, - 00378, 09794, 00337, - 008) 0897357 l = ( 073, - 04856, 033, 0648, 00865, 000, 00435) m = ( 0374, 059, 00664, - 030, - 004, 0738) 3 080788 l 3 = ( - 03549, 0988, - 080, 0780, 0039, - 04606, - 0779) m 3 = ( - 05, - 07858, 0, 090, 070, 08095) 4 070973 l 4 = ( 03058, - 034, 086, - 0764, 06633, - 4, - 0463) m 4 = ( - 737, 5, 03780, - 0943, 0098, - 0065) 5 039579 l 5 = ( - 0675, 033, - 048, - 0464, 075, 0074, 03688) m 5 = ( 0469, - 0058, 099, 0938, - 05376, - 0380) 6 03779 l 6 = ( - 348, 09988, 07534, 04939, - 06763, 0389, 0453) m 6 = ( - 0087, - 05066, 007, 0033, 859, - 0089) X, X, X 3 = - 3 0-3 7 0 0 0 X, X, X 3, X, X, 5, X X 03, X X 3 X, X,, X p R= X, X,, X p 4X, X, X 3, X 4,

78 9 R= 07346 0708 0697 07040 07086 0839 ( X, X ) ( X 3, X 4 )

905, 05, 00, 05 007 ( ) CK N C - K N - N C, ( ) - N ( 3) - 4! P 4 30 3- C N - N C N - C N C - N - N C N ( C 4 30), P 365 365 C N - N C N, ( A) 4( ) - ( 8 9 ) 5, ( ) - ( 7 9 ) 5 ( 3) - ( 8 9 ) 5 + ( 7 9 ) 5 5( ) ( 4) C 3 5( 9 ) 3 ( 8 9 ) 6 C K a C - b K C a + b 4 ( ) 3 8 7 4 0, 6, 5 6 8C 3 0 0 905 0 48 3! 3 0 0 3 a - a + b-, a - a + b-, a ( a - ) a ( a + b) + a ( b- ) 5 3, 6 3

80 4 7( ) 0, ( ) 4 90, ( 3) 4 70 8059 90056, 8 300455, 04 30087, 049 38036 39063 40086, 00497 ( B) 9 9, 0 9 ( - r - ), ( - ) - 3C N N - K ( ) N - K 4 mk - ( m- ) K K 505 6 p - q q, q p - q q 7p = 9 90 q= 0 6 8008, 06 X = F x( x ) = 40 Y= 0 0 x < 0 x 40 ( ) ( 00 - ) ( ) 3 ( A) 0x< 40 F Y( x ) = 0 30 0 x < 0 0x < 30 x 30

8 4( ) P { = k}= 03 k- 07 k=,, 3 4 ( ) p 7 0 5 4 5, 4 5, 7 0 6, 039 8( ), f ( x ) = 9( ), F ( x ) = 7 30 7 0 0 0 x < 0 ( e- - e - ), ( ), F ( x ) = x 0x < ( ) 0 x ex x < 0, - - e- x x 0 e- 0 x < 0 x 0x < x - x - x < x 4, - 9 4, 3, - 5 037, 009 30736 ( e- 7( ) P { = k}= C k ( 094) k ( 006) - k ( ) ( 094) 8p q k - e - ) ( 3) - ( 094) - ( 094) - 006 ( 4) 006, 00564 9,,, 0035 30000 6e 43e - - e - 3 6-85e - 5 700693, 0083 88645 046 9 p 0 04 03 0 004469 p 0 04 03 0

8 3F X ( x ) = 3F Y( x ) = f Y( x ) = 0 x < 0 x 0x< f X ( x) = y> 0 x< - e - ( x- ) x 0 x < e- 33 4 ( a + b) ( a + b ) 34000 (x - ) x 0< x< x 0 ( B) ( ) f ( x) = 337 ( ) f x( x ) = ( 3) f Y( x ) = 000 e- x 000 x 0 0 x < 0 [ - e- 000 x 000 ] - e - x 000 x 0 0 x < 0 000 e- x 000 x0 0 x< 0 ( A) ( ) ( ) X 5 4, 56, 3 8 X 3( ) k=, k= 0 p X 0 / 56 5/ 8 5/ 8 / 56 5/ 56 5/ 4 5/ 8 35/ 56 p Y j 3/ 8 5/ 8 5/ 4

83 ( ) f X ( x ) = g X ( x ) = 4( ) f ( x, y) = F ( x, y) = ( ) f X ( x ) = ( 3) F X( x ) = 3 5( ) f ( x, y) = ( ) f X ( x ) = 6( ) F ( x, y) = 3e - 3x x 0 0 x < 0 f Y ( y) = 0 4 e- 3x ( - e - 4x ) x 0 g Y ( y) = 0 x < 0 0x 0y 0 0 x < 0 y < 0 xy 0x 0y x 0x y> y x > 0y x > y> 0x f Y( y) = 0 0 x < 0 x 0x F Y( y ) = x > > y > x > 0 0 ( - x ) 0x f Y( y) = 0 0 x < 0 y< x( y- ) x 0x, y 0x, y y- x >, y x >, y > 0 x < 0 y< - 4e 4y y0 y< 0 0y 7e - 0 0 y< 0 y 0y y> y 0y 7y y0 0 y< 0 0 ( ) G(x, y) = x ( y - ) x 0x 4, y4 0x 4, y < 4 ( 3) 4 y - x > 4, y 4 x > 4, y< 4

84 8P {X = 0 X = }= 3 P {X= X = }= 3 9p j= p X j p j p Y j = j p j = p X j p j p j = p j p Y = j p X p j p X p j j 0p = 4, p = 0, p 3 = 4, p = ; X x x p Y j Y 3 9 36, 36 4f X Y ( x y) = f Y X ( y x ) = f X Y ( x y) = f Y X ( y x ) = 5f Y X( y x ) = f Y X( y x ) = f X Y( x y ) = f X Y( x y ) = y y y 3 p X 4 8 6 8 3 8 3e - 3x x 0 0 x < 0 4e - 4y y0 4 3 0 y< 0 3e - 3x + 3y x y 0 x < y ( y> 0) 4y 4 e- - e - 4x 0< yx ( x > 0) 0 0< y< x + x + ( 0< x < ) 0 x- < y < x + ( x) 0 0< x < y+ y + ( 0y) 0 y- < x < 3- y ( y< 3) 0 8X Y, X Y 9 e - y 0x, - < y< + 0f ( x, y) = 0 4 3 4

85 0 3 p 9 4 9 4 9, - - p 9 9 0 3 9 9 - - 0 0 0 0 9 9 9 0 0 0 0 0 0 0 3 4 5 6 7 8 9 0 4 3 4 5 6 5 4 3 p 36 36 36 36 36 36 36 36 36 36 36 6f X + X ( z ) = 7f X ( x ) = 9F ( s) = 300, 04 3 5 3 338 0 z 0 ze - z z > 0 e- ( x- 3) 0 s0 4-349[ - ( 8 9 ) 5 ] 36-5 4 370, 49 38, 0 f Y( y ) = 9 s 0< s f ( s) = s s> ( y- ) e- 0 0 s0 0< s 4 s s> 9 40 4 66 4( ) ( ) 3 x A- 6x A+ 9 ( 3) x A= 3 3, 0 x A 6 3 4 AB =, x= AB< m in( A, B ) max( A, B ) 43( ) 755% ( 3) 3% B, P B- A

86 x + 0< x< 44E [ Y X= x ] = x x X+ 0< X< E [ Y X] = X X 45E [ X Y] = E [ Y X] = 0 4600465 4765 480007, 440 ( B) ( ) X X 0 3 ( ) ( 3) X p X j 0 / 79 05/ 79 05/ 79 / 79 70/ 79 0/ 79 6/ 79 4/ 79 35/ 79 63/ 79 / 79 / 79 0 3 6/ 79 378/ 79 5/ 79 36/ 79 X 0 ( 4) ( 5) Y p p 0 6/ 36 0/ 36 0/ 36 p 0/ 36 0/ 36 6/ 36 p 5/ 36 8/ 36 3/ 36 p 3 / 36 4/ 36 / 36 0 3 4 5 / 79 75/ 79 350/ 79 0/ 79 35/ 79 / 79 X Y 0 0 3/ 66 / 66 6/ 66 5/ 66 0/ 66 0 0/ 66 0 0

87 X Y 0 3 0 0/ 990 59/ 990 5/ 990 3/ 990 5/ 990 70/ 990 0/ 990 0/ 990 5/ 990 75/ 990 45/ 990 5/ 990 ( ) P {Y= k, Yj = s}= Ck 5C s 5- k- s 5- k7 9 5 ( k+ s5) ( ) P {Y= k}= Ck 58 5-9 5 YYj k X X j 0 P x k 0 ( 7 9 ) 5 ( 8 9 ) 5 - ( 7 9 ) 5 ( 8 9 ) 5 ( 8 9 ) 5 - ( 7 9 ) 5 - ( 8 9 ) 5 + ( 7 9 ) 5 - ( 8 9 ) 5 p x s ( 8 9 ) 5 - ( 8 9 ) 5 X X j ( 3) cov ( X, X j ) = ( 7 9 ) 5 - ( 8 9 ) 50 ( j ) DX= ( 8 9 ) 5 - ( 8 9 ) 50 X X j = [ ( 7 9 ) 5 - ( 8 9 ) 50 ) ] / [ ( 8 9 ) 5 - ( 8 9 ) 50 ] ( 4) DX = 9 [ ( 8 9 ) 5 - ( 8 9 ) 50 ] + 7 [ ( 7 9 ) 5 - ( 8 9 ) 50 ] 3( ) P {Y= s X= k}= C s 5- kp s ( - p ) 5- k- s 0s5- k ( ) P {X = k, Y= s}= C s 5p k ( - p ) 5- k C s 5- kp s 5- k- s ( - p ) = 5! k! s! ( 5- k- s)! p k [ p ( - p ) ] s 5- k- s [ - p - p ( - p ) ] ( 3) E [ Y X] = ( 5- X) p E [ Y X ] = ( 5- X) p ( - p ) + ( 5- X) p ( 4) E Y= E [ E ( Y X) ] = ( 5-5p ) p = 5( - p ) p DY = E Y - ( E Y) = E ( E [ Y X] ) - ( E Y) = 5( - p ) p [ - p ( - p ) ] 4( 3) Yb ( 5, ( - p ) p ), Y= s X X Y= sb ( 5- s, p ( - p ) p ) 5( ) f ( x, y) = xe - xy x, y0 0

88 ( ) f Y( y ) = ( 3) f X Y( x y ) = ( 4) 474 ( 5) 08655 6( ) f X ( x ) = y ( e - y - e - y ) + e - y - e - y y0 0 x e x y y ( e - y - e - y ) + e - y - e - 0 x 0< x < x 3- x x3 0 3- y 0y f Y( y ) = 0 X Y 5 0 ( ) 0, X Y 44 ( 3) f Y X( y x ) = x 0yx ( 0< x < ) f Y X ( y x ) = f Y X( y x ) = f X Y( x y ) = ( 4) E [ Y X] = y x ( y> 0) 0 0y ( < x < ) 0 3- x 0y3- x ( x < 3) 0 y x 3- y 3- y ( 0y) 0 x 0< x < < x < 3- x x < 3 E [ X Y] = 3 ( 0Y) 8 X X 0 0 - e - 0 e - - e - e - cos ( X, X ) = e - - e - 3 X, X = e- e -

89 9 U 3( ) 0 V 3 ( ) f Z( z ) = 0 4 4 3 0 - z 0< z < U V= 0 3 ( A) f ( x, x,, x ) = e - (x + + x ), x 0, =,,, f ( x, x,, x ) =, 0x, =,,, ; 0 3P ( X = k, X = k,, X = k) = p - p ( - p ) k + k + + k, k, =,,, 4u= x- a b, s u= s x b 7P ( X ( ) x ) = - ( - F ( x ) ), P ( X ( ) x) = ( F ( x ) ) ;, X ( > 0), P {X( ) x }= - e - x, x 0; P {X ( ) x }= ( - e - x ), x 0 8( ) 40; ( ) 5 / 6; ( 3) 3 / 6 9E [ X ] = x e - 0 x dx = 0 t- e - t dt ( t= x ) = ( + ) = - - 3 ( ) 0f ( x) = = 35 ( - ) m m m x m - e - x, x> 0 c= 3, P ( S = k) = ( ) k e -, k= 0,,, ; k!, S N (, ),

90 x - P ( S x ) 0 3u 06= 053, u 08= 0846, u 09= 8, 444 u 095= 65 545, 07, 558, 309 6063, 00684, 009 7353, 3365, 45, 369 50 ( - 063, 063) ( B) 4 4, X f ( x) = exp ( - x T x) x= ( x, x,, x ) T, Y Axy F ( y, y,, y ) = exp ( - x T x ) dx dx dx t= Ax A, x = A - t = A T t, J = x tj = A T A, A T -, J = F ( y, y,, y ) = y = y y - - - y - exp - exp - ( A T t ) T ( A T t ) dtdtdt tt t dt dt d tn Y, Y,, Y, Y N ( 0, ), =,,, 5, 43X N ( 0, Yk= Y= ) ( - ) S = ( X - X) = =, X - = ( X ) ( X + X + + X k- X k+ ), k=,,, - ; k ( k+ ) ( X + X + + X ) = X

9 ( X, X,, X ) T ( Y, Y,, Y) T, Y= ( Y, Y,, Y) T, X= ( X, X,, X ) T, Y= AX, A, Y, Y,, Y, YN ( 0, ), =,,, A, Y + Y + + Y = Y T Y= ( AX) T AX= X T ( A T A) X= X T X = X + X + + X, ( - ) S = X - ( X) = =, 4 46 ( - ) S ( - ), ( - ) S Y, X 6 = Y - Y = Y + Y + + Y - X = X -, =,,, X, X,, X, X N ( 0, ), =,,,, 4 8 Y P, 0< <, lim P {- Y + } = A = {- Y+ }, lim P ( A) = 0 P ( X Yx ) = P ( X Yx, - d X P ( X x - ) + P ( A ) Y+ ) + P ( X Yx, A ) X, F ( x) X, lim P ( X Yx) lim P ( X x - ) = F 0, F ( x ) x -, lim P ( X Yx ) F ( x ) P ( X Yx) = P ( X x Y, - Y+ ) + P ( X Yx, A) P ( X x, - Y+ ) + P ( X Yx, A ) + = P ( X x + ) - P ( X x +, A ) + P ( X Y x, A ),, lim P ( A) = 0, lim 0, P ( X Yx ) lim P ( X x + ) = F x +

9, lim P ( X Yx ) F ( x ) F ( x ) lim P ( X Yx ) lim P ( X Yx ) F ( x ) limp ( X Yx ) = F ( x ), x R C= ( - ) 4 5 MLE= p MLE= X p MLE= X MLE = = X 6 5 8 7a ME = X- = lnx 3B, b ( X - X) ( A) ME = X + 3B, B = 8 X 9 ME= X-, MLE = X ( ), X () = min {x,, x } = ( X - X) 0( X ( ), 05 X () ( 7, 9) 3 5, 60 4( 45, 969) 55 6( 36, 097) 7( 9, 0788) 8( 04, 77) ),, - = -

93 ( B) X () - X ( ) 3( 766, 3795) ( A) e -, - e - c= 76 3 4 5( ) = 005, ; ( ) = 00, 6 7 8 0 9 ( ) H 0 = 000; ( ) H 0 5,, 5,, 0 3 4 / 6 5 5 6 7 8 ( B) W = = ( X - 0)

94 W F = ( S ) r ( S ), ( S ) = = ( X - F 3( ) = x = 0965 ), = j = ( Yj - ), F, ( ) U, = 0 4 X, p = P ( X= ), = 0,, 3 H 0 p 0= 3, p =, p 3= 6 0= = 005, 3, 005 ( r - ) = 005 ( ) = 599 0= < 005 ( r - ) = 599, 5- ( 3) 6

P{X= m}= m m! e- 0 0 03 04 05 06 07 08 m 0 0904837 08873 074088 067630 060653 05488 0496585 044939 0090484 063746 045 0688 030365 03987 034760 0359463 000454 006375 0033337 005366 007586 0098786 0663 043785 3 00005 00009 0003334 000750 00636 009757 008388 0038343 4 0000004 0000055 000050 000075 000580 000964 0004968 0007669 5 000000 000005 0000057 000058 0000356 0000696 0007 6 000000 0000004 000003 0000036 000008 000064 7 000000 0000003 0000008 000009 8 000000 000000 9 0 3 4 5 6 7 09 0 5 0 5 30 35 40 m 0 0406570 0367879 0330 035335 008085 0049787 003097 00836 03593 0367879 0334695 07067 005 04936 00569 007363 06466 083940 050 07067 05656 0404 084959 04655 3 0049398 00633 0550 080447 03763 0404 05785 095367 4 005 00538 0047067 00904 03360 06803 0888 095367 5 00000 0003066 0040 0036089 006680 00089 0369 05693 6 0000300 00005 0003530 00030 007834 0050409 0077098 00496 7 0000039 0000073 0000756 0003437 000994 00604 0038549 0059540 8 0000004 0000009 00004 0000859 000306 00080 006865 009770 9 000000 000004 00009 0000863 00070 0006559 0033 0 000004 0000038 00006 000080 00096 00059 0000007 0000049 0000 0000730 00095 000000 000000 0000055 00003 000064 3 000000 000003 0000057 000097 4 000000 000004 0000056 5 000000 0000003 000005 6 000000 0000004 7 000000

96 m 45 50 55 60 65 70 75 80 0 0009 0006738 0004087 000479 000503 000009 0000553 0000335 0049990 0033690 00477 004873 0009773 0006383 000448 000684 0479 00844 0068 004468 003760 0034 005556 000735 3 06878 040374 0333 008935 006884 0059 0038888 00866 4 089808 075467 05589 033853 08 0096 00797 00575 5 07087 075467 0700 06063 045369 0777 009374 009604 6 080 0463 0577 06063 057483 049003 03679 038 7 008363 004445 03449 037677 04634 049003 046484 039587 8 004639 006578 008487 00358 0885 030377 03738 039587 9 00365 003666 005866 0068838 00858 00405 0444 04077 0 00044 00833 00856 004303 0055777 0070983 0085830 00996 000464 00084 00463 0059 003959 00457 00585 00790 000599 0003434 0006537 0064 007853 006350 0036575 00487 3 0000554 0003 000766 000599 000897 00488 000 00966 4 000078 000047 000086 0008 000444 0007094 00305 00694 5 0000053 000057 0000399 000089 000796 00033 000565 000906 6 000005 0000049 000037 0000334 0000730 000448 000649 000453 7 0000004 000004 0000044 00008 000079 0000596 00069 0004 8 000000 0000004 000004 0000039 000000 00003 0000487 0000944 9 00000 0000004 00000 0000035 0000085 00009 0000397 0 00000 0000004 00000 0000030 000007 000059 000000 0000004 000000 000006 000006 000000 0000003 0000009 00000 3 000000 0000003 0000008 4 000000 0000003 5 000000 6 7 8 9

97 m 85 90 95 00 m 0 m 0 000003 00003 0000075 0000045 5 0000 0000 000730 000 00007 0000454 6 0000 3 0000 0007350 0004998 0003378 00070 7 00005 4 00005 3 00086 004994 000696 0007567 8 0003 5 0000 4 04455 0033737 005403 00897 9 0009 6 0009 5 007533 006077 004865 0037833 0 00058 7 00034 6 00658 009090 00764 0063055 0006 8 00057 7 0949 076 00374 0090079 0076 9 00089 8 037508 03756 0360 0599 3 007 0 0034 9 09869 03756 030003 050 4 0038 009 0 00303 08580 050 050 5 0057 006 0085300 009700 00666 03736 6 00646 3 0034 00604 007765 0084440 0094780 7 00760 4 0046 3 0039506 0050376 006706 007908 8 0084 5 0057 4 003986 003384 00487 005077 9 00888 6 00590 5 00359 00943 00659 003478 0 00888 7 00655 6 00070 000930 005746 00699 00846 8 0070 7 00036 0005786 0008799 00764 00767 9 0076 8 000705 000893 0004644 000709 3 00669 30 0076 9 000076 000370 0003 000373 4 00557 3 0703 0 000034 000067 00003 000866 5 00446 3 00659 00003 000064 0000433 0008989 6 00343 33 00599 0000050 000008 00006 0000404 7 0054 34 0059 3 000009 000004 000089 000076 8 008 35 00453 4 0000007 000006 000005 0000073 9 005 36 00378 5 000000 0000006 000004 000009 30 00083 37 00306 6 000000 000000 0000004 00000 3 00054 38 004 7 000000 000000 0000004 3 00034 39 0086 8 000000 000000 33 0000 40 0039 9 000000 34 000 4 000 30 4 00073 43 0050 35 00007 44 00035 36 00004 45 0003 37 0000 46 0005 38 0000 47 0000 39 0000 48 00006

98 x - e- t 0( x) = dt( x 0) x 000 00 00 003 004 00 05000 05040 05080 050 0560 0 05398 05438 05478 0557 05557 0 05793 0583 0587 0590 05948 03 0679 067 0655 0693 0633 04 06554 0659 0668 06664 06700 05 0695 06950 06985 0709 07054 06 0757 079 0734 07357 07389 07 07580 076 0764 07673 07703 08 0788 0790 07939 07967 07995 09 0859 0886 08 0838 0864 0 0843 08438 0846 08485 08508 08643 08665 08686 08708 0879 08849 08869 08888 08907 0895 3 09030 090490 090678 09084 090988 4 0994 09073 090 09364 09507 5 09339 093448 093574 093699 0938 6 09450 094630 094738 094845 094950 7 095543 095637 09578 09588 095907 8 096407 096485 09656 096638 0967 9 0978 09793 09757 09730 09738 0 09775 097778 09783 09788 09793 0984 09857 098300 09834 09838 09860 098645 098679 09873 098745 3 09898 098956 098983 09900 099036 4 09980 0990 0994 09945 09966 5 099379 099396 09943 099430 099446 6 099534 099547 099560 099573 099586 7 099653 099664 099674 099683 099693 8 099745 09975 099760 099767 099774 9 09983 09989 09985 09983 099836 30 099865 099869 099874 099878 09988 3 099903 099906 09990 09993 09996 3 09993 099934 099936 099938 099940 33 09995 099953 099955 099957 099958 34 099966 099968 099969 099970 09997 35 099977 099978 099978 099979 099980 36 099984 099985 099985 099986 099986 37 099989 099990 099990 099990 09999 38 099993 099993 099993 099994 099994 39 099995 099995 099996 099996 099996

99 x 000 00 00 003 004 40 099997 099997 099997 099997 099997 4 099998 099998 099998 099998 099998 4 099999 099999 099999 099999 099999 43 099999 099999 099999 099999 099999 44 099999 099999 00000 00000 00000 x 005 006 007 008 009 00 0599 0539 0579 0539 05359 0 05596 05636 05675 0574 05753 0 05987 0606 06064 0603 064 03 06368 06406 06443 06480 0657 04 06736 0677 06808 06844 06879 05 07088 073 0757 0790 074 06 074 07454 07486 0757 07549 07 07734 07764 07794 0783 0785 08 0803 0805 08078 0806 0833 09 0889 0835 08340 08365 08389 0 0853 08554 08577 08599 086 08749 08770 08790 0880 08830 08944 0896 08980 08997 09047 3 0940 09309 09466 096 09774 4 09647 09785 099 093056 09389 5 093943 09406 09479 09495 094408 6 095053 09554 09554 09535 095449 7 095994 096080 09664 09646 09637 8 096784 096856 09696 096995 09706 9 09744 097500 097558 09765 097670 0 09798 098030 098077 0984 09869 0984 09846 098500 098537 098574 098778 098809 098840 098870 098899 3 09906 099086 099 09934 09958 4 09986 099305 09934 099343 09936 5 09946 099477 09949 099506 09950 6 099598 099609 0996 09963 099643 7 09970 0997 09970 09978 099737 8 09978 099788 099795 09980 099807 9 09984 099846 09985 099856 09986 30 099886 099889 099893 099897 099900 3 09998 0999 09994 09996 09999 3 09994 099944 099946 099948 099950 33 099960 09996 09996 099964 099965 34 09997 099973 099974 099975 099976 35 09998 09998 09998 099983 099983 36 099987 099987 099988 099988 099989 37 09999 09999 09999 09999 09999 38 099994 099994 099995 099995 099995 39 099996 099996 099996 099997 099997 40 099997 099998 099998 099998 099998 4 099998 099998 099998 099999 099999 4 099999 099999 099999 099999 099999 43 099999 099999 099999 099999 099999 44 00000 00000 00000 00000 00000

300 3 P{ ( n) > ( n) }= = 0995 099 0975 095 090 075 000 0004 006 00 000 000 005 003 0 0575 3 007 05 06 035 0584 3 4 007 097 0484 07 064 93 5 04 0554 083 45 60 675 6 0676 087 37 635 04 3455 7 0989 39 690 67 833 455 8 344 646 80 733 3490 507 9 735 088 700 335 468 5899 0 56 558 347 3940 4865 6737 603 3053 386 4575 5578 7584 3074 357 4404 56 6304 8438 3 3565 407 5009 589 704 999 4 4075 4660 569 657 7790 065 5 460 59 66 76 8547 037 6 54 58 6908 796 93 9 7 5697 6408 7564 867 0085 79 8 665 705 83 9390 0865 3675 9 6844 7633 8907 07 65 456 0 7434 860 959 085 443 545 8034 8897 083 59 340 6344 8643 954 098 338 404 740 3 960 096 689 309 4848 837 4 9886 0856 40 3848 5659 9037 5 050 54 30 46 6473 9939 6 60 98 3844 5379 79 0843 7 808 879 4573 65 84 749 8 46 3565 5308 698 8939 657 9 3 457 6047 7708 9768 3567 30 3787 4954 679 8493 0599 4478 3 4458 5655 7539 98 434 5390 3 534 636 89 007 7 6304 33 585 7074 9047 0867 30 79 34 650 7789 9806 664 395 836 35 79 8509 0569 465 4797 9054 36 7887 933 336 369 5643 9973 37 8586 9960 06 4075 649 30893 38 989 069 878 4884 7343 385 39 9996 46 3654 5695 896 3737 40 0707 64 4433 6509 905 33660

30 = 0995 099 0975 095 090 075 4 4 906 55 736 9907 34585 4 38 3650 5999 844 30765 3550 43 859 4398 6785 8965 365 36436 44 3584 548 7575 9787 3487 37363 45 43 590 8366 306 33350 389 = 05 00 005 005 00 0005 33 706 384 504 6635 7879 773 4605 599 7378 90 0597 3 408 65 785 9348 345 838 4 5385 7779 9488 43 377 4860 5 666 936 07 833 5086 6750 6 784 045 59 4449 68 8548 7 9037 07 4067 603 8475 078 8 09 336 5507 7535 0090 955 9 389 4684 699 903 666 3589 0 549 5987 8307 0483 309 588 370 775 9675 90 475 6756 4845 8549 06 3337 67 899 3 5984 98 36 4736 7688 989 4 77 064 3685 69 94 339 5 845 307 4996 7488 30578 380 6 9369 354 696 8845 3000 3467 7 0489 4769 7587 309 33409 3578 8 605 5989 8869 356 34805 3756 9 78 704 3044 385 369 3858 0 388 84 340 3470 37566 39997 4935 965 367 35479 3893 440 6039 3083 3394 3678 4089 4796 3 74 3007 357 38076 4638 448 4 84 3396 3645 39364 4980 45559 5 9339 3438 3765 40646 4434 4698 6 30435 35563 38885 493 4564 4890 7 358 3674 403 4394 46963 49645 8 360 3796 4337 4446 4878 50993 9 337 39087 4557 457 49588 5336 30 34800 4056 43773 46979 5089 5367 3 35887 44 44985 483 59 55003 3 36973 4585 4694 49480 53486 5638 33 38058 43745 47400 5075 54776 57648 34 394 44903 4860 5966 5606 58964 35 403 46059 4980 5303 5734 6075 36 4304 47 50998 54437 5869 658 37 4383 48363 59 55668 5989 6883 38 4346 5953 58384 56896 66 648 39 44539 50660 5457 580 648 65476 40 4566 5805 55758 5934 6369 66766 4 5669 5949 5694 6056 64950 68053 4 47766 54090 584 6777 6606 69336 43 48840 5530 59304 6990 67459 7066 44 4993 56369 6048 640 6870 7893 45 50985 57505 6656 6540 69957 7366

30 4F P {F ( m, n) > F ( m, n) }= = 00 m 3 4 5 6 7 8 9 3986 4950 5359 5583 574 580 589 5944 5986 853 900 96 94 99 933 935 937 938 3 554 546 539 534 53 58 57 55 54 4 454 43 49 4 405 40 398 395 394 5 406 378 36 35 345 340 337 334 33 6 378 346 39 38 3 305 30 98 96 7 359 36 307 96 88 83 78 75 7 8 346 3 9 8 73 67 6 59 56 9 336 30 8 69 6 55 5 47 44 0 39 9 73 6 5 46 4 38 35 33 86 66 54 45 39 34 30 7 38 8 6 48 39 33 8 4 3 34 76 56 43 35 8 3 0 6 4 30 73 5 39 3 4 9 5 5 307 70 49 36 7 6 09 6 305 67 46 33 4 8 3 09 06 7 303 64 44 3 5 0 06 03 8 30 6 4 9 0 3 08 04 00 9 99 6 40 7 8 06 0 98 0 97 59 38 5 6 09 04 00 96 96 57 36 3 4 08 0 98 95 95 56 35 3 06 0 97 93 3 94 55 34 05 99 95 9 4 93 54 33 9 0 04 98 94 9 5 9 53 3 8 09 0 97 93 89 6 9 5 3 7 08 0 96 9 88 7 9 5 30 7 07 00 95 9 87 8 89 50 9 6 06 00 94 90 87 9 89 50 8 5 06 99 93 89 86 30 88 49 8 4 05 98 93 88 85 40 84 44 3 09 00 93 87 83 79 60 79 39 8 04 95 87 8 77 74 0 75 35 3 99 90 8 77 7 68 7 30 08 94 85 77 7 67 63

303 = 00 m 0 5 0 4 30 40 60 0 609 607 6 674 600 66 653 679 6306 6333 939 94 94 944 945 946 947 947 948 949 3 53 5 50 58 58 57 56 55 54 53 4 39 390 387 384 383 38 380 370 378 376 5 330 37 34 3 39 37 36 34 3 30 6 94 90 87 84 8 80 78 76 74 7 7 70 67 63 59 58 56 54 5 49 47 8 54 50 46 4 40 38 36 34 3 9 9 4 38 34 30 8 5 3 8 6 0 3 8 4 0 8 6 3 08 06 5 7 0 08 05 03 00 97 9 5 0 06 04 0 99 96 93 90 3 4 0 05 0 98 96 93 90 88 85 4 0 05 0 96 94 9 89 86 83 80 5 06 0 97 9 90 87 85 8 79 76 6 03 99 94 89 87 84 8 78 75 7 7 00 96 9 86 84 8 78 75 7 69 8 98 93 89 84 8 78 75 7 69 66 9 96 9 86 8 79 76 73 70 67 63 0 94 89 84 79 77 74 7 68 64 6 9 87 83 78 75 7 69 66 6 59 90 86 8 76 73 70 67 64 60 57 3 89 84 80 74 7 69 66 6 59 56 4 88 83 78 73 70 67 64 6 57 53 5 87 8 77 7 69 66 63 59 56 5 6 86 8 76 7 68 65 6 58 54 50 7 85 80 75 70 67 64 60 57 53 49 8 84 79 74 69 66 63 59 56 5 48 9 83 78 73 68 65 6 58 55 5 47 30 8 77 7 67 64 6 57 54 50 46 40 76 7 66 6 57 54 5 47 4 38 60 7 66 60 54 5 48 44 40 35 9 0 65 60 55 48 45 4 37 3 6 9 60 55 49 4 38 34 30 4 7 00

304 = 005 m 3 4 5 6 7 8 9 64 995 57 46 30 340 368 389 405 85 900 96 95 930 933 935 937 938 3 03 955 98 9 90 894 889 885 88 4 77 694 659 639 66 66 609 604 600 5 66 579 54 59 505 495 488 48 477 6 599 54 476 453 439 48 4 45 40 7 559 446 407 384 369 358 350 344 339 8 53 446 407 384 369 358 350 344 339 9 5 46 386 363 348 337 39 33 38 0 496 40 37 348 333 3 34 307 30 484 398 359 336 30 309 30 95 90 475 389 349 36 3 300 9 85 80 3 467 38 34 38 303 9 83 77 7 4 460 374 334 3 96 85 76 70 65 5 454 368 39 306 90 79 7 64 59 6 449 363 34 30 85 74 66 59 54 7 445 359 30 96 8 70 6 55 49 8 44 355 36 93 77 66 58 5 46 9 438 35 33 90 74 63 54 48 4 0 435 349 30 87 7 60 5 45 39 43 347 307 84 68 57 49 4 37 430 344 305 8 66 55 46 40 34 3 48 34 303 80 64 53 44 37 3 4 46 340 30 78 6 5 4 36 30 5 44 339 99 76 60 49 40 34 8 6 43 337 98 74 59 47 39 3 7 7 4 335 96 73 57 46 37 3 5 8 40 334 95 7 56 45 36 9 4 9 48 333 93 70 55 43 35 8 30 47 33 9 69 53 4 33 7 40 408 33 84 6 45 34 5 8 60 406 35 76 53 37 5 7 0 04 0 39 307 68 45 9 7 09 0 96 384 300 60 37 0 0 94 88

305 = 005 m 0 5 0 4 30 40 60 0 49 439 459 480 49 50 5 5 533 543 940 94 943 945 945 946 947 948 949 950 3 879 874 870 866 864 86 859 857 855 853 4 596 59 586 580 577 575 57 569 566 563 5 474 468 46 456 453 450 446 443 440 436 6 406 400 394 387 384 38 377 374 370 367 7 364 357 35 344 34 338 334 330 37 33 8 335 38 3 35 3 308 304 30 97 93 9 34 307 30 94 90 86 83 79 75 7 0 98 9 85 77 74 70 66 6 58 54 85 79 7 65 6 57 53 49 45 40 75 69 6 54 5 47 43 38 34 30 3 67 60 53 46 4 38 34 30 5 4 60 53 46 39 35 3 7 8 3 5 54 48 40 33 9 5 0 6 07 6 49 4 35 8 4 9 5 06 0 7 45 38 3 3 9 5 0 06 0 96 8 4 34 7 9 5 06 0 97 9 9 38 3 3 6 07 03 98 93 88 0 35 8 0 08 04 99 95 90 84 3 5 8 0 05 0 96 9 87 8 30 3 5 07 03 98 94 89 84 78 3 7 0 3 05 0 96 9 86 8 76 4 5 8 03 98 94 89 84 79 73 5 4 6 09 0 96 9 87 8 77 7 6 5 07 99 95 90 85 80 75 69 7 0 3 06 97 93 88 84 79 73 67 8 9 04 96 9 87 8 77 7 65 9 8 0 03 94 90 85 8 75 70 64 30 6 09 0 93 89 84 79 74 68 6 40 08 00 9 84 79 74 69 64 58 5 60 99 9 84 75 70 65 59 53 47 39 0 9 83 75 66 6 55 50 43 35 5 83 75 67 57 5 46 39 3 00

306 = 005 m 3 4 5 6 7 8 9 6478 7995 864 8996 98 937 948 9567 9633 385 3900 397 395 3930 3933 3936 3937 3939 3 744 604 544 50 488 473 46 454 447 4 065 898 960 936 90 907 898 890 5 00 843 776 739 75 698 685 676 668 6 88 76 660 63 599 58 570 560 55 7 807 654 589 55 55 5 499 490 48 8 757 606 54 505 48 465 453 443 436 9 7 57 503 47 448 43 40 40 403 0 694 546 483 447 44 407 395 385 378 67 56 463 48 404 388 376 366 359 655 50 44 4 389 373 36 35 344 3 64 497 435 400 377 360 348 339 33 4 630 486 44 389 366 350 338 39 3 5 60 477 45 380 358 34 39 30 3 6 6 469 408 373 350 334 3 3 305 7 60 46 40 366 344 38 36 306 98 8 598 456 395 36 338 3 30 30 93 9 59 45 390 356 333 37 305 96 88 0 587 446 386 35 39 33 30 9 84 583 44 38 348 35 309 97 87 80 579 438 378 344 3 305 93 84 76 3 576 435 375 34 38 30 90 8 73 4 57 43 37 338 35 99 87 78 70 5 569 49 369 335 33 97 85 75 68 6 566 47 367 333 30 94 8 73 65 7 563 44 365 33 308 9 80 7 63 8 56 4 363 39 306 90 78 69 6 9 559 40 36 37 304 88 76 67 59 30 557 48 359 35 303 87 75 65 57 40 54 405 346 33 90 74 6 53 45 60 59 393 334 30 79 63 5 4 33 0 55 380 33 89 67 5 39 30 50 369 3 79 57 4 9 9

307 = 005 m 0 5 0 4 30 40 60 0 9686 9767 9849 993 997 00 006 00 04 08 3940 394 3943 3945 3946 3946 3947 3948 3949 3950 3 44 434 45 47 4 408 404 399 395 390 4 884 875 866 856 85 846 84 836 83 86 5 66 65 643 633 68 63 68 6 607 60 6 546 537 57 57 5 507 50 496 490 485 7 476 467 457 447 44 436 43 45 40 44 8 430 40 40 400 395 389 384 378 373 367 9 396 387 377 367 36 356 35 345 339 333 0 37 36 35 34 337 33 36 30 34 308 353 343 333 33 37 3 306 300 94 88 337 38 38 307 30 96 9 85 79 7 3 35 35 305 95 89 84 78 7 66 60 4 35 305 95 84 79 73 67 6 55 49 5 306 96 86 76 70 64 59 5 46 40 6 99 89 79 68 63 57 5 45 38 3 7 9 8 7 6 56 50 44 38 3 5 8 87 77 67 56 50 44 38 3 6 9 9 8 7 6 5 45 39 33 7 0 3 0 77 68 57 46 4 35 9 6 09 73 64 53 4 37 3 5 8 04 70 60 50 39 33 7 4 08 00 3 67 57 47 36 30 4 8 04 97 4 64 54 44 33 7 5 08 0 94 5 6 5 4 30 4 8 05 98 9 6 59 49 39 8 6 09 03 95 88 7 57 47 36 5 9 3 07 00 93 85 8 55 45 34 3 7 05 98 9 83 9 53 43 3 5 09 03 96 89 8 30 5 4 3 0 4 07 0 94 87 79 40 39 9 8 07 0 94 88 80 7 64 60 7 7 06 94 88 8 74 67 58 48 0 6 05 94 8 76 69 6 53 43 3 05 94 83 7 64 57 48 39 7 00

308 = 00 m 3 4 5 6 7 8 9 465 49995 5403 566 5764 5859 598 598 60 9850 9000 997 995 9930 9933 9936 9937 9939 3 34 308 946 87 84 79 767 749 735 4 0 800 669 598 553 5 498 480 466 5 66 37 06 39 097 067 046 09 06 6 375 09 978 95 875 847 86 80 798 7 5 955 845 785 745 79 699 684 67 8 6 865 759 70 663 637 68 603 59 9 056 80 699 64 606 580 56 547 535 0 004 756 655 599 564 539 50 506 494 965 7 6 567 53 507 489 474 463 633 693 595 54 506 48 464 450 439 3 907 670 574 5 486 46 444 430 49 4 886 65 556 504 469 446 48 44 403 5 868 636 54 489 456 43 44 400 389 6 853 63 59 477 444 40 403 389 378 7 840 6 58 467 434 40 393 379 368 8 89 60 509 458 45 40 384 37 360 9 88 593 50 450 47 394 377 363 35 0 80 585 494 443 40 387 370 356 346 80 578 487 437 404 38 364 35 340 795 57 483 43 399 376 359 345 335 3 788 566 476 46 394 37 354 34 330 4 78 56 47 4 390 367 350 330 36 5 777 557 468 48 385 363 346 33 3 6 77 55 464 44 38 359 34 39 38 7 768 549 460 4 378 356 339 36 35 8 764 545 457 407 375 353 336 33 3 9 760 54 454 404 373 350 333 30 309 30 756 539 45 40 370 347 330 37 307 40 73 58 43 383 35 39 3 99 89 60 708 498 43 365 334 3 95 8 7 0 685 479 395 348 37 96 79 66 56 663 46 378 33 30 80 64 6 4

309 = 00 m 0 5 0 4 30 40 60 0 6056 606 657 600 635 66 687 633 6339 6336 9940 994 9943 9945 9946 9947 9947 9948 9949 9950 3 73 705 687 669 660 650 64 63 6 63 4 455 437 40 40 393 384 375 365 356 346 5 005 989 97 955 947 938 99 90 9 90 6 787 77 756 740 73 73 74 706 697 688 7 66 647 63 66 607 599 59 58 574 565 8 58 567 55 536 58 50 5 503 495 486 9 56 5 496 48 473 465 457 448 440 43 0 485 47 456 44 433 45 47 408 400 39 454 440 45 40 40 394 386 378 369 360 430 46 40 386 378 370 36 354 345 336 3 40 396 38 366 359 35 343 334 35 37 4 394 380 366 35 343 335 37 38 309 300 5 380 367 35 337 39 3 33 305 96 87 6 369 355 34 36 38 30 30 93 84 75 7 359 346 33 36 308 300 9 83 75 65 8 35 337 33 308 300 9 84 75 66 57 9 343 330 35 300 9 84 76 67 58 49 0 337 33 309 94 86 78 69 6 5 4 33 37 303 88 80 7 64 55 46 36 36 3 98 83 75 67 53 50 40 3 3 3 307 93 78 70 6 54 45 35 6 4 37 303 89 74 66 58 49 40 3 5 33 99 85 70 6 54 45 36 7 7 6 309 96 8 66 58 50 4 33 3 3 7 306 93 78 63 55 47 38 9 0 0 8 303 90 75 60 5 44 35 6 7 06 9 300 87 73 57 49 4 33 3 4 03 30 98 84 70 55 47 39 30 0 40 80 66 5 37 9 0 0 9 80 60 63 50 35 0 03 94 84 73 60 0 47 34 9 03 95 86 76 66 53 38 3 8 04 88 79 70 59 47 3 00

30 = 0005 m 3 4 5 6 7 8 9 6 0000 65 500 3056 3437 375 395 409 985 990 99 99 993 993 994 994 994 3 5555 4980 4747 469 4539 4484 4443 443 4388 4 333 68 46 35 46 97 6 35 4 5 78 83 653 556 494 45 40 396 377 6 863 454 9 03 46 07 079 057 039 7 64 40 088 005 95 96 889 868 85 8 469 04 960 88 830 795 769 750 734 9 36 0 87 796 747 73 688 669 654 0 83 943 808 734 687 654 630 6 597 3 89 760 688 64 60 586 568 554 75 85 73 65 607 576 55 535 50 3 37 89 693 63 579 548 55 503 494 4 06 79 668 600 556 56 503 486 47 5 080 770 648 580 537 507 485 467 454 6 058 75 630 564 5 49 469 45 438 7 038 735 66 550 507 478 456 439 45 8 0 7 603 537 496 466 444 48 44 9 007 709 59 57 485 456 434 48 404 0 994 699 58 57 476 447 46 409 396 983 689 573 509 468 439 48 40 388 973 68 565 50 46 43 4 394 38 3 963 673 558 495 454 46 405 388 375 4 955 666 55 489 449 40 399 383 369 5 948 660 546 484 443 45 394 378 364 6 94 654 54 479 438 40 389 373 360 7 934 649 536 474 434 406 385 368 356 8 98 644 53 470 430 40 38 365 35 9 93 640 58 466 46 398 377 36 348 30 98 635 54 46 43 395 374 358 345 40 883 607 498 437 399 37 35 335 3 60 849 579 473 44 376 349 39 33 30 0 88 554 450 39 355 38 300 93 8 788 530 48 37 335 309 90 74 6

3 = 0005 m 0 5 0 4 30 40 60 0 44 446 4630 4836 4940 5044 548 553 5359 5465 994 994 994 994 995 995 995 995 995 995 3 4369 4339 4308 478 46 447 43 45 499 483 4 097 070 044 07 003 989 975 96 947 93 5 36 338 35 90 78 60 53 40 7 4 6 05 003 98 959 947 936 94 9 900 888 7 838 88 797 775 765 753 74 73 79 708 8 7 70 68 66 650 640 69 68 606 595 9 64 63 603 583 573 56 55 54 530 59 0 585 566 547 57 57 567 497 486 475 464 54 54 505 486 476 465 455 444 434 43 509 49 47 453 443 433 43 4 40 390 3 48 464 446 47 47 407 397 387 376 365 4 460 443 45 406 396 386 376 366 355 344 5 44 45 407 388 379 369 358 348 337 36 6 47 40 39 373 364 354 344 333 3 3 7 44 397 379 36 35 34 33 3 30 98 8 403 386 368 350 340 330 30 30 99 87 9 393 376 359 340 33 3 3 300 89 78 0 385 368 350 33 3 3 30 9 8 69 377 360 343 34 35 305 95 84 73 6 370 354 336 38 308 98 88 77 66 55 3 364 347 330 3 30 9 8 7 60 48 4 359 34 35 306 97 87 77 66 55 43 5 354 337 30 30 9 8 7 6 50 38 6 349 333 35 97 87 77 67 56 45 33 7 345 38 3 93 83 73 63 5 4 9 8 34 35 307 89 79 69 59 48 37 5 9 338 3 304 86 76 66 56 45 33 30 334 38 30 8 73 63 5 4 30 8 40 3 95 78 60 50 40 30 8 06 93 60 90 74 57 39 9 9 09 96 83 69 0 7 54 37 9 09 98 87 75 6 43 5 36 9 00 90 79 67 53 36 00

3 5t - P( tn> t ( n) ) = 00 005 005 00 0005 3078 634 706 38 63657 886 90 4303 6965 995 3 638 353 38 454 584 4 533 3 776 3747 4604 5 476 05 57 3365 403 6 440 943 447 343 3707 7 45 895 365 998 3499 8 397 860 306 896 3355 9 383 833 6 8 350 0 37 8 8 764 369 363 796 0 78 306 356 78 79 68 3055 3 350 77 60 650 30 4 345 76 45 64 977 5 34 753 3 60 947 6 337 746 0 583 9 7 333 740 0 567 898 8 330 734 0 55 878 9 38 79 093 539 86 0 35 75 086 58 845 33 7 080 58 83 3 77 074 508 89 3 39 74 069 500 807 4 38 7 064 49 797 5 36 708 060 485 787 6 35 706 056 479 779 7 34 703 05 473 77 8 33 70 048 467 763 9 3 699 045 46 756 30 30 697 04 457 750 40 303 684 0 43 704 60 96 67 000 390 660 0 89 658 980 358 67 8 645 960 36 576

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