,, AB,, ( CIP) /, 000 ( /, ) ISBN 704009448 F47 CIP ( 000) 86786 55 00009 0064054588 ht tp www hep edu cn ht tp www hep com cn 006404048 787960/ 6 05 370 000 730,,
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3,,,,, 4,,, A, B, A, B ( ), 999, ( ),,,,,, 000 3
8 3 4 7 5 4 ( A) 9 ( B) 3 33 33 4 3 50 4 56 5 6 ( A) 66 ( B) 69 70 3 70 3 78 33 86 34 94 35 03 ( A) 09 ( B) 3 5 4 5 4 9 43 44 3 ( A) 36 ( B) 38 39
5 39 5 43 53 48 54 50 55 56 ( A) 63 ( B) 65 66 6 66 6 7 63 78 64 90 65 94 ( A) 05 ( B) 07 0 7 0 7 73 8 ( A) 8 ( B) 9 30 8 30 8 37 83 45 84 49 85 54 ( A) 60 ( B) 6 63 9 63 9 7 77 79 95 95 98 3 300
3 4F 30 5t 3 33
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,,,, ; ;, ( ) ; ( ), ; ( 3),,,,,,, ( ) 05,, 05, ( ) ( r n) De Mor gan 048 06 058 Buffon 4 040 048 05069 Pearson K 000 6 09 0506 Pearson K 4 000 0 05005,,,,,,,,, = {, } =, =, = {, },, 6,, 6 = {,,, 6 },,, 6,, = {,,, 6} 3, r n
3, = 0,,,, = = {0,,, } {0,,, } 4, (! ) t, 0t<, = {t 0 t < + }, = {t 0 t < } = [ 0, + ),,,,,,, 7,,,,, 7,,,,,,, A, B, 5, 6 A 5 B 5 C A, B, C, A,,,, 6 ( 6 ), 6,,, B C, C 4,,,
4,,,, ( ),,,, A, B,,, A A 6 5, = {,, 3, 4, 5, 6}, B C B = {,, 3, 4} C = {, 4},,,,,,,,,,, A B, A B, B A, A B, A B B A A B, A B A, A A B, B A, A B ( ), A = B,, A = B A B, 3 ( ) A B A B ( ),
5 A B, A + B, A B A B, 7, A B= {,, 3, 4, 5} 4 ( ) A = B = 5 A B A B ( ), AB ( AB), AB A B, 7, 5 A B A B = {, 3} A B A B, A- B 7, A B 6 A - B = {5} A B,, AB, AB=, A B,, 3 4 7 A, A, A,, A = - A, A = = A,, A, A, A, A, A A, A A =, A A =, A, A 8 A, A,, A, A, A,, A A, A,, A, A A A, A = A, A,, A A, A,, A, A A A, A =
6 A, A,, A,, A, A,, A,, A, A,, A,,, A, A, = A,, A, A, A,, A,, A = 9 A, A,, A,, ( ) A A j =, j,, j =,, ( ) A = A, A,, A,, A A ) 0 ( ) ( 8, A, B, C, D, P, F ( ) A B ( [ 90, 00] ), D ( [ 60, 70) ), ( [ 80, 90) ), P ( [ 60, 00] ), C ( [ 70, 80) ), F ( [ 0, 60) ), A, B, C, D, F ; P F, P = F ; ABCD P, P = A BCD 9, A, B, C, ( ) A ; ( ) AB ; ( 3) ABC ; ( 4) AB C + A BC + A B C; ( 5) A + B+ C; ( 6) A + B + C ; ( 7) ABC + AB C+ A BC; ( 8) AB + AC+ BC; ( 9) A B C ; ( 0) AB C + A BC + A B C+ A B C ; ( ) ABC;, ( 9 ( 6) ( ) ),
7,, 9,,, ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( ) ( ) A B= BA ( ) ( ) ( AB) C= A( BC) A BC( )
8 3 ( ) A ( BC) = ( AB) ( AC) ( ) ( ) A ( BC) = ( AB) ( AC) ( ) 4 ( A = ) = A( ) 5 ( ) A B= A B ( ) ( ) A B= A B ( ),,,,,,, A?,,,,,?, A, A,,,,,,,,,,,, A ( ), A, P ( A), A, A
9,,,,,,,,, A, r ( A ), f ( A) r ( A) f ( A ) = F, F A F, f ( A ) F A,, f ( A) ( ) f ( ) = ; ( ) A, f ( A ) 0; ( 3) A, A,, A,, f = A = = f ( A ),, f ( A),, f ( ) = 0, f ( A),, A P ( A), A, P ( A), F,,, ( ) P ( ) = ; ( ) A, P ( A ) 0; ( 3) A, A,, A,, P = A = = P ( A ),,,,,
0,,,,,, 933,,,, A A, F P ( A) F, A P ( A ) P ( A), A P ( ), ( ),, P ( ), F,, P ( ) P ( ) = ; A, P ( A ) 0; 3 A, A,, A,, P = A = = P ( A ), A, P ( A ) A, P ( ), (, P ),,,,, P ( ) = 0 F,, (, F, P ),,, F
, A i=, =,,,, A = = 3 P ( ) = P = A = P ( ), P ( ) = 0 = P ( A ) = P ( ) = ( ) A, A,, A, P = A = = P ( A ) A =, = +, +,, 3 3P ( A ) = - P ( A ) 4P ( A- B) = P ( A) - P ( A B) A B, ( ) P ( A - B) = P ( A) - P ( B) ( ) P ( A ) P ( B) 50P ( A) 6P ( AB) = P ( A) + P ( B) - P ( AB) 6,, P ( A BC) = P ( A ) + P ( B) + P ( C) - P ( AB) - P ( AC) - P ( BC) + P ( ABC) 0 5, A, P ( A ) = P ( A 0), =,, 3, 4, 5 ( ) 5 ; ( ) ; ( 3) A 0, A,, A 5 A =, = 0 5 = P ( ) = P = 0 A = 5 = 0 P ( A ) = 5 5 P ( A 0) + P ( A 0) = 6P ( A 0), = P ( A 0) = 6, P ( A ) =, ( =,, 3, 4, 5) 6 ( ), ( ), ( 3) A, B, C, ( ) P ( A) = P ( A 0) = 6,
5 5 ( ) P ( B) = P ( A ) = - P ( A 0) = = 6, ( 3) P ( C) = P ( 3 A ) = P ( A 0) + P ( A ) + P ( A ) + P ( A 3) = 7 = 0 6 P ( A ) = 05, P ( A B) = 0, P ( B) = 04, ( ) P ( AB) ; ( ) P ( A- B) ; ( 3) P ( A B) ; ( 4) P ( A B ) ( ) AB+ A B= B, AB AB, P ( AB) + P ( A B) = P ( B) P ( AB) = P ( B) - P ( AB) = 04-0 = 0 ( ) P ( A ) = - P ( A ) = - 05= 05, P ( A - B) = P ( A) - P ( AB) = 05-0= 03, ( 3) P ( A B) = P ( A) + P ( B) - P ( AB) = 05+ 04-0= 07, ( 4) P ( A B ) = P ( AB) = - P ( AB) = - 07= 03 3,,,,,,,,, ( ) ; ( ) ( ), = {,, }; ( ), P { }= P { }= = P { } = P ( ) = P ( ), = { } = = P { } =, ( =,,, ) P { },
3 3,, A, P ( A) A, A m,,,, m, A = {,,, m } = m 3 P ( A) = { j = j } m P { j } = m j =, P ( A ) = A A A =, 0, 3, 7, ( ), ; ( ), ( ) 0, C 0= 0, 0 3, C 3= 3,, A P ( A) = C 3 C 0 = 3 0 ( ) 0 C 0,, C 3C 7, C 3, B, C P ( B) = C 3C 7 C 0 P ( C) = C 3 C 0 = = 45 = 7 5, 3 45 = 5 3,, 3, 4, ( ),, 3, 4 ; ( ) ; ( 3) ; ( 4) ( ) 4 4! = 4, 4
4 ( ), ( ), ( 3), ( 4) ABCD ( ) A, P ( A) = 4 = ( ) B ( 3! ) =, P ( B) = 4 = ( 3),, 3,,!, C 3=, P ( C) = 4 = ( 4),,,, P ( D ) = 4 N ( N ),, ( ) ; ( ) ; ( 3) ; ( 4) k( k) N, N, ( ), ( ), ( 3), ( 4) A, B, C, D ( ),,!, P ( A) =! N ( ),, C N (! ) ( P N ), P ( B) = P N N ( 3) C C, N -, ( N - ), P ( C ) = ( N - ) N, P ( C) = - P ( C ) = N - ( N - ) N ( 4) k ( C k ), - k
3 5 N - ( ( N - ) - k ), k C k ( N - ) - k, P ( D) = Ck ( N - ) - k N 5 a + b, a, b, ( ), ( ) ; ( ) ; ( 3),, ( a + b) ( a + b)!, ( ), ( ), ( 3) A, B, C ( ) a,, a + b-, ( a + b- )!, A a [ ( a + b- )! ], P ( A) = a [ ( a + b - )! ] ( a + b)! = a a + b ( ) a, - b - ( P - b ) a + b- ( ( a + b- )! ), B a P - b [ ( a + b- )! ], P ( B) = a P - b [ ( a + b - )! ] ( a + b)! ( 3) C C P b [ ( a + b- )! ] P ( C ) = P b [ ( a + b - )! ] ( a + b)! = a P - b P a + b C,, P ( C) = - P ( C ) = - = P b P a + b = C b C a + b C b C a + b 5 ( a + b) ( a + b) ( a + b),, 5, ( ), a a + b,, a, a + b,,,
6, ( ),,, ( 3) ( 3),,,,,,,,,,, t= S ( ) ( ) A A S ( A ), A A, P ( A ) = ts ( A) t P ( ) = ts( ) = S ( ) P ( A) = S( A ) S( ) ( ) ( ),, ( ) ;, 6,,,,, ( ) 0, 0, 60, ( 0, 60), 0 A, = ( 0, 60), A = ( 50, 60),
4 7 P ( A) = 0 60 = 6 7 ( ) 7 8, 0,, 7 0,, x, y, = {( x, y ) 0 x 60, 0 y 60} A, A = {( x, y ) ( x, y ), x - ( 3) y 0},, P ( A) = S( A ) S( ) = 60-40 = 5 60 9 3 4,, A, B, 8, 475 644 9 5 56 8 500 700 00, 8 00 = 683%,?, 500, 500 5
8,,, 5 500 = 5% A, B P ( B), A, B, A, B, P ( B A), P ( B A ) = 5 5/ 00 = 500 500/ 00 = P ( A B) P ( A),,, P ( A) > 0, P ( B A) = P ( AB) P ( A) ( ), ( 4), A, B P ( B A) = S ( AB) S ( A) 4 P ( B A) = P ( AB) P ( A) = S ( AB) / S ( ) S ( A) / S ( ) = P ( AB) P ( A),,, 3 (, P ), A, B, P ( A) > 0, P ( B A) = P ( AB) ( ) P ( A ) A, B B F, P ( B A) F, A, A, P ( A) > 0, P ( A) ( ) P ( A) =
4 9 ( ) B, P ( B A) 0 ( 3) A, A,, A,, P ( = A A) = = P ( A A ), (, P ) A, P ( A) > 0, P ( A ), A=, P ( A ) P ( ), P ( A),, ( ) 9 0, 3, 7, ( ) ( ), ; ( ), A ( =, ) ( ) A,, 7 9,, 9, P ( A A ) = 9 ( ) A,,, ( ) ( ) P ( A A ) ( ) P ( A A ) = P 3 = P 0 5, P ( A ) = 3 0 P ( A A ) = P ( A A ) P ( A ) = 9 9,,,,, a b? 9, 0,, 7, 9
0, ( ) ( ), P ( B) > 0, P ( A B) = P ( A) P ( B A) ( P ( A ) > 0) ( 3) P ( A B) = P ( AB) P ( B) ( 4) P ( AB) = P ( B) P ( A B) ( P ( B) > 0) ( 5) ( 3) ( 5), P ( B A ) P ( A B) P ( A B) 0, 60%, 0%,, A, B,, P ( A) = 60% P ( B A) = 0% P ( A B) = P ( A) P ( B A) = 60%0% = 6% 0, 3 7, ( ),,,,,, A ( =, ), A A, P ( A ) = 3 0 P ( A A ) = 9 P ( A A ) = P ( A ) P ( A A ) = 3 0 9 = 5,, ( 3) ( 5) P ( A A A ) = P ( A ) P ( A A ) P ( A 3 A A ) P ( A A A A - ) ( 6)
4,,,, 0, 3, 7, ( ),,,, A ( = ), P ( A ) = P ( A A ) + P ( A A ) = P ( A ) P ( A A ) + P ( A ) P ( A A ) P ( A ) = 3 0, P ( A ) = 7 0, P ( A A ) = 9, P ( A A ) = 3 9, P ( A ) = 3 0 9 + 7 0 3 9 = 3 0 3,, 60%, 40%,,, 80%,, 40%, A, A, B P ( A ) = 60%, P ( A ) = 40%, P ( B A ) = 80%, P ( B A ) = 40%, P ( B) = P ( AB) + P ( A B) = P ( A ) P ( B A ) + P ( A ) P ( B A ) = 60%80% + 40%40% = 64%, B
5, A ( A A ) B AB A B, ( A A ) B, P ( B A ) P ( B A ) 5 5 ( ) {A }, A =, B, P ( B) = P ( B) = P ( B ) = P [ B ( A ) ] = P [ ( B A ) ] = = P ( A ) P ( B A ) P ( A ) P ( B A ) ( 7) P ( B A ) 3,,,,, 4 ( ), 9 ( ),,,,,,,,
4 3 A, B, P ( A B), P ( A B) = P ( AB) P ( B) P ( AB) = P ( A) P ( B A) P ( B) = P ( A) P ( B A) + P ( A ) P ( B A ) P ( A B) = P ( A) P ( B A) P ( A ) P ( B A ) + P ( A ) P ( B A ) P ( A) = 3 0, P ( B A ) = 9 P ( A ) = 7 0, P ( B A ) = 3 9 P ( A B) = 3 0 9 3 0 9 + 7 0 3 9 = 9 ( 8) ( 8),,, ( Bayes T homas) {A }, A = =, B, P ( B) > 0, P ( A B) = P ( A B) P ( B) = P ( A ) P ( B A ) P ( A ) P ( B A j ) j,, ( 9) 5, 45%, 35%, 0%, 4%, %, 5%,, ( ) ; ( ), A ; A ; A 3; B
4 P ( A ) = 45%, P ( A ) = 35%, P ( A 3) = 0%, P ( B A ) = 4%, P ( B A ) = %, P ( B A 3) = 5% ( ) P ( B) = P ( A ) P ( B A ) = 3 = 45%4% + 35%% + 0%5% = 35% ( ) ( ) P ( A B) = P ( A B) P ( B) = P ( A ) P ( B A ) P ( B) = 45%4% 35% = 54% 5,,,,,, 5, B A, P ( B) > 0, A B, P ( A) > 0, B A P ( A B) = P ( A ) ( 0) P ( B A) = P ( B) ( ) P ( A) > 0, P ( B) > 0, ( 0) ( ) P ( AB) = P ( A ) P ( B) ( ),, A B B A, A B P ( A ) = 0, P ( B) = 0 ( ),, 4 (, P ), A, B, P ( AB) = P ( A ) P ( B) A B, A B 6 ( ) A 5, B, P ( A) = 4 6, P ( B) = 3 6, P ( AB) = 6 = 3
5 5 P ( AB) = P ( A ) P ( B) = 3 A B ( ) A 4, B ( ), P ( A) = 3 6, P ( B) = 3 6, P ( AB) = 6 = 3 A B P ( AB) P ( A ) P ( B) 5 ( ) A, A,, A, < j, A, A, P ( A A j ) = P ( A ) P ( A j ) ( 3), A,,, A, A, A 3,,, P ( A A j ) = P ( A ), j,, j =,, 3,,, P ( A A A 3) P ( A ), 6 A, A,, A ( ) k ( k ) A, A,, A k ( < < < k) P ( A A A k ) = P ( A ) P ( A ) P ( A k ) ( 4) A, A,, A, =,, >,,,, 5,, k ( k)
6,, A, A,, A, m ( m ), A, A,, A, P ( A ) = - P ( A ) = - ( - P ( A ) ) ( 5) = = = m=,, 7,, 05, 06, 08, ( ) ( ) A ( =,, 3), A A A 3+ A A A 3+ A A A 3, A + A + A 3 ( ) P ( A A A 3+ A A A 3+ A A A 3) = P ( A A A 3) + P ( A A A 3) + P ( A A A 3) = P ( A ) P ( A ) P ( A 3) + P ( A ) P ( A ) P ( A 3) + P ( A ) P ( A ) P ( A 3) = 05( - 06)( - 08) + ( - 05)06( - 08) + ( - 05) ( - 06)08 = 06 ( ) P ( A + A + A 3) = - P ( A ) P ( A ) P ( A 3) = - 05040 = 096 ( Bernoulli),, 7,,,,,
5 7, A, A, A,, 0,, 000, 000, 8,,, P ( A) = p, P ( A ) = q, p > 0, q> 0, p + q=,,, A p, A 3 ( ), A p ( 0< p < ),, A k ( b ( k;, p ) ) q= - p b( k;, p ) = C k p k q - k A A, =,,,, A k ( Bk) C k A A A k A j A j A j - k,,, k,, k ( C k ), j, j, j - k,,, k - k,, k, P ( A ) = p, P ( A A A k A j A j A j - k ) = P ( A ) P ( A ) P ( A k ) P ( A j ) P ( A j - k ) = p k q - k b( k;, p ) = P ( B k) = C k p k q - k 4, A p, A k ( k) g ( k, p ) = q k- p A k k k A k- k A, 3 4
8 8 0, 3 7,, ( ) 0, 0 0 3 ( ),, 3 3 A P ( A ) = 3, =,,, 0 ( ) B 0, Bk 0 k k= 0,,, 0 P ( B) = - P ( B ) = - P ( B0) = - 3 7 3 7 P ( B3) = C 3 0 0 0 ( ) C 3, D 3 4 P ( C) = 7 0 3 0, P ( D ) = P ( A A ) = P ( A ) P ( A ) = 9 5 9, 9 ( ),, j, j A k k k=,,, 5, 5, B, C j, BC j 7 0 7 0 0, P ( B) = - 8 9 5, P ( C) = - 8 9 5 B ( B ), 8, j 8, P ( C B ) = - P ( C B ) P ( C), B C, B C 7 8 5
9 ( A) ( ), ; ( ) ( ) ; ( 3), ; ( 4) 0, 3 7,,,,,? 3,, A, B 5, C 5,, A, B, C, A+ B, A- B, B - A, AB, AC, A + B 4 9 ( 5), ( 8), ( 0), ( ) 5? ( ) x - a < x - a ; ( ) x > 0 x 0; ( 3) x > 0 x < 8; ( 4) x > 0 x ; ( 5) 0 0 ; ( 6) 0 0 ; ( 7) 0 0 6 5, A =,, 3, 4, 5 B 5 = 0,,, ( 3, 4, 5),, ( ) ( 3) ( 5) 5 A 5 = A 5 = = 3 B 5 = 0 = B ; ( ) A; ( 4) 5 = A B = ; B B ; = = 3 B = 3 ; ( 6) A A A 3A 4A 5 B 3; ( 7) A B 5; ( 8) 5 A B 5 = 7 AB= A( B - A) = ( A- B) ( B - A) ( AB) 8 3 6 6 5 5
30 9 P ( A) = 04, P ( B) = 05, P ( A- B) = 05, P ( AB), P ( AB), P ( B - A), P ( A B ) 0 P ( A) = 04, P ( BA ) = 0, P ( CA B ) = 0, P ( A+ B+ C) N N, ( ), N ( ) k ( kkn ) ; ( ) ; ( 3) N,, 30, 6, ( ) ; ( ), 3 ( 365), ( 365 ) 4 5, 9, 9,, ( ) ; ( ) j ; ( 3) j ; ( 4) 3 ; 5 4, 6 5, k 7, 0, 4,,, ( ) 3 ; ( ) 3 ; ( 3) 3 8 0, 3 9 5, 3,, 3,, 0 3 AAACE H I I MMN T T, MAT H E MAT I CI AN 0,, 3,,,, 3 a, b ( ) ( ), ; ( ), ; ( 3),
3 4 A (, P ), P ( A) > 0, ( ) BA, P ( B A) = ; ( ) B, B (, P ), B B =, P ( B + B A) = P ( B A) + P ( B A) 5, ( ), ; ( ), 6,, 4 70 4 3,,, ( ) ; ( ) ; ( 3) 8 3, 0, 7 A, 3 B; 5 ; 8,, A, ; B,,, 9 30, 0, 00, 006, 0 005 ( ),, ; ( ), 30 9, 3, 3,, 3 ( ) ; ( ) 3, 3 3 ELISA ( Enzyme Linked Imriunosorbent Assay ) EL ISA 95%, % 000, ( ),?, 00,,? 3,,,, 80%, 08? 33,,,, AB C,, ABC?? 34 P ( A) = 0, A B
3 35 P ( A) > 0, P ( B) > 0, AB, A B?? 36ABC, A, B, C 37 38, 09, 095, 08,, 39,,, 00, 0, 03,, 00 03, 40 ( ),, 0;, 06; ( ) ; ( ), ( B), 0, 0,,,, A, B, AB r?, A B, AB r 3( ), N,,, k? 4 k,,,, m 5 l, 6, p, p,,,? 7 0 5 5, 3 7 5, ( ) p ( ), q 8, AAAA, BBBB, CCCC,, 06, 0, ( ) ABCA ; ( ) ABCA, AAAA
,,,,,,,,,,,,,,,,,,, 3, 4, 5, 6, ;,, [ 0, + ),,,,,, -,,, X = X ( ),, y= f ( x ), x, y x, x X = X ( ),,
34 ( ), X,, (, P ), X = X ( ),, (, P ) X, X a, { X ( ) = a}, I R, X I X,,,, X,, {X = a }, {a< X b}, X, = {,, 3, 4, 5, 6} X X ( ) =,,,, X, X = {, } X ( ) =, =, -, =,, X (, P ), X, X x, x, X, x, {X = x },, X x 3 X, {x, =,, }, p ( x ) = P {X = x } =,, ( ) {p ( x ), =,, } X p ( x ) p, X
35 X x x x P p p p, {p ( x ) } ( ) p ( x ) 0, =,,, ( ) ( ) p ( x ) = ( 3), X, X ( ) =, =, 0, = P {X = }= P { }= P {X = 0}= P { }= X X 0 P X =, ; 0, X ( ) P {X = }= a, =,, 3; 3 ( ) P {X = }= a 3, =,, a ( ), a = 7 38 ( ) 3 3 = P {X = } = = = = P {X = } = = = a 3 a 3 = a 38 7, 3 = a - 3 = a,, a = X {p ( x ) }, X
36, P {a X b} = P ( {X = x }) = a x b P {X = x }, I, a x b = p ( x ) ( a b) ( 4) a x b 3 X ( ), P {X < }= 0, P {X }= P {X = }= 7 38 3 = 9 9, P {X < }= P {X = }= 9 9, P {X I } = p ( x ) ( 5) x I P {X < 5}= P {X = }+ P {X = }= 9 9 + 7 38 4 9 = 5 9, P {X 3}= P {X = }+ P {X = }+ P {X = 3}=, P {X 4}= P {X = }+ P {X = }+ P {X = 3}=,,,,,, ;,, 0,, ( ),,, 4 [ a, b], X ( ) X = [ a, b],, c[ a, b], B= ( c, d ] [ a, b], X ( ) =, [ a, b], ( 6) P {X = c}= P { = c}= 0 P {c< X d }= P { B }= d - c b- a ( 7)
37, P {c< X d }= P {X d }- P {X c}, ( 8) X, X ( -, x ] P {X x } F ( x ) = P {X x } x ( -, + ), F ( x ) ( -, + ), F ( x ), ( 8),, 4 X, F ( x ) = P {X x } x ( -, + ) ( 9) X, X F ( x ) 5 4 X x < a, {X x },, F ( x ) = P {X x }= 0 a x < b, {X x }= {ax x }, [ a, x ] [ a, b],, x b, F ( x ) = P {X x }= P {a X x }= x - a b- a {X x } = {a X b}, F ( x ) = P {X x }= P {a X b}= b- a b- a =, X F ( x ) = 0, x < a ; x - a, a x < b; b- a, x b 4, ( ) x x, F ( x ) F ( x ) ; ( ) F ( - ) = lim x - F ( x ) = 0; F ( + ) = lim F ( x ) = ; x + ( 3) F ( x + ) = F ( x ),,,
38, F ( x ),, X,,, 6 X, X, P {X = 0}= P {X = }=, x < 0, 0x <, F ( x ) = P {X x }= 0 F ( x ) = P {X x }= P {X = 0}= x, F ( x ) = P {X x }= P {X = 0}+ P {X = }= + =, X F ( x ) = 0, x < 0,, 0x <,, x, X, F ( x ), X 0,,,,, X F ( x ), X, F ( x ) F ( x ) 6 X, X 7 X
39 F ( x ) = 0, x <, 9, x <, 9 5, x < 3, 9, x 3 F ( x ), X, F ( x ),, 9 3, 9, 6 9, 4 9, X P {X = }= 9 9, P {X = }= 6 9, P {X = 3}= 4 9 4, [ a, b] 7 X,, X [ a, b], X [ a, b], [ a, b], [ a, b] b- a, X [ a, b], X [ a, b], X [ a, b] b- a, [ c, d ] [ a, b], P {c X d } = d c b - a dx = d - c b - a [ a, b] X, 0, f ( x ) = a x b; b- a 0, ( 0) f ( x ) X, X F ( x ) = x - f ( t) dt, 5 X, f ( x ), F ( x ) = P {X x } = x - f ( x ) X, f ( t) dt ( )
40 5, ( ) f ( x ) 0 x ( -, + ) ; ( ) + - f ( x ) dx =,,, X,, 5,,, X 3) P {a < X b} = F ( b) - F ( a ) = b a f ( x ) dx ( ), F ( x ) P {a< X b} ( 3 x, ( ), P {X = x }= 0 ( 3), ( ), f ( x ), F ( x ) = f ( x ) ( 4) 8 X [ a, b], 5, X, 5, F ( x ) = x = a x = b, f ( x ) = F ( x ) = 0, x < a, x - a, a x < b, b- a, x b 0, x < a, b- a, a < x < b, 0, x > b ( 5) F ( x ), f ( x ) =, x = a x = b ( 6) b- a
4 (, f ( a) f ( b), ), ( 0) X [ a, b], b- a [ a, b] 0, [ a, b], X [ a, b], 4 X,,,,,,,, 0 ( x ) 0 3 4 5 6 7 8 9 0 ( ) 3 3 ( f ) 0 0 0 0 3 0 3 0 0 0 0 0 0, x, 0 x 0 x = 0 = = x f = 0 = 0 0 + 0 + 0 + + 9 0 + 0 0 = 5,,,,, x, X ( ) x P {X = }= p, = 0,,,, 0,
4 0 x p = 0 6 X x ( =,, ), P {X = x }= p, =,,, = x p < ( 7), X, EX, EX = = x p ( 8) = =, EX = x p, x p, = x p <,?,,,,, =, x p,, x p = 9 5,, 3, 3 X, E X X 0,,,,, P {X = 0}= C3 = C 3 5 0, P {X = }= C 3C = 6 C 3 5 0, P {X = }= C 3C = 3 C 3 5 0 E X = 0 0 + 6 0 + 3 0 =, ( ),
43, ( ),, [ a, b],,, X, f ( x ),, f ( x ) [ a, b], x / [ a, b], f ( x ) = 0 a = x 0< x < < x + = b, X x = ( x, x + ) x + P {X x } = f ( x ) dx ( 9) x x,, P {X x }f ( x ) x ( 0) x x 0 x x p f ( x 0 ) x 0 f ( x ) x f ( x ) x X, x f ( x ) x ( ) = 0 X,,, ( ) b a x f ( x ) dx = + - x f ( x ) dx, X X,,, 7 X, f ( x ), + - X, EX, EX = + - 0 X f ( x ) = x f ( x ) dx <, x f ( x ) dx ( ) cos x, x ; 0,
44 E X f ( x ) -, 0,, E X, E X = + x f ( x ) dx = - -, E X = 0, E X x cos x dx, X f ( x ) = E X, x, 0x <, - x, x, 0, + EX = x f ( x ) dx = x dx + - 0 x ( - x ) dx = x 3 3 + x - 0 3 x 3 =,, X, g ( x ), X g ( x ) Y= g ( X ) Y, g ( x ), X Y, X, Y g ( x ), Y X,, Y X, X Y, 4,, Y= g ( X ),,, Y, X Y= g ( X ), X, g ( x ) ( ) X, P {X = x }= p =,,
45 = g ( x ) p <, Eg ( X ), Eg ( X ) = g ( x ) p ( 3) = + - ( ) X, f ( x ), g ( x ) f ( x ) d x <, Eg ( X ), Eg ( X ) = + - g ( x ) f ( x ) dx ( 4) (,, ) Y = g ( X ), Y, y j, j =,, P {Y = y j } = P ( g ( x ) = y j {X = x }) = P {X = x } g ( x ) = y j E g ( X ) = y j P {Y = y j } j = = y j j = = j = g ( x ) = y j P {X = x } g ( x ) = y j g ( x ) P {X = x } = g ( x ) P {X = x } = X 9, X X 0, E ( X - EX ) 9 ( ) P E X =, E ( X - EX ) = ( 0 - ) 0 + ( - ) 6 0 + ( - ) 3 0 = 44 0 + 004 6 0 + 064 3 0 = 036 3 X, 0 6 0 3 0
46 E X - E X f ( x ) = EX =, x, 0x <, - x, x, 0, ( ) + E X - E X = E X - = x - f ( x ) dx - = x - x dx + x - ( - x ) dx 0 = ( - x ) x dx + 0 = ( - x ) x dx + 0 ( x - ) dx ( x - ) ( - x ) dx = 3,,,,, ( ) a, Ea = a ( ),, g ( x ), g ( x ), Eg ( X ), E g ( X ), E [ g ( X ) + g ( X ) ] = E g ( X ) + Eg ( X ) ( 5) ( 3) E X, a, ) E ( X + a ) = E X + a ( 6) (,, ( ) g ( x ) a, ( ), g ( x ) + g ( x ) g ( x ) + g ( x ) g ( x ) + g ( x ) p g ( x ) p + g ( x ) E g ( X ), Eg ( X ), E [ g ( X ) + g ( X ) ] g ( x ) = g ( x ) + g ( x ), ( 3) ( 5) ( 3) ( ) g ( x ) = x, g ( x ) a, ( 3) p
47 4 E X, E X, E( X - EX ) = E X - ( E X ) ( 7) ( X - EX ) = X - X E X + ( E X ), ( 5) E ( X - E X ) = E [ X - X E X + ( E X ) ] = E X - EX E X + ( E X ) = E X - ( E X ),,,, ( ),,,,,,,, 8 X, E X, X - E X X,, E ( X - E X ), DX VarX,, E ( X - EX ) D X X X X, X - EX,, X, E ( X - EX ) = 0, X E X,, ( X - EX ) X E X, DX = E ( X - E X ) X E X, X - E X, E X - E X, ( ), ( 3) ( 4), g ( x ) = ( x - E X ), X, P {X = x }= p, =,,, DX = E ( X - E X ) = ( x - E X ) p ( 8) X, f ( x ),, DX = E ( X - E X ) = + - 4, ( 7), ( x - E X ) f ( x ) dx ( 9) DX = E X - ( E X ) ( 30), X DX, a, ( ) D a = 0; ( 3) ( ) D ( X + a ) = DX ; ( 3)
48 ( 3) D ( a X ) = a DX ( 33) 5 X, X DX = E ( X - EX ) = 036, EX = 9 ( 30) DX EX = 0 0 + 6 0 + 3 0 = 8, D X = EX - ( E X ) = 8- ( ) = 036 6 X,, E X = + = x 4 4 f ( x ) = x, 0x <, - x, x, 0, EX =, - x f ( x ) dx = 0 x 3 dx + x ( - x ) dx + 0 3 x 3 - x 4 4 = 7 6, DX = E X - ( E X ) = 7 6 - = 6 7X,, l ( C) = E ( X - C) ( 34) C= EX, l( C), DX l( C) = E ( X - C) = E [ ( X - E X ) + ( E X - C) ] = E [ ( X - E X ) + ( X - E X ) ( EX - C) + ( EX - C) ] = E ( X - EX ) + ( EX - C) E ( X - EX ) = D X, C= EX,, l( C) C= E X, DX 7 X, C X, X C X - C, E ( X - C) ( ),, C E ( X - C), 6 C= E X X, DX
49, E X k 9X, k> 0 EX k ( E X k < ), X k, E X k X k X t, s ( 0< s< t) X ( ), f ( x ), E X s x = x s f ( x ) dx s x + x s f ( x ) dx s > f ( x ) dx s x + x t f ( x ) dx s x > P { X s } + E X t < k, E X k, E ( X + C) k,, E ( X - E X ) k E X ) k 0X, k, EX k, E ( X - X k, E X - E X k X k E X X, D X = E ( X - E X ) X, EX <, X, 3 h( x ) x, X, E h( X ), > 0, P {h( X ) } E h( X ) ( 35), X f ( x ), Eh ( X ) = + - h( x ) f ( x ) dx = h ( x ) f ( x ) dx h + h( x ) f ( x ) dx h ( x ) ( x ) < h ( x ) f ( x ) dx h ( x ) h ( x ) f ( x ) dx = P {h( x ) } k ( ) X k, E X <,
50 > 0 = k E X P { X } ( 36) k ( ) X, > 0 P { X - E X } DX ( 37) 3 X 0 a, P {X = a }, { X - EX > 0}= { X - EX > = } P { X - E X > 0} = P { X - EX > = P { X - E X > = } DX = 0,,, P { X - E X > }DX / = 0 P { X - EX > 0}= 0 P { X - EX = 0}=, 3, a EX 3, X, P {X = a }= X a 3 3, X DX = 0 X a, E X = a,
3 5, X x, x, X x =, P {X = x }= p, P {X = x }= - p ( 0< p < ) ( 38) p EX = p x + ( - p ) x ( 39) D X = p ( - p ) ( x - x ) ( 40) x = 0 p, P {X = }= p, P {X = 0}= - p ( 0< p < ) ( 4) X p X p 0, X p EX = p, DX = p ( - p ) ( 4), A, P ( A) = p, P ( A ) = - p, x, x X ( ) = x, A A ( ) x, A ( A ) ( 43) p, A, A X, X, 0, X p 0- n,,, P {X = x }=, =,,,, ( 44) X {x, x,, x } E X = x = x ( 45) = DX = ( x - x ) ( 46) =,,, = {,, }, P { }=
5, =,,, X, X X,, = {,,, 6}, X ( ) =, X {,,, 6}, A p ( 0< p < ), X A, X 0,,,, k, 0k, {X = k} A k,, P {X = k}= C k p k ( - p ) - k k= 0,,,, ( 47), X ( 47), X, p, X b(, p ), b( k;, p ) = C k p k ( - p ) - k b( k;, p ), k= 0,,,, ( p + q) ( q= - p ),, =, b(, p ) p 0 8 N, N, N ( N + N = N ),,,, X,,, p = N N, X b, N N k N P {X = k}= C k N, N N - k 0k ( 48), EX = kb( k;, p ) = k= 0 k= = pc k - - p k- ( - ) - ( k - ) q k= - = pc k - p k ( - ) - k q k= 0 - = pb( k; -, p ) k= 0 k C k p k q - = p ( 49) k
3 53, b( k; -, p ), ( 0k- ) - p EX = k= 0 k b( k;, p ) = [ k( k- ) + k] b( k;, p ) k= 0 = k ( k- ) b( k;, p ) + k= k= 0 kb ( k;, p ) = ( - ) p b ( k- ; -, p ) + p k= - = ( - ) p b( k; -, p ) + p k= 0 = ( - ) p + p = p + p q ( 50) DX = E X - ( EX ) = p + p q- p = p q ( 5), A p, X A, X, 4 g ( k, p ) = q k- P {X = k}= q k- p g ( k, p ) k ( 5) p ( ), ( 5) p,, E X E X = p q - = = p = p q - = q + = p p ( 53) ( 54) DX = EX - ( EX ) = q p ( 55) 9 X, m,, P {X > m+ X > m}= P {X > } ( 56) P {X > m+ } P {X > m+ X > m}=, ( 5) P {X > m} P {X > m} = k= m + q k - p = q m j = q j - p = q m
54, P {X > m+ }= q m+, P {X > }= q + P {X > m+ X > m}= qm = q = P {X > } q m ( 56), m,, ( 56),, N, N, N ( N = N + N ),, X, X ( ) N P {X = k}= Ck N C - C N k 0k ( 57) a < b, C b a = 0 ( 57) 8,,, N, N N,,, ( 8), C k N C - C N N k C k N N k N N - k ( 58) N, N, N, N N p, N N - p, k,, lim N C k N C - C N N k = C k p k ( - p) - k ( 59), ( Poisson) E X = N N, ( 60) D ( X ) = N N N N N - N - ( 6) X
3 55 P {X = k}= k ( k! e- k0) ( 6) > 0, X, X P ( ), ( 6) k k! e ( ), K = 0 k k! e- = e e - = ( 6) EX = K = 0 k k k! e- = K = k - ( k - )! e- E X = + = K = 0 k k! e- = ( 63) D X = EX - ( EX ) = ( 64),,,,, 0 = 0, 95%, ( )? X, a, X a,, a P {X a }095 X = 0, a k= 0 0 k k! e- 0 095 4 k= 0 5 k= 0 0 k k! e- 0 0966 < 095 0 k k! e- 0 0953 > 095, 5 95% 4( ), A p ( ),, p ( > 0
56 ), k, lim b( k,, p ) = k k! e- ( 65), b(, p ), p, = p, b( k;, p ) ( p ) k e - p ( 66) k! 800, 0005 ( ), X 800, X b ( 800, 0005), = 8000005= 4, P {0 X } = P {X = k} = b( k; 800, 500) K = 0 K = 0 K = 0 4 k k! e- 4 = e - 4 + 4 + 6! 038 P {X > }= - P {0X }- 038= 0769 4,,,,, [ a, b],,, 5 8,, X, X X, f ( x ) =, a x b b- a 0, ( 67)
4 57 X [ a, b], X U[ a, b] [ a, b] [ a, b] 0, [ a, b] + - b- a f ( x ) dx = b a f ( x ) dx = ( 67) F ( x ) = 0, x < a, x - a, a x b b- a, x b, ( 68) EX = a + b, ( 69) DX = ( b- a ) ( 70),,, ( ),,,, A,, A A ( ), X, f ( x ) = x e- x 0 0, x < 0 > 0, X, X e( ) ( 7) X F ( x ) = - x e- x 0, 0, x < 0 ( 7) ( 7) E X =, ( 73)
58 DX = ( 74),, 5 X r s, P {X > r + s X > s}= P {X > r } ( 75),, X, 000, 3 000,, E X = 000, X F ( x ) = = EX = 000 - e- x 000, x 0, 0, x < 0 P {X > 000}= - P {X 000}= - F ( 000) = e - 000, 3 000 e - 3, - e - 3 X, ( x ) = ( x - ) e- ( - < x < ) ( 76),, > 0 X, X N (, ), + + - e - x dx = ( 77) ( x ) dx =, N (, ) ( ), - E X =, DX = ( 78), ( x ) 4,
4 59 ( x ), x ; x =, x = = 0, =, X N ( 0, ), X, 0( x ), 0( x ) = x e- ( 79),,,, 4,,,,,, ( ),,,, ( x ), ( ( x ) ) x ( x ) = - ( t) dt = x - e - ( t - ) dt ( 80) x,, x,, ( 0 ( x ),,,, x 0 0( x ) 0( x ), (, 0( x ) x = 0! ) 0( x ) 0( x ) x < 0 0( x ), 0( - x ) = 0( x ), x < 0, 0( x ) = 0( - x ), 0( - x ) 0( x ) 0( x ), 0( x ) x = 0, - x 0( - x ) = - 0( t) dt = + x 0( t) dt = - 0( x ) ( 8)
60 0( x ) + 0( - x ) = ( 8), 0( x ) = 05, x < 0, 0( x ) = - 0( - x ), 0( - x ), 0( x ) 3 X N ( 0, ), ( ) P {X 96}, P {X - 96}, P { X 96 }, P {- < X } ( ) P {X a }= 0709, P { x < b}= 094, P {X c}= 098, a, b, c ( ) 0( x ), P {X 96}= 0( 96) = 0975, P {X - 96}= 0( - 96) = - 0( 96) = - 0975= 005, P { X 96}= P {- 96X 96}= 0( 96) - 0( - 96) = 0( 96) - = 0975- = 095, P {- < X }= 0( ) - 0( - ) = 0( ) - [ - 0( ) ] = 0( ) + 0( ) - = 09775+ 0843- = 08855, ( ) a = 053, b78 P { X < b}= 0( b) - = 094, 0( b) = P {X c}= 098< 05, - c= 053, c= - 053 ( + 094) = 096 c< 0,, 0( - c) = - 0( c) = 0709, 3 6 X N (, ), Y= a x + b, ab, a 0, Y N ( a + b, a ) Y F Y ( x ), f Y ( x ), X ( x ), ( x ), a > 0, F Y( x ) = P {Yx }= P {a X + bx } ( 83) F Y( x ) = P X x - b a f Y ( x ) = F Y ( x ) = a x - b a = x - b a = a x - b a ( 84) ( 85)
4 6 a < 0,, f Y ( x ) = F Y ( x ) = P f Y ( x ) = F Y ( x ) = - a x - b a = a YN ( a + b, a ) X x - b a = [ x - ( a + b) ] e- a x - b a a e- = - x - b a = - x- b - a X N (, ), = X - N ( 0, ) X a x - b a ( 86) ( 87) a ( 88) X N (, ) N ( 0, ), X = + 3 X N (, ), 0( x ), ( x ), 0( x ), 0( x ), x - ( x ) = 0 ( x ) = x - 0 ( 89) ( 90) 3, 6 ( 84) ( 85) 4, 6,,,,, 4 X N ( 8, 05 ), ( ) ( 9), ( 7) ; ( ) P {75X 0}; ( 3) P { X - 8 }; ( 4) P { X - 9 < 05} ( ) ( 9) = P {X 9}= P X - 8 05 9-8 05 = P X - 8 05 = 0( ) = 09775; ( 7) = P {X 7}= P X - 8 05 7-8 = P X - 8 05 05 - = 0( - ) = - 0( ) = 0075;
6 ( ) P {75X 0}= P 75-8 05 X - 8 05 0-8 = P - X - 8 05 05 4 = 0( 4) - 0( - ) = 0( 4) + 0( ) - = 09 4 6833+ 0843-0843; X - 8 ( 3) P { X - 8 }= P < = 0( ) - = 09775-05 = 09545; ( 4) P { X - 9 < 05}= P {85< X < 95}= P < X - 08 < 3 05 = 0( 3) - 0( ) = 09 8650-08430573 5 X N (, ), 50 350 936%, - x + x 09, x? P {X > 50}= P {X < 350}, x =, = 350+ 50 = 300, P {X < 350}= P X - 300 350-300 < = 50 0 = 0936 50 43, 35 X N ( 300, 35 ), X - P { - x < X < + x }= P < x = x 0-09, 0 x 9 x = 095, 645, x 64535= 5758 5,, N (, ) X, Y= ax + b, a 0, Y X, X, Y X,, Y, Y, g ( x ), X, Y Y= g ( X ) ( 9) Y X,,,,,,
5 63 X Y= g ( X )?, ( ) B, C= {x g ( x ) B}, {YB}= {g ( X ) B}= {X C} ( 9) P {YB}= P {g ( X ) B}= P {X C} ( 93) ( 93), X Y 6 X, Y= X, x 0, P {Yx }= P {X x }= P {- x X x } ( 94) P {Y= x }= P {X = x }= P ( {x = x }{x = - x }) ( 95) X Y, X Y, ( 93) X Y= g ( X ),, X Y Y= X 7 X P {X = - }= 4, P {X = 0}=, P {X = }= 4 Y 0,, ( 9), P {Y= 0}= P {X = 0}= P {X = 0}= P {Y= }= P {X = }= P ( {X = }{X = - }) = P {X = }+ P {X = - } = 4 + 4 =, X Y, Y y, =,, C= {x j g ( x j ) = y }, ( 9) ( 93), {Y= y }= {g ( X ) = y }= {X C} ( 96) P {Y = y } = P {X C} = x j C P {X = x j } ( 97) Y, Y X
64,,,,,,,, N (, ) X Y= a X + b, 6 Y, a + b a,, X F X ( x ) f X ( x ), Y= g ( X ), ( 93), Cx= {t g ( t) x } F Y( x ) = P {Yx }= P {g ( X ) x }= P {X Cx} ( 98) P {X Cx } X F X ( x ) f X ( x ), Y, F Y ( x ) P {X Cx } = Cx f X ( t) dt ( 99) 8 X, F ( x ), F ( X ) [ 0, ] F ( x ),, F - ( x ) Y= F ( X ) F Y( x ), F Y ( x ) = P {Yx }= P {F ( X ) x } F ( x ) [ 0, ], x < 0, F Y ( x ) = P {F ( X ) x } = 0, x >, F Y ( x ) = P {F ( X ) x }=, 0x, F ( x ), F Y( x ) = P {F ( X ) x }= P {X F - ( x ) }= F ( F - ( x ) ) = x, F Y( x ) = 0 x < 0 x 0x x > Y= F ( X ) [ 0, ] 9 F ( x ),,
5 65 9( ( ) ) X N ( 0, ), Y= X Y F Y ( x ), F Y( x ) = P {Yx }= P {X x }, x < 0, x 0, F Y( x ) = P {X x }= 0 F Y( x ) = P {X x }= P {- x < X < x }= 0( x ) - Y= X F Y ( x ) = 0( x ) - x 0 0 x < 0 f Y ( x ) = F Y ( x ) = x 0( x ) x 0 = 0 x < 0 x x e- x 0 ( 00) 0 x < 0 ( 00) ( ), ( ) = ( ) 30 ( ) X,, Y= lnx N (, ) Y= ln X N (, ), X = e Y, YN (, ),, x > 0, F X ( x ) = P {X x }= P {e Y x }= P {Yln x }= ( ln x ) x 0, F X ( x ) = 0 X f X ( x ) = F X ( x ) = x ( ln x ) x > 0 0 x 0 x e- ( ln x - ) = x > 0 x 0 0,,, Black_ Sch oles, P 0,,, P, r,
66 P = P 0e r r = ln P = lnp - lnp 0 P 0 P 0,, P r, 30 ( ) X Y= lnx N (, ), X = e Y, + E X k = E e ky = e ky ( y - ) - e- dy = + - exp - ( y - ) - k ( y - ) + k 4 +, k=,, k+ k dy = e k + k E X = e + DX = E X - ( EX ) = e + - e + = e + ( e - ) ( A) 30 0,,, 0 0, 0 0,,, X ( ) P {X= } = a, =,,, 00; ( ) P {X= } = a, =,,, ( ) ( ) a 3 X F ( x ) = 0, x < - 5 5, - 5x < - 3 0, - x< 0, 0x <, x
67 X 4 0, 7, 3,,, ( ) ; ( ) 5 X 3, P {X > - 3 }, P { X < 3 }, P { X + > } 6 X A P {05< X08} 7 F ( x) = F ( x ) = 0, x 0 Ax, 0< x, x > + x, ( ) - < x< + ; ( ) 0< x < +, ; ( 3) - < x< 0, 8X,,, 0, x < 0 0, x< 0 ( ) F ( x ) = x, 0x <, ( ) F ( x ) = x, 0x <, x, x 9 X x, 0x < ( ) f ( x ) = ae - x, ( ) f ( x) = a F ( x ), P - < X 0, P X x = - x, x < a 0, F ( + x) + F ( - x ) =, - < x < + ( ) 3 X EX R, X, P {X> }, R % % 3% 4% 5% 6% p 0 0 0 03 0 0 0,?? 3 (, ) 00,,, K %, K f ( x ) = 5, 0x5 0,
68 4 ( 5) 5, ( 3) ( 33) 6 3, ( 37) 7 07, 03, 08, 0, ( ) ( ) ( ) ; ( ) ; ( 3) ; ( 4) 8 P,,, 9, P { = }= P { = }, E, D, E, P { = 3} 0,, 3, ( ) [ 0, 5], 3,,, 500, 0 000 005, 800, 3 5 43, 3,, 000, 000 5 N (, ) 6 X N ( 0, 0 ), 00, 96, 7 X N ( 0, 5 ), X 00V, 00V< X < 40V X > 40V, 0, 00, 0, X; ( ), 00V40V 8 N ( 70, 00), 5%,? 9 R, R 0 3 p 0 04 03 0 30 X [ a, b], X+ ( > 0) [ a +, b + ] 3 X [ -, ], X
69 DX= 4 3X, Y= X + ( > 0) 33, [ a, b], 34 6 [ 500, 500] E L 35X, Y= - e - X 36 X, 0, E X = 5, 37, f ( x ), = f ( y) = yf ( y ), y > 0 0 y 0 38 X f X ( x), Y= a X+ b, ( a 0) Y f Y( y) = f X a y- b a = a ( B) + - F ( x ), a> 0, [ F ( x + a) - F ( x ) ] dx, E = 000 ( ), ; ( ), X ; ( 3), Y 3 X N (, ), = 40 ( ), 30( ) 50( ) 05 70 00, 4 m,,, E = m - - m 5 b a f ( x ) dx ( b - a ) b a f ( x ) dx, f ( x ) d [ a, b]
3,,,, ( ),, 3 3 X, X,, X (, ), ( X, X,, X ) (, ) n,,, n 3 ( X, X,, X ) (, ), F ( x, x,, x ) = P {X x, X x,, X x } ( 3) ( X, X,, X ) X, X,, X,, ( X, Y), ( x, y ), F ( x, y ) = P {X x, Yy } ( X, Y) {( t, s) tx, sy }( 3 ), ( X, Y) 3 P {x < X x, y < Yy }, P {x < X x, y < Y y } = P {( X, Y) } = P {( X, Y) } ( ) } - P {( X, Y) ( ) }
3 7 - P {( X, Y) ( ) } + P {( X, Y) } = F ( x, y ) - F ( x, y ) - F ( x, y ) + F ( x, y ) ( 3) 3 3, ( ) ( ) 0F ( x, y) ; ( ) F ( x, y ) x y ; ( 3) F ( -, y) = lim x - F ( x, y ) = 0, F ( x, - ) = F ( -, - ) = F ( +, + ) = lim F ( x, y) = 0, y - lim F ( x, y) = 0, ( x, y) ( -, - ) lim F ( x, y) = ( x, y) ( +, + ), ( X, Y) F ( x, y), F ( x, y ) X Y F X ( x ) F Y( y ), F X ( x ) = P {X x }= P {X x, Y< + }= F ( x, + ) ( 33) F Y( y ) = P {X y }= F ( +, y) ( 34) F X ( x ) F Y( y ) ( ) F ( x, y ), ( X, X,, X ) F ( x, x, x ), F ( x ) = F ( +,, +, x, +,, + ) ( 35) F ( x, x,, x ) =,,, 33 ( X, Y), ( X, Y)
7 3,,, ( X, Y), X, Y,, 34 ( X, Y) ( x, y j ),, j =,, P {X = x, Y= y j }= p j,, j =,,, ( 36) ( 36) ( X, Y), X Y p j ( ) p j 0,, j =,,, ( ) p j = j, ( 3), 3 X Y y y y j P {X = x } x p p p j p j j x p p p j p j j x p p p j p j j P {Y= y j } p p p j X Y, X, Y p X, =,, ; p Y j, j =,,,, p X = P {X = x } = P { j {X = x, Y = y j }) = j P {X = x, Y = y j } = j p j, =,, ( 37) p Y j = P {Y = y j } = p j j =,, ( 38) ( 37), ( 38) P {X = x, Y= y j }= p j,, j =,,,,, 3 3 3, X, Y,
3 73, X Y X, Y 0,,,, ( X, Y) (, ), (, ), (, ), 0, ( X, Y),, P {X = 0, Y= 0}= 3 = 9, P {X = 0, Y= }= 3 = 9, P {X =, Y= }= 3 = 9, X, Y ( ) X Y 0 p X 0 9 9 9 4 9 9 9 0 4 9 9 0 0 9 p Y j 4 9 4 9 9,, ( X, Y) D,, P {( X, Y) D } = p j ( 39) ( x, y ) D j, F {( x, y) } = P {X x, Y y} = 3 ( X, Y), P {X 0, Y = 0}, P {X 0, Y 0 }, P {X Y= 0}, P {X = Y}, P { X = Y } X Y p j ( 30) x x, y y j - 0 0 0 0 0 03 005 0 05 0 0 P {X 0, Y= 0}= P {X =, Y= 0}+ P {X =, Y= 0} = 005+ 0= 005,
74 3 P {X 0, Y0}= P {X = 0, Y= - }+ P {X = 0, Y= 0} = 0+ 0= 03, P {X Y= 0}= P ( {X = 0}{Y= 0}) = P {X = 0}+ P {X 0, Y= 0} = 0+ 0+ 0+ 005= 035 P {X = Y}= P {X = 0, Y= 0}+ P {X =, Y= } = 0+ 0= 03, P { X = Y }= P {X = 0, Y= 0}+ P {X =, Y= - }+ P {X =, Y= } = 0+ 03+ 0= 06 35 ( X, Y), F ( X, Y), f ( x, y ), ( x, y ), F ( x, y) = x y - - f ( s, t) dsdt, ( 3) ( X, Y), f ( x, y ) ( X, Y) ( ), X Y 35, f ( x, y), ( ) f ( x, y) 0, ( ) + + - - f ( x, y ) dxdy =, f ( x, y),,,,, ( 3) D, P {( X, Y) D } = D f ( x, y ) dx dy ( 3), ( 3), F X ( x ) F X ( x ) = P {X x } = P {X x, Y < + } = x - = x - + - f ( s, t) dsdt + f ( s, t) dt ds ( 33) - ( 33), X, f X ( x ) = + - f ( x, y) dy ( 34)
3 75, Y, f Y( y) = + - f ( x, y ) dx ( 35) ( 34), ( 35) f X ( x ) f Y( y ) ( X, Y) f ( x, y ) 33 ( ) G, S ( G), ( X, Y) G, G, G, ( X, Y) f ( x, y) = C=, S ( G), + + - - f ( x, y ) = C, ( x, y ) G, 0, f ( x, y ) dxdy =,, ( x, y) G, S( G) 0, ( 36) ( X, Y) ( 36), ( X, Y) G ( 3), ( X, Y) G, D, P {( X, Y) D }= D f ( x, y ) dx dy = S( G) dxdy = S ( D G) S ( G) S ( D G) D G D G ( 37) 34 ( X, Y) f ( x, y ), ( X, Y ) g ( x, y ) f ( x, y) = g ( x, y ) = k xy, 0x, 0y, 0, ; k xy, 0x y, 0, ( ) k ( X, Y ) ; ( ) k ( X, Y ) ( ), + + - - f ( x, y ) dx dy = 0k x ydx dy =, 0
76 3 k = 4, f ( x, y ) = 4xy, 0x, 0y, 0, ( X, Y) f X ( x ), f Y ( y ), 0x, x < 0 x <,, f X ( x ) = + f X ( x ) = f X ( x ) = f Y ( y ) = - 0 f ( x, y) dy, 4x ydy = x f X ( x ) = 0 x, 0x, 0, y, ( ), + + - -, k = 8, g ( x, y) dx dy= g ( x, y ) = 0y, 0, 0x y = 8 k = kx ydx dy = 0dx x 8xy, 0x y, 0, kx ydy ( X, Y) g X ( x ), g Y ( y ), 0x, x < 0 x >, g X ( x ) = + - g ( x, y ) dy g X ( x ) = x 8xy dy = 4x ( - x ), g X ( x ) = 0, g X ( x ) = 4x ( - x ), 0x, 0,,
3 77 g Y ( y) = 4y 3, 0y, 0,,,, ( X, Y) ( x, y ) = - e- ( - ),,,,, > 0, > 0, <, ( X, Y) (, ;, ; ), ( X, Y) N (, ;, ; ) 33, (, ) ( x - ) ( x - ) ( y - ) ( y - ) - + ( 38), 33,,, ( X, Y) N (, ;, ; ), X ( x ) Y( y ), u= x -, t= X N YN X ( x ) = + = + = = y-, - - ( x, y ) dy - e - e - ( x - e- ) + - ( x - ), (, ), (, ) Y( y) = ( - [ u - u t+ t ] ) e - - ( y- ), dt e- ( t - u) ( - ) dt ( x, y) X ( x ), Y ( y ),, = 0, ( x, y),
78 3 ( x, y) = X ( x ) Y( y ) ( 39) ( ), 4 ( ),,,,,, = 0, ( 39), ( 3) (, 5 6 ) 3 X,, F X ( x ) = P {X x }, - < x < + A, A {X x } x, P {X x A }F ( x A ), F ( x A ), - < x < + A, X 35 X [ 0, ], X >, X, F ( x X > ) = P {X x, X > } P {X > } X [ 0, ], P {X > }= x, P {X x, X > }= 0 x >, P {X x, X > }= F ( x ) - F ( ) = F ( x ) - F ( x ) X,,,
3 79 F ( x ) = 0, x < 0, x, 0x,, x >, x > P {X x, X > }= x - < x, x > F ( x X > ) = 0 x x - < x x > A Y A= {Yy }, P {Yy}> 0, F ( x Yy) = P {X x, Yy} = F ( x, y) P {Yy } F Y( y) ( 30), X Y, ( 30) X Y, x y, {X x } {Yy }, P {X x, Yy }= P {X x }P {Yy} F ( x, y) = F X ( x ) F Y( y) ( 3) F ( x Yy ) = F X ( x ) x, y, {X x } {Yy }, X Y, ( 3) 36 X, Y F ( x, y ), F X ( x ), F Y ( y ), x y, F ( x, y) = F X ( x ) F Y( y) X Y,, X Y X Y,, 3 X Y X
80 3 Y,, A B, P {X A, YB}= P {X A}P {YB} ( 3) 3,,, A = ( x, x ], B= ( y, y ] 3 X Y, g ( x ), g ( y), g ( X ) g ( Y) = g ( X ), = g ( Y), x, y, D x= {t g ( t) x }, D y = {t g ( t) y}, 3, P { x, y }= P {g ( X ) x, g ( Y) y} 36 = P {X D x, Y D y } = P {X D x }P {Y D y } = P { x }P { y} 37 X, X,, X, F ( x, x,, x ), F ( x ), =,,,, x, x,, x F ( x, x,, x ) = F ( x ) F ( x ) F ( x ) X, X,, X,, ( ), ( X, Y), P {X = x, Y= y j }= p j, (, j =,, ), P {Y= y j }> 0, P {X = x Y= y j }= P {X = x, Y= y j } = p j P {X = y j } p Y j ( 33) P {X = x Y= y j } Y= y j, X = x, p j j, P {X = x Y= y j }= p j, =,,, Y= y j, A, B Borel, {X A}, {X B }
3 8, X x ( =,, ),, p j, =,, ( ) p j 0 ( ) p j =, P {X = x Y= y j }= p j, =,, Y= y j, X,, P {X = x }> 0, X = x, Y P {Y= y j X = x }= p j = p j, p X j =,, ( 34) ( 33), ( 34),, j =,,, p j = p X p j = p Y j p j ( 35) 36 X Y 3, Y= 0, X X = 0, Y P {Y= 0} = 0+ 005+ 0= 05, Y= 0, X P {X = 0 Y= 0}= P {X = Y= 0}= P {X = Y= 0}= P {X = 0, Y= 0} = 0 P {Y= 0} 05 = 08 P {X =, Y= 0} = 005 P {Y= 0} 05 = 0 P {X =, Y= 0} = P {Y= 0} 0 05 = 0 P {X = 0}= 0+ 0+ 0= 03, X = 0, Y P {Y= - X = 0}= P {Y= 0 X = 0}= P {Y= X = 0}= P {Y= -, X = 0} = 0 P {X = 0} 03 = 3 P {Y= 0, X = 0} = 0 P {X = 0} 03 = 3 P {Y=, X = 0} = 0 P {X = 0} 03 = 0 3, X Y,, j, P {X = x, Y= y j }= P {X = x }P {Y= y j } ( 36),, j, ( 36), X Y, ( 36), x y, P {X x, Y = y j }= P ( {X = x, Y = y j }) x x
8 3, x, y, = P {X = x, Y = y j } x x = P {X = x }P {Y = y j } x x = P ( {X = x }) P {Y = y j } x x = P {X x }P {Y = y j }, P {X x, Y y}= P ( {X x, Y = y j }) y y j, 33X, = P {X x, Y = y j } y y j = P {X x }P {Y = y j } y y j = P {X x }P ( {Y = y j }) y y j = P {X x }P {Y y } Y, P {X = x, Y= y j } = p j,, j =,,, p X p Y j,, j =,,, X Y p j = p X p Y j,, j =,, ( 37) 37 X Y 3, X Y? P {X = 0}= 0+ 0= 03, P {Y= - }= 0+ 03+ 05= 055 P {X = 0, Y= - }= 0,, P {X = 0, Y= - }P {X = 0}P {Y= - }, X Y,,, 3, X Y,, p j = p X p Y j, X
3 83 Y P {X = 0}= P {X = }= P {X = }= 3 P {Y= 0}= P {Y= }= P {Y= }= 3 X Y, X Y ( 37), 3 3 X 0 Y 0 p X 9 0 p Y j 9 3 9 9 0 3 0 9 9 3 3 3 3 Y X 0 p Y j 0 p X 9 9 9 3 9 9 9 3 9 9 9 3 3 3 3 ( X, Y), F ( x, y ) f ( x, y ) Y= y, X {Y= y }, P {X x Y= y}= P {X x, Y= y } P {Y= y} ( 38) 0, X, ( 38) 0 0, P {X x Y = y}= y0 lim P {X x y - y < Y y} P {X x, y - y < Y y } = y0 lim P {y - y < Y y} y y - = y0 lim x - = y x - y y - f ( u, y ) du f Y( y) f ( u, t) du dt f Y ( t) dt y ( 39), y, f Y ( y ) > 0, P {X x Y= y }Y= y X, F X Y ( x y ) ( 39)
84 3 F X Y ( x y ) = x f ( u, y) du ( 330) - f Y ( y ), f ( x, y) f Y( y) Y= y, X, f X Y( x y ), x, f X ( x ) > 0, X = x, Y F Y X ( y x ) = y - f Y X ( y x ) = f ( x, y) f x ( x ) f ( x, t) dt ( 33) f X ( x ) ( 33),,, f ( x, y) = f X ( x ) f Y X ( y x ) = f Y ( y ) f X Y( x y ) ( 333) 38 ( X, Y) D = {( x, y ) x + y } f Y X ( y x ) f X Y( x y ) ( X, Y) f ( x, y ) = f X ( x ) f X ( x ) = +, f Y ( Y) -, x x <, f ( x, y ) dy = f Y( y ) = f Y X ( y x ) = f ( x, y) f X ( x ) =, y y <, f X Y ( x y ) =, x + y 0, - x, x, - y, y, 0, 0, - x, y - x 0,, x - y - y 0, 39 ( X, Y) N (, ;, ; ), f X Y ( x y ) f Y X ( y x ) 3, X N (, ), YN (, )
3 85 f X Y ( x y ) = f ( x, y ) f Y ( y ) = - e- ( - ) ( x - ) e - ( y- ) - ( x - ) ( y - ) + ( y - ) = - e- ( - ) x - y- -, ) ) = - e- ( - ) x- - ( y- ) Y= y, X N ( + ( y- ), ( - ) ), X = x, Y N ( + ( x - ), ( -, 34 ( X, Y) f ( x, y ), f X ( x ) f Y( y ), X Y f ( x, y ) = f X ( x ) f Y( y ) ( 334) X Y x, y, F ( x, y ) = F X ( x ) F y ( y ) = x f - X ( u) du y f Y ( t) dt - = x - y - f X ( u) f Y( t) dudt, f X ( x ) f Y( y ) ( X, Y), f ( x, y) = f X ( x ) f Y( y) f ( x, y) = f X ( x ) f Y ( y ), F ( x, y) = x - y - f X ( u) f Y ( t) dudt X Y = x - f X ( u) du y - = F X ( x ) F Y( y ) f Y ( t) dt 30 34 X Y, X Y 34, X Y
86 3 X f ( x, y) = Y 34, X Y 4xy, f X ( x ) = f Y ( y ) = 0x, 0y 0, x, 0x 0, y, 0y 0, f ( x, y ) = f X ( x ) f Y ( y ) g ( x, y ) = g X ( x ) = g Y ( y ) = 8xy, 0x y 0, 4x ( - x ), 0x 0, 4y 3, 0y 0, > x 0> y 0> 0 ( x 0, y 0), g ( x 0, y 0) = 0, g X ( x 0) g Y ( y 0) = 6x 0( - x 0) y 3 0 0, X Y g ( x 0, y 0) g X ( x 0) g Y ( y 0) 3 ( X, Y) N (, ;, ; ), X Y = 0 3 X N (, ), YN (, ), N (, ;, ; ) N (, ), N (, ) ( x, y ), X ( x ), Y ( y ), = 0, ( x, y) = X ( x ) Y( y ) 33 ( X, Y), g ( x, y ), g ( X, Y) ( X, Y) ( X, Y)
33 87 P {X = x, Y= y j }= p j,, j =,, zk, k=,,, Z= g ( X, Y), ( 39), Z P {Z = zk} = P {g ( X, Y) = zk} = g( x, y j ) = z k P {X = x, Y = y j } ( 335) k=,, 3 ( X, Y) 3, = X + Y = X Y = X + Y -, 0,,, 3, 4, = X Y -, -, 0, 4 ( 335), P {= - }= P {X + Y = - } = P ( X = 0, Y = - } = 0, P {= 0}= P {X + Y = 0} = P {X = 0, Y = 0} + P {X =, Y = - } = 0 + 03 = 05, P {= }= P {X + Y = } = P {X =, Y = 0} + P {X =, Y = - } = 005 + 05 = 0, P {= }= P {X + Y = } = P {X = 0, Y = } + P {X =, Y = 0} = 0 + 0 = 0, P {= 3}= P {X + Y = 3} = P {X =, Y = } = 0, P {= 4}= P {X + Y = 4} = P {X =, Y = } = 0 P {= - }= P {X =, Y = - } = 05, P {= - }= P {X = Y = - } = 03, P {= 0}= P {X = 0, Y = - } + P {X = 0, Y = 0} + P {X = 0, Y = } + P {X =, Y = 0} + P {X =, Y = 0}, = 0 + 0 + 0 + 005 + 0 = 035, P {= }= P {X =, Y = } = 0, P {= 4}= P {X =, Y = } = 0 ( X, Y) X Y - 0 0 0 0 0 03 005 0 05 0 0
88 3, ( x, y j ) g ( x, y j ) ( X + Y, X Y ) Y X 0-0 0 0 0-0 Y X 0-0 0 0 0 0 0 0 03 0 005 0 3 03-005 0 0 05 0 0 4 05-0 0 0 4 g ( x, y ), g ( X, Y),, P { = - }= 0, P { = 0}= 05, P { = }= 05+ 005= 0, P { = }= 0, P { = 3}= 0, P { = 4}= 0, P { = - }= 05, P { = - }= 03, P { = 0}= 0+ 0+ 005= 035, P { = }= 0, P { = 4}= 0 33 X, Y,, = X + Y P { = k}= P {X + Y= k} k = P {X =, Y = k - } = 0 k = P {X = }P {Y = k - } = 0 k = = 0 e- = e - ( + = ( + ) k k k - ( k - ) e- ) = 0 ( + ), k e- k= 0,,, k- ( k- ) = X + Y + ( X, Y), f ( x, y), g ( x, y ), g ( X, Y) ( X, Y)
33 89 Z= g ( X, Y) F Z( z ) F Z( z ) = P {Z z} = P {g ( X, Y) z } = P {( X, Y) D z } = D z f ( x, y) dx dy ( 336) D z = {( x, y) g ( x, y ) z }, f Z ( z ), z, f Z( z) = F Z ( z ) ( 337) 34 ( ) ( X, Y) f ( x, y), X + Y 34, z, D z = {( x, y ) x + y z },,, F Z( z ) = P {Z z } = P {X + Y z } = f ( x, y) dx dy D z = + = + = z - dy z - y - f ( x, y ) dx - dy ẕ f ( u - y, y ) du - + - f ( u - y, y) dy du ( 338) f Z( z ) = + -, f Z( z) = + - 34D z= {( x, y) x + y z } f ( z - y, y ) dy ( 339) f ( x, z - x ) dx ( 340), X Y, ( 339) ( 340) f Z( z) = + - = + - f X ( x ) f Y ( z - x ) dx f X ( z - y) f Y( y ) dy ( 34) ( 34) f X ( x ) f Y( y), f X f Y( z ), ( 34) f Z ( z) = f X f Y ( z ) = f Yf X ( z ) ( 34) ( 34) ( 34),
90 3 35( ) X N (, ), YN (, ), X Y, X + YN ( +, + ) f X + Y ( z ) = + f - X ( x ) f Y ( z - x ) dx = + - = + - = + - e - e - e - ( x - ) e - ( z - x - ) dx ( x - ) + ( z - x - ) dx + x - - ( z - - ) + ( z - - ) - ( + ) dx ( z - - = e - ( + + ) ( 343) X + YN ( +, + ) ) 35, 35 X, Y N (, ) N (, ),, a, a X + byn ( a + b, a + b ) ( 344) b 0,, 36( ) ( X, Y) f ( x, 35D z= {( x, y ) = x y z } f Z( z ) = F Z ( z) = + 0 y ), y f ( zy, y ) dy - 0 Z= X Y z, D z = {( x, y) x y z }( 35) F Z ( z) = P X Y z = D z f ( x, y ) dx dy = + z y 0 + 0 - - + z y f ( x, y ) dx dy f ( x, y ) dx dy Z - yf ( zy, y) dy
33 9 = + - y f ( zy, y ) dy ( 345) 37( ) X, Y F ( x ), G( x ), f ( x ), g ( x ), X Y, M= max {X, Y}, N = min {X, Y}M N F M ( z ) = P {Mz }= P {X z, Yz }= P {X z }P {Yz} = F ( z ) G( z) ( 346) M f M ( z) = F M ( z) = F ( z ) G( z ) + F ( z) G ( z ) = f ( z ) G( z) + F ( z) g ( z) N ( 347) F N ( z ) = P {N z } = P ( {X z } {Y z }) = - P {X > z, Y > z } = - P {X > z}p {Y > z } = - [ - F ( z ) ] [ - G( z ) ] ( 348) N f N ( z ) = F N ( z ) = f ( z ) [ - G( z) ] + g ( z ) [ - F ( z ) ] ( 349) 38 ( X, Y) G= {( x, y) 0x, 0y}, X Y S f ( s) ( X, Y) f ( x, y ) = F ( s) S,, ( x, y ) G, 0, ( x, y ) G F ( s) = P {S s} = f ( x, y ) dx dy, x ys s0, F ( s) = 0, s F ( s) =, 0< s<, 36, f ( x, y ) dx dy = - xy s s dx s dy x = s ( + ln - ln s) F ( s) = s 0, s0, ( + ln - ln s), 0< s<, s 36
9 3 f ( s) = F ( s) = ( ln - ln s), 0< s<, 0, ( X, Y) Z= g ( X, Y), EZ,, Z (, Z ), Z= g ( X, Y) ( X, Y), P {X = x, Y= y j } = p j,, j =,,, EZ = Eg ( X, Y) = g ( x, y j ) p j ( 350), j ( X, Y), f ( x, y),, EZ = Eg ( X, Y) = + g ( X, Y) = X Y, EX Y = 39 ( X, E X Y E X Y = + + + - - + - - g ( x, y ) f ( x, y ) dx dy ( 35) x y j p j, ( X, Y),, j x yf ( x, y) dx dy, ( 35) ( X, Y) Y) 34 g ( x, y ), g ( x, y ) = + - - 8xy, x yg ( x, y) dx dy = dx x y 8xy dy 0 x = 4 9 0 x y, 0, 30 ( X, Y) 3, E X Y E X Y = 0( - )0+ 000+ 00 + ( - )03+ 005+ 0 + ( - )05+ 00+ 0 = 0 3, X Y, [ 0, 0], 000,,
33 93, 500, Z, Z= g ( X, Y) = ( X, Y) E Z= 0 f ( x, y) = 0 0 0 = 0 dy 0 0 y 000Y, YX, 000X + 500( Y- X ) = 500( X + Y), Y> X, 0x 0, 0y 0, 00 0, g ( x, y) f ( x, y ) dx dy 000y 00 dx + 0 = 0 0 y( 0 - y ) dy + 5 0 0 = 0000 3 + 5500 4 6667( ) 0 0 dy y 0 500( x + y ) 00 dx 3 y - 0y - 50 dy,,,, ( ) X, Y,, E ( X + Y) E ( X + Y) = E X + EY; ( 353) ( ) X, Y,, E XY, EX Y= E X EY ( 354),,,, ( X, Y) f ( x, y ), F X ( x ) F Y ( y ), ( 35),,,, E ( X + Y ) = E X + E Y, E X + Y E ( X + Y ), E ( X + Y) E X Y = E ( X Y ) = E X E Y E X Y
94 3 + - - = + + - - = + = + ( ) E ( X + Y) = + - x + - = E X + E Y ( x + y ) f ( x, Y) dx dy x f ( x, y) dx dy + + + - - - f ( x, y) dy dx + + x f X ( x ) dx + + - yf Y( y) dy y f ( x, y ) dx dy y - + - ( ) X, Y, f ( x, y) = f X ( x ) f Y ( y), E X Y= + + - - = + - = E X EY x yf X ( x ) f Y( y ) dx dy xf x ( x ) dx + - yf Y ( y ) dy f ( x, y) dx dy ( ) X, X,, X ;, E ( X + X + + X ), E ( X + X + + X ) = EX + E X + + EX ( 355) ( ) X, X,, X,, E ( X X X ), E ( X X X ) = E X EX EX ( 356) 34,,,, 37 ( X, Y), E X, EY, E [ ( X - E X ) ( Y- E Y) ], X Y, cov( X, Y) cov( X, Y) = E[ ( X - E X ) ( Y- EY) ] ( 357) ( 350) ( 35), ( X, Y),
34 95 P ( X = x, Y= y j }= p j,, j =,, X Y cov( X, Y) =, j ( x - EX ) ( y j - E Y) p j ( 358) ( X, Y), f ( x, y ), X Y cov( X, Y) = + + - -, ( 357) ( x - E X ) ( y - EY) f ( x, y ) dx dy ( 359) coo( X, Y) = EX Y- E X EY ( 360),, 35 ( ) cov( X, X ) = DX ; ( ) cov( X, Y) = cov( Y, X ) ; ( 3) cov( a X, by) = a bcov( X, Y), a, b ; ( 4) cov( C, X ) = 0, C ; ( 5) cov( X + X, Y) = cov( X, Y) + cov( X, Y) ( 6) X Y, cov( X, Y) = 0 X, Y,, X + Y,, X Y, D ( X + Y) = DX + DY+ cov( X, Y) ( 36) ( ), X, D ( X + Y) = DX + DY ( 36) Y) Y, cov( X, cov( X, Y) E ( X - E X ) ( Y- EY) D X DY ( 363) ( ) ( 5) D( X + Y) = cov( X + Y, X + Y) = cov( X, X ) + cov( X, Y) + cov( Y, Y) = D X + D Y + cov( X, Y) X Y, ( 6) cov( X, Y) = 0, D( X + Y) = DX + D Y 3 ( X, Y) 3,
96 3 cov( X, Y) X P {X = 0}= 03, P {X = }= 045, P {X = }= 05; Y 30, P {Y= - }= 055, P {Y= 0}= 05, P {Y= }= 0, E X = 003+ 045+ 05= 095, EY= ( - )055+ 005+ 0= - 05 E X Y= 0, cov( X, Y) = EX Y- E X EY= 09505= 045 33 ( X, Y) 34 g ( x, y ), cov( X, Y) D ( X + Y) ( X, Y) g ( x, y ) = 8xy, 0x y 0, 34,, E X = + E Y= + g X ( x ) = g Y( y) = 39, 4x ( - x ), 0x 0, 4y 3, 0y 0, - x g X ( x ) dx = 0 x 4x ( - x ) dx = 8 - yg Y( y) dy = 0 y 4y 3 dy = 4 5 E X Y= 4 9 cov( X, Y) = EX Y- E X EY= 4 9-4 5 8 5 = 4 5 EX = + 5 - x g X ( x ) dx = 0 x 4x ( - x ) dx = 3
34 97, E Y = + - y g Y( y ) dy = 0 y 4y 3 dy = 3 DX = E X - ( EX ) = 3-8 5 DY= E Y - ( E Y) = 3-4 5 = 5 = 75 D ( X + Y) = DX + DY + cov( X, Y) = 5 + 75 + 8 5 = 9 ( 36) ( 36) 36 ( X, X,, X ), X, =,,,, (,,, D = X = = DX + < j, X, X,, X, D = X = ), = X, j cov( X, X j ) ( 364) DX ( 365) = ( 36) ( 36), 38 ( X, X,, X ), X DX, =,,,, j = cov ( X, Yj ) (, j ), j =,,, ( j ) ( X, X,, X ), X= ( X, X,, X ), D X ( 364) = (,,, ), ( 366), D ( X) = DX ( 366) = (,,, ), DX = D ( X) 0 D X,,
98 3,, X Y K, X = K X, Y= K Y, X Y X Y, K, cov( X, Y) = K cov( X, Y),, X = X - EX DX, Y = Y- EY cov( X, Y ), cov( X, Y ) X Y,, cov( X, Y ) = DY cov( X, Y) D X DY ( 367) 39 ( X, Y), X Y,, X, Y= X Y cov( X, Y) D X ( 363), X, Y DY ( 368) 34 X, Y, Y= a X + b, ( a 0, b ), D X, X, Y cov( X, Y) = cov( X, ax + b) = a cov( X, X ) = ad X, X, Y = D Y= D( ax + b) = a DX, cov( X, Y) DX DY = a DX DX a DX = a a ( 369), a > 0,, X, Y= ; a < 0, X, Y = - 34, X Y, X, Y 34, P {Y= ax + b}= ( 370), X Y, X, Y =, ( 370), X Y, X Y X, Y = X Y?
34 99 37 ( X, Y), D X, DY, X, Y = X Y,, a 0 b,, a > 0, X Y = ; P {Y= a x + b}= a < 0 XY = -, X, Y = X, Y= 0, X Y, DX, DY, 35( 6) cov( X, Y) = 0, X, Y= 0 X, Y= 0, X Y ( 35 ),, X, Y = 0 X Y,, X Y, DX, DY, X Y ( ) cov( X, Y) = 0 ( ) E X Y= E XE Y ( 3) D ( X + Y) = DX + DY 35 [ -, ], X = sin, Y= cos X Y, EX = D X = - sin d = 0, E Y = - cos d= 0, - sin d =, D Y = E Y = EX Y = sin cos d = 0 - cos d =, E X Y= E XE Y, X Y X Y X Y X + Y = X, Y = X, Y= 0 X, Y <,, X Y,,, X Y - Y= a X + b+ ( 37) a 0, b, X, Y X, Y X, Y X, X, Y X a X
00 3 + b, Y X ( ),, X Y ( 37), X, Y 0< < X, Y,, X, Y 0,,,,, ( ) 30 ( ) ( X, X Y= y j X Y= y j Y), Y= y j, X P {X = x Y= y j }= p j =,,, x p j <, E [ X Y = y j ] = x p j ( 37) ( ) ( X, Y), Y= y, X f X Y ( x y ), + - x f X Y( x y) dx < X Y= y, E [ X Y = y ] = + - x f X Y ( x y ) dx ( 373) X Y= y 36 ( X, Y) 3, Y= 0, X 36, Y= 0, X P {X = 0 Y= 0}= 08P {X = Y= 0}= 0P {X = Y= 0}= 0 E [ X Y= 0] = 008+ 0+ 0= 0 37 ( X, Y) 38, E [ X Y= y ] ( y < )
34 0 38, y <, y <, f X Y ( x y ) = E [ X Y = y ] = + - - y, x - y ; 0, x f ( x y ) dx = - y x - - y - y dx = 0 ( ) C, E [ C Y= y ] = C ( 374) ( ) k, k, E [ X Y= y ], =,,, E [ k X + k X Y= y ] = ke [ X Y= y ] + k E [ X Y= y] ( 375) ( 3) X Y, E [ X Y= y] = E X ( 376) E [ X Y= y] Y, Y y, E [ X Y= y ], E [ X Y], Y,, E [ X Y= y ], E [ X Y] ( ) C, E [ C Y] = C ( 377) ( ) k, k, E [ k X + k X Y] = ke [ X Y] + ke [ X Y], ( 378) ( 3) X Y, E [ X Y] = E X ( 379), ( 4) g ( x ), E [ g ( Y) X Y] = g ( Y) E [ X Y],, E [ g ( Y) Y] = g ( Y) ( 380) ( 5) E ( E[ X Y] ) = E X ( 38) ( 4) Y= y, g ( Y) g ( y) ( ), E [ g ( Y) X Y= y ] = E [ g ( y ) X Y= y] = g ( y) E[ X Y= y ] E [ g ( Y) X Y] = g ( Y) E [ X Y] ( 5), g ( y ) = E [ X Y= y], g ( Y) = E[ X Y], E ( E [ X Y] ) = E ( g ( Y) ) = + = + = + - g ( y) f Y ( y) dy - E [ X Y = y] f Y ( y ) dy - + - x f X Y ( x y ) dx f Y ( y ) dy
0 3, = + + - - + - - = + = + x - + - = + - = E X [ x f X Y( x y) f Y( y ) ] dx dy [ x f ( x, y) ] dx dy xf X ( x ) dx f ( x, y ) dy dx 39 ( X, Y) N (,,,, ) E [ X Y] Y= y, X 330 X, N + ( y - ), ( - ) E [ X Y= y ] = + ( y - ) ( 38) E [ X Y] = + ( Y- ) ( 383) Y Y= a X + b+ ( 384) X, 0, E [ Y X ] E [ Y X ] = E ( ax + b+ X ] = E [ ax X ] + E [ b X ] + E[ ] = a X + b+ E = a X + b ( 385),,, E X X, E ( X - E X ) = mine ( X - C) ( 386) C, X, X Y, Y Y, X Y, X, Y, X Y g ( Y) ( ), g 0, g 0( Y) X, g 0( Y) X
3503 E [ X - g 0( Y) ] = mine [ X - g ( Y) ] ( 387) g, Y y, X Y, X, Y= y X,, X, E [ X Y= y ], E [ X Y],, ( 387),, E ( X - E ( X Y) ) = min g E ( x - g ( Y) ) ( 386) E ( X - g ( Y) ) = E ( ( X - E [ X Y] ) + ( E [ X Y] - g ( Y) ) ) = E ( X - E [ X Y] ) + E ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) + E ( E [ X Y] - g ( Y) ) ( 38) ( 380) E ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) = E ( E [ ( X - E [ X Y] ) ( E [ X Y] - g ( Y) ) Y] ) = E ( ( E [ X Y] - g ( Y) ) E[ ( X - E [ X Y] ) Y] ) = E ( ( E [ X Y] - g ( Y) ) ( E [ X Y] - E [ X Y] ) = 0 E( X - g ( Y) ) = E ( X - E [ X Y] ) + E ( E [ X Y] - g ( Y) ) E( X - E[ X Y] ) 35 a,, a lim a= a > 0, N, > N a- a <,,, X, X,, X,,, X, X,, X, X, > 0, N, > N,
04 3 X - X < X, X, X - X,,,,,, > N, X - X <,, X - X >, 33 X, X, X, 0, {X } X, X, X,, > limp { X - X > }= 0 ( 388) P X P - lim X = X,, A p, A ( ) p,, A, A,, p ( A ) P p,,, P 37 ( ) A A p ( 0< p < ), p P - lim = p lim P, E = p, D > 0 { - p < } = ( 389) = D ( ) = p q, ( q= - p ), > 0, P - p > p q
3505 = = = 0lim P lim P - p > lim p q = 0 - p > = 0 A =,,, 0 A, E = p, ( 305) lim P - = E < = ( =,, ) 0, = 38 ( ),,,,, E D,, C, DC=,,, > 0 limp - = =, = P D - = = = lim P E < = ( 390) D E = = D( ) C = = - = E < = C 0,,,,, E =, > 0, limp = = - < = ( 39) P,, 39 ( ),,,,
06 3,, E =, limp = - < = ( 39),, = E,,,?,,,,,, 3, 4,,,,,, =,,,,,,, ( ),,, = 30 ( ),,,,, E =, D ( ) = > 0, =,,, lim P = - x = x - e- t dt ( 393),, 30 N ( 0, ), =
, a N (, ), =, = - a N ( 0, ) ( 394) = a N,,,, > 0, P > 0,, = = = - > 0,, - >?, P = = a - N 0, - > = - P = - - = - P = / / - 0 / - = - 0 ( 395) / 0( x ) N ( 0, ), 0 ( 395) / P = - > 0,, - > = 33 00,, 00g 0g, 0kg 3507, =,,, 00, 0 0 =, E = 00, D ( ) = 0, E= i= 00E = 0000, D= 00,
08 3 P { > 000}= P - 0000 000-0000 > 00 00 = P - 0000 > 00 = - P - 0000 00-0( ) = - 09775= 0075, 3 X B(, p ), 0< p <, limp X - p P q x = x e - - t d t ( 396) 3,,, X B (, p ), P {a X < b} = C k p k q - k,, a k< b X - p,, 3, P q a N ( 0, ) X a N ( p, p q), P {a X < b}= P a - p p q X - p p q b- p 0 p q - a - p 0 p q, < b- p p q P {a X < b} 33 0 000, 07,, 6 800 7 00, B ( 0000, 07), E = 000007= 7000, D = 00000703= 00, P {6800< < 700}= P = P - 00 00 < - 7000 00 < 00 00-7000 < 436 4583 0( 436) - = 099999 333, [ 0, 000] ( ) 0%,, 00 095,? X, X, A = {X
09 00}, A =,,,, P {A }= 09, B(, 09), {X 00}= 64, 5 3 P {X 00}= P 08, 08 = P { > 08} = P = - 0 - - 09 090 > 3 = 0 08-09 090 3 095 ( A) F ( x, y) ( X, Y), x < x, y < y, F ( x, y ) - F ( x, y ) - F ( x, y ) + F ( x, y ) 0 5, 3,,, X, =, ( ) ( X, X ) ; ( ) P {X = 0, X 0}, P {X = X }, P {XY= 0} 3 ( X, Y) ( X, Y) f ( x, y) g( x y ) f ( x, y) = g( x, y ) = ( ) k, k, ( ) k e - 3x- 4y, x > 0, y > 0 0, k e - 3x- 4y, x > y> 0 0, 4 ( X, Y) G= {( x, y) 0x, 0y }, ( ) ( X, Y) ; ( ) X Y ; ( 3) P {Y< X } 5 ( X, Y) G= {( x, y ) > y > x > 0}, ( ) ( X, Y), ( ) X Y 6 X Y F x( x ), F Y( y) F X( x ) = 0, x< 0 x, 0x, F Y( y ) =, x> 0, y< y-, y, y> X Y, ( ) ( X, Y) ; ( ) = X, = Y, (, ) G( x y ) ; ( 3) P X<, y > 3 7 X Y, x, x, y, y ( x < x ; y < y ), {x < Xx }
0 3 {y < Yy } 8 X = X 9 ( X, Y), X P {X= x }= p X, =,,,, =,,, X= x, Y P {Y= y j X = x }= p j, j = ( ) ( X, Y) P {X= x, Y= y j}= p j,, j =,, ; ( ) Y P {Y= y j }= p Y j ; ( 3) Y= y j X P {X = x Y= y j }= p j, =,, 0 ( X, Y) X Y - 0 p X 0 p p p 3 0 p 0 p Y j 4 4 ( ) ( X, Y) p, p, p 3, p ; ( ) X Y X Y, ( X, Y), X x Y y y y 3 p X 8 x 8 p Y j 6 ( X, Y), P {X = x, Y= y }= p j, =,,, m, j =,,,, ( p j ) m X Y ( p j ) m 3, B, C, x + Bx + C= 0 p q 4 3 f X Y ( x y), f Y X ( y x ), g X Y ( x y), g Y X ( y x ) 5 D y = x -, y= x +, x = ( X, Y) D f Y X ( y x ) f X Y( x y) 6 D= {( x y) } a x b, ( x ) < y < ( x ) }, ( x ), ( x ), ( x ) < ( x ) [ a, b], ( X Y) D 7 ( X Y), X f X ( x ) x, X= x