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

Transcription

1 ,, AB,, ( CIP) /, 000 ( /, ) ISBN F47 CIP ( 000) ht tp www hep edu cn ht tp www hep com cn / ,,

2 996,,,,,,, 997 7, 40 ; 998 4,,, 6, 8, 3, 5, ( ),, 3,,, ;, ;,,,,,,,,,

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

4 ( A) 9 ( B) ( A) 66 ( B) ( A) 09 ( B) ( A) 36 ( B) 38 39

5 ( A) 63 ( B) ( A) 05 ( B) ( A) 8 ( B) ( A) 60 ( B)

6 3 4F 30 5t 3 33

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

8 ,,,, ; ;, ( ) ; ( ), ; ( 3),,,,,,, ( ) 05,, 05, ( ) ( r n) De Mor gan Buffon Pearson K Pearson K ,,,,,,,,, = {, } =, =, = {, },, 6,, 6 = {,,, 6 },,, 6,, = {,,, 6} 3, r n

9 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,,,

10 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 ( ),

11 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,, 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 =

12 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) ( ) ),

13 7,, 9,,, ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( ) ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( )

14 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

15 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 ),,,,,

16 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

17 , 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,

18 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) = = 07, ( 4) P ( A B ) = P ( AB) = - P ( AB) = - 07= 03 3,,,,,,,,, ( ) ; ( ) ( ), = {,, }; ( ), P { }= P { }= = P { } = P ( ) = P ( ), = { } = = P { } =, ( =,,, ) P { },

19 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

20 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

21 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,,,

22 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),

23 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( ) = = ,, A, B, 8, , 8 00 = 683%,?, 500, 500 5

24 8,,, = 5% A, B P ( B), A, B, A, B, P ( B A), P ( B A ) = 5 5/ 00 = / 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) =

25 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

26 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 ) = = 5,, ( 3) ( 5) P ( A A A ) = P ( A ) P ( A A ) P ( A 3 A A ) P ( A A A A - ) ( 6)

27 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 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

28 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 ( ),,,,,,,,

29 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) = = 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

30 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

31 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)

32 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) = = 096 ( Bernoulli),, 7,,,,,

33 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

34 8 8 0, 3 7,, ( ) 0, ( ),, 3 3 A P ( A ) = 3, =,,, 0 ( ) B 0, Bk 0 k k= 0,,, 0 P ( B) = - P ( B ) = - P ( B0) = P ( B3) = C ( ) C 3, D 3 4 P ( C) = , 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 , P ( B) = , P ( C) = B ( B ), 8, j 8, P ( C B ) = - P ( C B ) P ( C), B C, B C 7 8 5

35 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) , 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)

36 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),

37 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,, ,,, ( ) ; ( ) ; ( 3) 8 3, 0, 7 A, 3 B; 5 ; 8,, A, ; B,,, 9 30, 0, 00, 006, ( ),, ; ( ), 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

38 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,,,? , 3 7 5, ( ) p ( ), q 8, AAAA, BBBB, CCCC,, 06, 0, ( ) ABCA ; ( ) ABCA, AAAA

39 ,,,,,,,,,,,,,,,,,,, 3, 4, 5, 6, ;,, [ 0, + ),,,,,, -,,, X = X ( ),, y= f ( x ), x, y x, x X = X ( ),,

40 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

41 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

42 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 = }= = 9 9, P {X < }= P {X = }= 9 9, P {X I } = p ( x ) ( 5) x I P {X < 5}= P {X = }+ P {X = }= = 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)

43 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 ),,,

44 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

45 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 ( )

46 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

47 4 (, f ( a) f ( b), ), ( 0) X [ a, b], b- a [ a, b] 0, [ a, b], X [ a, b], 4 X,,,,,,,, 0 ( x ) ( ) 3 3 ( f ) , x, 0 x 0 x = 0 = = x f = 0 = = 5,,,,, x, X ( ) x P {X = }= p, = 0,,,, 0,

48 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 E X = =, ( ),

49 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 = X f ( x ) = x f ( x ) dx <, x f ( x ) dx ( ) cos x, x ; 0,

50 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 x ( - x ) dx = x x 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 =,,

51 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 + ( - ) ( - ) 3 0 = = X,

52 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

53 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)

54 48 ( 3) D ( a X ) = a DX ( 33) 5 X, X DX = E ( X - EX ) = 036, EX = 9 ( 30) DX EX = = 8, D X = EX - ( E X ) = 8- ( ) = 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 x 3 - x 4 4 = 7 6, DX = E X - ( E X ) = = 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

55 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 <,

56 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,

57 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 { }=

58 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

59 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

60 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

61 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 k= 0 5 k= 0 0 k k! e < k k! e > 095, 5 95% 4( ), A p ( ),, p ( > 0

62 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), = = 4, P {0 X } = P {X = k} = b( k; 800, 500) K = 0 K = 0 K = 0 4 k k! e- 4 = e ! 038 P {X > }= - P {0X }- 038= ,,,,, [ a, b],,, 5 8,, X, X X, f ( x ) =, a x b b- a 0, ( 67)

63 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)

64 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 = e- x 000, x 0, 0, x < 0 P {X > 000}= - P {X 000}= - F ( 000) = e - 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,

65 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)

66 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) = = 005, P { X 96}= P {- 96X 96}= 0( 96) - 0( - 96) = 0( 96) - = = 095, P {- < X }= 0( ) - 0( - ) = 0( ) - [ - 0( ) ] = 0( ) + 0( ) - = = 08855, ( ) a = 053, b78 P { X < b}= 0( b) - = 094, 0( b) = P {X c}= 098< 05, - c= 053, c= ( + 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)

67 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 = P X = 0( ) = 09775; ( 7) = P {X 7}= P X = P X = 0( - ) = - 0( ) = 0075;

68 6 ( ) P {75X 0}= P X = P - X = 0( 4) - 0( - ) = 0( 4) + 0( ) - = ; X - 8 ( 3) P { X - 8 }= P < = 0( ) - = = 09545; ( 4) P { X - 9 < 05}= P {85< X < 95}= P < X - 08 < 3 05 = 0( 3) - 0( ) = X N (, ), %, - x + x 09, x? P {X > 50}= P {X < 350}, x =, = = 300, P {X < 350}= P X < = 50 0 = , 35 X N ( 300, 35 ), X - P { - x < X < + x }= P < x = x 0-09, 0 x 9 x = 095, 645, x 64535= ,, N (, ) X, Y= ax + b, a 0, Y X, X, Y X,, Y, Y, g ( x ), X, Y Y= g ( X ) ( 9) Y X,,,,,,

69 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 = - } = =, 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

70 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 ),,

71 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,

72 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

73 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 ,?? 3 (, ) 00,,, K %, K f ( x ) = 5, 0x5 0,

74 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, , 800, , 3,, 000, 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 X [ a, b], X+ ( > 0) [ a +, b + ] 3 X [ -, ], X

75 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( ) , 4 m,,, E = m - - m 5 b a f ( x ) dx ( b - a ) b a f ( x ) dx, f ( x ) d [ a, b]

76 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) ( ) }

77 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)

78 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,

79 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 p Y j ,, ( 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 P {X 0, Y= 0}= P {X =, Y= 0}+ P {X =, Y= 0} = = 005,

80 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} = = 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= } = = ( 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)

81 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

82 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,,

83 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),

84 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,,,

85 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

86 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 }

87 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} = = 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}= = 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

88 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= - }= = 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

89 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 Y X 0 p Y j 0 p X ( 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)

90 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 (, )

91 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 X Y, X Y 34, X Y

92 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)

93 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 = - } = = 05, P {= }= P {X + Y = } = P {X =, Y = 0} + P {X =, Y = - } = = 0, P {= }= P {X + Y = } = P {X = 0, Y = } + P {X =, Y = 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}, = = 035, P {= }= P {X =, Y = } = 0, P {= 4}= P {X =, Y = } = 0 ( X, Y) X Y

94 88 3, ( x, y j ) g ( x, y j ) ( X + Y, X Y ) Y X Y X g ( x, y ), g ( X, Y),, P { = - }= 0, P { = 0}= 05, P { = }= = 0, P { = }= 0, P { = 3}= 0, P { = 4}= 0, P { = - }= 05, P { = - }= 03, P { = 0}= = 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)

95 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) = D 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),

96 ( ) 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 z y f ( x, y ) dx dy f ( x, y ) dx dy Z - yf ( zy, y) dy

97 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

98 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 ) = xy, x yg ( x, y) dx dy = dx x y 8xy dy 0 x = x y, 0, 30 ( X, Y) 3, E X Y E X Y = 0( - ) ( - ) ( - ) = 0 3, X Y, [ 0, 0], 000,,

99 33 93, 500, Z, Z= g ( X, Y) = ( X, Y) E Z= 0 f ( x, y) = = 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 = ( ) 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

100 = = + = + ( ) 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),

101 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,

102 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 = = 095, EY= ( - ) = - 05 E X Y= 0, cov( X, Y) = EX Y- E X EY= 09505= ( 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 5 EX = x g X ( x ) dx = 0 x 4x ( - x ) dx = 3

103 34 97, E Y = + - y g Y( y ) dy = 0 y 4y 3 dy = 3 DX = E X - ( EX ) = DY= E Y - ( E Y) = = 5 = 75 D ( X + Y) = DX + DY + cov( X, Y) = = 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,,

104 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?

105 ( 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

106 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] = = 0 37 ( X, Y) 38, E [ X Y= y ] ( y < )

107 , 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

108 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

109 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,

110 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

111 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 ( ),,,,

112 06 3,, E =, limp = - < = ( 39),, = E,,,?,,,,,, 3, 4,,,,,, =,,,,,,, ( ),,, = 30 ( ),,,,, E =, D ( ) = > 0, =,,, lim P = - x = x - e- t dt ( 393),, 30 N ( 0, ), =

113 , 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,

114 08 3 P { > 000}= P > = P > 00 = - P ( ) = = 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} , 07,, , B ( 0000, 07), E = = 7000, D = = 00, P {6800< < 700}= P = P < < < ( 436) - = , [ 0, 000] ( ) 0%,, ,? X, X, A = {X

115 09 00}, A =,,,, P {A }= 09, B(, 09), {X 00}= 64, 5 3 P {X 00}= P 08, 08 = P { > 08} = P = > 3 = ( 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 }

116 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

117 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 EX Y, E ( X+ Y) 3,,, [, ], 3 ( 353) ( 354) 33 60, 40, 0, 34 9, 35 33

118 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 X Y X, Y 4 A B, r A, r B, r A r B V V= 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 % % % ( ) ; ( ) r f = 5%, 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,, , 07,, 5,, 95%? 48, ( ),, [ - 05, 05]

119 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

120 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}

121 4, ;,,,,, 43 F t,,, 4,,,,,,,,,,,,, X ( ),

122 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,

123 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 ),

124 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

125 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,,,

126 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

127 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 ), ( )

128 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

129 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) = 005, 005 0( u 005) = - 005= 095, u 0 05= u 00 5, 0( u 0 05) = - 005= 0975

130 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

131 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 X m X m +, X Y

132 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

133 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+

134 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}= F, X F ( m, ),, ( 4), F 095( m, ) = F ( m, ) = X - F (, m) F - (, m), ( 4) F 005(, m), F ( m, ) =, X F ( 5, 0), P X < 474 = 005, F 005(, m)

135 43 9 P X 44 = 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,

136 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,

137 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,

138 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 )

139 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 / ( - )

140 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,

141 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 ),

142 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 ( ) ; ( 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 )

143 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)

144 38 4 ( B) X ? X ( X, X,, X ), X X = 4, X 00, X- ( X ) 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 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 ( )

145 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 ( )

146 X ( ) f ( x ; ) = ( x > 0), > 0 68, 30, 69, 43, 74, 98, 08,, 5, e- x, X, = E X, X = X x = ( ) = 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 =,

147 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

148 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 ( )

149 ( 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 )

150 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

151 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( ( ), ) = ( x - ) = 0, 4 0= = A = B= lnl( x = x,, ) 0= ( 0, lnl ( ( ), ) C= lnl(, ) ( ) A < 0 AC- B = ( - 0, ) ( ) = - = = ( x - x ) 0 = 0 (, 0 0 ) ( 0, 0 ) = - ) > = 0 = X ( X - X ) =, = x, = = ( x - x ) MLE, MLE MLE,,

152 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 ( ) P {X > 80} E X L( ) = = e- x, ( x > 0, =,, ) lnl ( ) = - ln - n x i= d d ln L = x = 0, = 0= x, = X x = 77 P {X > 80}= e - 80 P ( X > 80) X, P ( X > 80) 7 7 = 0353 e e X [ 0, ],, > 0 ( X,,

153 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 ( )

154 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 ) =

155 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 )

156 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,,, ( -, + )

157 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,

158 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

159 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

160 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 ) / } , 08 p 95% 0, p, p p -

161 54 55 ( ( X - u X ( - X ) /, X + u X ( - X ) / ) ) = 400, X = = 07, = 005, u 005 = 96, ( / 400, / 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, )

162 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, = 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

163 X N (, ),, = , x = % = 36, = 450, = 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, ( ) 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,, - ;, -,, ;,,,,

164 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 - ) =

165 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 = = 406, 90% ( 86406) ( 889, 90) ( - ) S - ( - ), ( - ) S - ( - ) 0 0 ( 5- ) = 364, ( 5- ) = 385, = 974, 90% ( 974, 580) = 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)

166 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,

167 556 S = ( - ) S + ( - ) S = 68, S= 846, + - ( 09-06) = - 359, ( 09-06) = % ( - 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, ( 034, 6), 5 5 S

168 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 / + /

169 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,

170 ,, 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 ; ( ) = , 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%

171 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%

172 6,,,,,, 44, ,, 65,,,, 6,, 6, X F ( x ; ),,,,, X N (, ), =, ; =,

173 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, ;,,,?,,

174 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)

175 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/ 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),

176 70 6,, 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 ; ),,,,,,,,,,,,

177 6 7,,,, H 0, H 0,, ;, H 0,, H 0,, ( 66),, H 0,,,,,,,,,,,,,,,, ;, 6 X N (, ),,,,,

178 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)

179 P {( X, X,, X ) C H 0}= P P = X - 0 0/ > u H 0 X - 0/ > u H 0 6, X N (, ),, 500,, 5, ,, 5? H H > 500, 0= 500, 0= 00, = 5 = 005, u = 645, x = 575, u= x = 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)

180 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

181 5, t / ( - ) = t 00 5 ( 8) = x = 9998, s = 9 8 ( x - x = s=, t= x - 0 = s, t = 0495< 306= t 0 05 ) = 9 8 = x - 9x = 469, 9 = ( 8), H 0, , 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 = s/ 0, t= = - t = - 6 ( 5), H 0, X N (, ),,,, ( X, X,, X ), 6 75 W = - S ( 67)

182 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 s > ( - ) ( 6) 0 S - S > ( - ) - S > ( - ) ( 6) 0

183 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, ? ( = 005) H 0 = 0 H 0, = 005, 3, - ( - ) = ( 6) = 37, ( - ) = 00 5 ( 6) = = 436, ( - ) s = x 6 77 ( x - x ) = 006 =

184 78 6, w = ( - ) s = 006 = 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 = = = 00 5 ( 9) 0 64, H 0,,,, 6 63,, X N (, ), YN (, ), = -, r = /,, r, ( X, X,, X ) X, ( Y, Y,,

185 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 / H 0 0 H < 0 X N (, ),, 0 H 0 = 0 H 0 T = X - S/ T 0= X - 0 S/ - t

186 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

187 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

188 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 = = = 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 =

189 , y = 704, u = x - y = = = 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 , 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

190 84 6 3, T ( 0= 0 ), ( 66), x ( x, x,, x ; y, y,, y ) - y > - t ( + - ) C= s ? ( = 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 = = ( ) = = 005, 5, t ( - ) = t 005 ( 5) = 060 t = = 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 (, ),,,

191 63 85 = 7, = 8, = + = 5, x = 5, s = 0003; y = 44, s = 0005 ( 65) s= 00646, t= S x - y = = 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)

192 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

193 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 = = 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 (, )

194 88 6 = 7, = 9, x = 5057, s = 0745; y = 5033, s = f = s = 0745 s = 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 <

195 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 <

196 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)

197 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

198 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, % H 0 p 05 = 00 U U

199 64 93 u = = 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

200 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= 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=

201 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

202 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 -

203 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,

204 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)

205 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

206 ( = 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) = = 3< 005( 4) = 9488,,, ( 68) F 0( x ),, F 0 4 H 0 F ( x ) = F 0( x ;,,, t ), ( 657), t,,, t, H 0

207 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 ( 3),,, r N, N,, N r,, 0 ( 660),,, C= {( n, n,, n r) 0> ( r- t - ) } ( r - t- ) r- t-,,, t+, r t ,, x = 360 ( g), s 0= 4655 ( g ), 450g, 3700g 3 ( 450, 700], ( 700, 950], ( 950, 300], ( 300, 3450], ( 3450, 3700] 5 7 0

208 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 ) = 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, p, p, ( - p, 0) ( - p, 0) / p, , 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,

209 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

210 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, % % % ( = 00) A A ; B B, r = q=, = 43, = 6, = 3, =

211 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) cm,, 0, 0053, 0003, ( ) = 005, ( ) = 00, 6, ( kg/ cm ),, 0, 0533, 064, 0688, 057, 0793, 079, 0600, 0683, 07, 0570 = 005,

212 06 7, N ( 80000, 4000 ) ( kg/ cm ), 00, ( ) = X,, 7, 44, 405, 4357, 4394, 436, 487, ( = 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, ( = 00)? 6 0, 0,,,, 9,, 800, 0,,,, 9

213 ? ( = 005) 7 99, ? ( = 005) 8 3 ABC 4, 00, A B C ( = 00)? ( B) X N ( 0, ), 0, ( X, X,, X ) X, ( 67) W,, XN (, ) YN (, ),, r = / ( X, X,, X ) ( Y, Y,, Y )

214 08 X Y ( 69) F,, ( ) H 0 r = ; ( ) H 0 r; ( 3) H 0 r, 3,,,, 4000,, ,, ( ) ; ( ) H 0 = 0 H 00 4, a, b, c, A, B, C,, a ( b, c) A ( B, C) ( = 005) X ) 5 X p ( 0< p < ), p ( X, X,, X H 0 p = p 0 H p p 0 ( p 0 )

215 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, , X? p ( x ) = x, 0< x < ; 0,

216 7 t, ( A nalysis of V ariance, A NOV A ) ( R A Fish er ) MBA, MBA MBA ( ) A A A A ,, A 4,, MBA?,,,

217 7,,,,,, ;,,, 7,,,,,,,, 7 7 MBA, MBA, MBM 7, MBA, A, =,, 3, 4,, B, j =,, 3, 4, 5,, MBA,,,,, ;,,, MBA,

218 7, 0, 5, 4, 0,,,,, ( B j, A ), j =,, 5, =,,, 4,, 7 7- MBA ( ) B B B 3 B 4 B 5 A A A A A r, A, A,, A r A X, A X ( =,,, r)

219 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)

220 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

221 7 5, =,, 3, 4 = 938, = 9, 3 = 995, 4= 08, = 97075, = , = , 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,

222 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

223 7 7, ( 7) ( 77) ;, ( 79) Q Q ; 3 ( 7) F ; 4, F F ( r-, r ( k- ) ), F, 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- ), Microsoft Excel SS df MS F P Value F crit , SS df MS, F 74 F, 75 P Value F E xcel 97,, 999

224 8 7 F,, F > F P Valne< P Value F, , 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,

225 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,,

226 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 ) ]

227 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- ) ) }

228 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- ) ) }, ( ) 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, F A F B, F A F B, F A F A F B F B, Excel97, 78, 79, 005

229 MBA ( ) B B B 3 B 4 B 5 A A A A A, B (, 78 ) ; SS df MS F P Value F Cr it , 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

230 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 ) =

231 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- ) ) },

232 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, , A, B,,

233 ( 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 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)

234 8 7 = X - X = = 4 = X - X = = = X 3- X = = 0 4= X 4- X = = 088 5= X 5- X = = - 85, B MBA,,, B MBA ( A) 6, ( cm ) , ( = 005), 5 500, A B C ( = 005) 3 5, 35, 7, ,?

235 9 4 A A A A = 005 ( )? ( )? ( 3)? ( B) 7 H 0 = = 3= 4 t t = 005, H 0? ( 74) =,, r A =,, r 3 ( 75) 4, A, F t

236 8,,,,,,,, 8,,, s v t s= vt, v t, s,, 8, 8 8 ( )

237 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 +

238 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,

239 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)

240 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,

241 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= , = , 0 Y = x 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

242 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)

243 837 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)

244 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 =

245 = = ( 0+ x 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 =

246 - = ( 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)

247 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, Excel97, df S S M S F Significance F Excel97, 83 Significance F F P, , 005 H 0 = 0,,, 8 F 005 F 00 5(, 8) = 53, F, F = 9347> F 005 (, 8),

248 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)

249 843 > F, F F (, - ), r > F ( 848) d d = F (, - ) (, - ) + ( - ) F F (, - ) (, - ) + ( - ) ( 950) r d H 0, d r, r ( - ) = d,, 8, r = , = 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)

250 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 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= , = , ( 857) ( 853) = 43404, = 3054 t t 0 = 736 t = 35964, 0 8 t , t 0 t, 005 H 0 0= 0 H 0 = 0 Excel97,, t 0 t P, t 0 t P 989

251 , P 005,, H 0 = 0 F t F t P ,,, 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)

252 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 ( 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

253 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 = ( 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 =

254 48 8 x 0= 55, Y0 Y0 Y 0= = 53353, E ( Y0) 53353, Y0 E ( Y 0 ) 95% ( , ) = ( , ), Y0 ( 55) = 3396, 95% ( 40034, ), x 0= 365= x, 95% ( , ) = ( , ), 95% ( 873), ( , ) = ( 37384, 5476) u , 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 ),

255 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= , , Y x ( ( 84) ),, Y x,, 9 8, Y x,, x Y,,, Y x, Y -

256 x ( ) Y ( ) lny Y = 0e x ( 876) Y = 0+ x ( 877) 86 lny x,, lny x, ( 876) 85 86, 86 ( 876), lny x ( 876) Y x, ( ), ( 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) ( 4934) x t Y x

257 Y = 3708e x ( 88) ( 877),, w = x, ( 877) Y w Y = 0+ w ( 88), w = x ( =,,, ), ( w, Y) ( =,,, ) ( 88) 0 ( 877) Y x 94 Y = 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,

258 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,

259 , R S 85,,,,

260 R S Y = x Y = 3708e 0003x Y = x ,,,,, 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)

261 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 =

262 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

263 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

264 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)

265 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 p x 0p + 0, Y 0= 0+ x 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 -,

266 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, ( ) , ( ) ; ( ) Y x, ; ( 3), %,,, ( = 005),?

267 , Y ( % ) x x ( ) ; ( ) ( = 005) ; ( 3) ( = 005), 4 0 Y x,, x, Y x x Y x x ( ) ( ) ( / ) x y 8

268 6 8 x y , ( ) ; ( ) 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

269 9 ( ),,,, 9,,,,,,,,,, 9, 3,, 3, 3 3, 3,

270 K K K 3 K4 K 5 K 6 K 7 K 8 K 9 K 0 K K K ( ) K K7 7, K 8K3 6 ; ( ) K K ( 3) 3,

271 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)

272 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,

273 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 = ( , - 08, 0790) ; 3= 03384, e3 = ( 083, , 0085) ; 0 Y= 0034X X X 3; Y= X - 08X X 3; Y3= 083X X X 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

274 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= = 7559, e = ( 0664, 0707, 0500) ; = 0000, e = ( , 00000, 09354) ; 3= 044, e3 = ( 0664, , 0500) ; ( 9) Y= 0664( X - ) + 05( X - ) ( X 3-3) ; Y= ( X - ) ( X 3-3) ; Y3= 0664( X - ) - 05( X - ) ( X 3-3) ;,, 3 X, X, X 3, Z, Z, Z 3, Y / 3= 5853%, Y Y ( + ) / 3= 986% 9, X X,,,, p

275 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)

276 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,

277 y = e X, =,,, p, k X k, y k= e Xk, =,,, p ; k=,,, y k= ( y k, y k,, y p k) ( k=,,, ), ( k, k,, k3), 3, k k S = , S = , k 6 S 6= , 9, 93 y y y y y y y y , 8 95%, 3,, 70%, = 5768 y = 0349z + 035z + 07z z z z z z z9+ 094z z+ 044z z 3 ( 95)

278 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 ( )

279 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

280 ( 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

281 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)

282 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 k k k k k k R 94,, ,, l m,, u k 09648, v k 09794,,, k k

283 77 94 l m l = ( 00863, 09648, , 0077, , 0094, 036) m = ( 0047, 00396, , 09794, 00337, - 008) l = ( 073, , 033, 0648, 00865, 000, 00435) m = ( 0374, 059, 00664, - 030, - 004, 0738) l 3 = ( , 0988, - 080, 0780, 0039, , ) m 3 = ( - 05, , 0, 090, 070, 08095) l 4 = ( 03058, - 034, 086, , 06633, - 4, ) m 4 = ( - 737, 5, 03780, , 0098, ) l 5 = ( , 033, - 048, , 075, 0074, 03688) m 5 = ( 0469, , 099, 0938, , ) l 6 = ( - 348, 09988, 07534, 04939, , 0389, 0453) m 6 = ( , , 007, 0033, 859, ) X, X, X 3 = 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,

284 78 9 R= ( X, X ) ( X 3, X 4 )

285 905, 05, 00, ( ) CK N C - K N - N C, ( ) - N ( 3) - 4! P C N - N C N - C N C - N - N C N ( C 4 30), P 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 ( ) , 6, 5 6 8C ! a - a + b-, a - a + b-, a ( a - ) a ( a + b) + a ( b- ) 5 3, 6 3

286 80 4 7( ) 0, ( ) 4 90, ( 3) , , , , ( B) 9 9, 0 9 ( - r - ), ( - ) - 3C N N - K ( ) N - K 4 mk - ( m- ) K K p - q q, q p - q q 7p = 9 90 q= , 06 X = F x( x ) = 40 Y= 0 0 x < 0 x 40 ( ) ( 00 - ) ( ) 3 ( A) 0x< 40 F Y( x ) = x < 0 0x < 30 x 30

287 8 4( ) P { = k}= 03 k- 07 k=,, 3 4 ( ) p , 4 5, 7 0 6, 039 8( ), f ( x ) = 9( ), F ( x ) = 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, , ( e- 7( ) P { = k}= C k ( 094) k ( 006) - k ( ) ( 094) 8p q k - e - ) ( 3) - ( 094) - ( 094) ( 4) 006, ,,, e 43e - - e e , p p

288 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 ( a + b) ( a + b ) (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 < 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

289 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 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

290 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 , 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 e - 3x x 0 0 x < 0 4e - 4y y 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) =

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

292 86 x + 0< x< 44E [ Y X= x ] = x x X+ 0< X< E [ Y X] = X X 45E [ X Y] = E [ Y X] = , 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 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 / / 79 75/ / 79 0/ 79 35/ 79 / 79 X Y 0 0 3/ 66 / 66 6/ 66 5/ 66 0/ /

293 87 X Y / / 990 5/ 990 3/ 990 5/ / 990 0/ 990 0/ 990 5/ / / 990 5/ 990 ( ) P {Y= k, Yj = s}= Ck 5C s 5- k- s 5- k7 9 5 ( k+ s5) ( ) P {Y= k}= Ck 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

294 88 ( ) f Y( y ) = ( 3) f X Y( x y ) = ( 4) 474 ( 5) ( ) 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 x 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 e - 0 e - - e - e - cos ( X, X ) = e - - e - 3 X, X = e- e -

295 89 9 U 3( ) 0 V 3 ( ) f Z( z ) = 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 ) = ( + ) = ( ) 0f ( x) = = 35 ( - ) m m m x m - e - x, x> 0 c= 3, P ( S = k) = ( ) k e -, k= 0,,, ; k!, S N (, ),

296 90 x - P ( S x ) 0 3u 06= 053, u 08= 0846, u 09= 8, 444 u 095= , 07, 558, , 00684, , 3365, 45, ( - 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

297 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 +

298 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 a 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) ),, - = -

299 93 ( B) X () - X ( ) 3( 766, 3795) ( A) e -, - e - c= ( ) = 005, ; ( ) = 00, ( ) H 0 = 000; ( ) H 0 5,, 5,, / ( B) W = = ( X - 0)

300 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

301 P{X= m}= m m! e m m

302 96 m

303 97 m m 0 m

304 98 x - e- t 0( x) = dt( x 0) x

305 99 x x

306 300 3 P{ ( n) > ( n) }= =

307 30 = =

308 30 4F P {F ( m, n) > F ( m, n) }= = 00 m

309 303 = 00 m

310 304 = 005 m

311 305 = 005 m

312 306 = 005 m

313 307 = 005 m

314 308 = 00 m

315 309 = 00 m

316 30 = 0005 m

317 3 = 0005 m

318 3 5t - P( tn> t ( n) ) =

319 [ ] V K, 983 [ ], 983 [ 3 ],, 988 [ 4 ], 979 [ 5 ],,, ( 3), 994 [ 6 ],, 3, 998 [ 7 ],,, 990 [ 8 ], 98 [ 9 ],,, 994 [ 0] Alder H L, Roessler E B,, 984 [ ], 98 [ ],, 989 [ 3], 998 [ 4], 998 [ 5],, 998 [ 6], 990 [ 7], 999 [ 8], 993