( CIP).:,3.7 ISBN TB CIP (3) ( ) ISBN O78 : 3.

Transcription

1

2 ( CIP).:,3.7 ISBN TB CIP (3) ( ) ISBN O78 : 3.

3 995,.,.,.,. :,,,,.. :,,,,,,.,,,,.,,.

4 ,,.,,,.,,,.,,,,.,.,,, ;,,.,,, ;,,,.,.,,,.,,.,,,,. : [email protected] 3 6

5 ( 3 ). ( 3 ). ( 5 ).3 ().4 ().5 (6) (3) (6). (6). (8).3 (9).4 (34).5 (38) (4) 3 (44) 3. (44) 3. (48) 3.3 (49) (53) 4 (56) 4. (56) 4. (6) 4.3 (67) 4.4 (7) 4.5 (75) (8) 5 (83) 5. (83) 5. (87) 5.3 (9) 5.4 (93) 5.5 (99) (3)

6 (6) (5). (5). (7).3 ().4 (3).5 (7).6 (3).7 (36).8 (39) (4) (44). (44). (45).3 (53).4 (55).5 (64) (7) 3 (73) 3. (73) 3. (78) 3.3 (8) 3.4 (86) 3.5 (87) (95) 4 (98) 4. (98) 4. (5) 4.3 (9) 4.4 () 4.5 (7) (5) 5 (9) 5. (9) 5. (33) (4)

7 3 6 (4) 6. (4) 6. (43) 6.3 (44) 6.4 (45) 6.5 (5) 6.6 (5) 6.7 (54) (57) 7 (6) 7. (6) 7. (63) 7.3 (7) 7.4 (77) 7.5 (8) (83) (86) (95) (37)

8 ,.,.... m m a a a a a a a m a m a m m ( ). A, B,. a a a A = a a a a m a m a m A i j a ij A ( i, j).a ij i, j, A [ a ij ] m A = [ a ij ], m A A m.,.,,... A a a a a a a a a a a, a,, a A,. a a a B =. a a a

9 4 a a a A = a a. a = = diag[,,, ]. a. a ( ). a, a =,, I E. A = [ a, A = [ a, a,, a ]. a a ],. b B = b, m. b m, O....3 IT ( : ): WorkPad Tablet PC NC a a a 3 a a a 3 A = a a a 3 a a a 3 a ij i j.

10 5 b b B = b b b 3 b 3 b i i, b i i ( i =,, 3). a x + a x + + a x = b a x + a x + + a x = b a m x + a m x + + a m x = b m m a a a A = a a a a m a m a m., m( + ) a a a b A = a a a b a m a m a m b m., m x b x = x b = b x b m.,. m A = [ a ij ]B = [ b ij ],, A B, A = B... a i j = b ij ( i =,,, m, j =,,, ) A = [ aij ] m A A,

11 6 A = A= [ a ij ] m A. A = O; A = A (A m, ) : () () A = (A) ; () (+ ) A = A + A ; 3 m A = [ a ij ], B = [ b ij ], A B A + B,. A + B = [ a ij + b ij ] m (ABC m, ) : () A + B = B + A ; () ( A + B) + C = A + ( B + C) ; (3) ( A + B) = A + B.,. A - B = A + ( - ) B = [ a ij - b ij ] m,. A = , B = - 3 A = 3 B = = ( - ) A - 3 B = ( - 3) , A - 3 B. = = ,, a b + a b + a 3 b 3, a i b + a i b + a i3 b 3 i ( i =, )., : C = a b + a b + a 3 b 3 a b + a b + a 3 b 3 a b + a b + a 3 b 3 a b + a b + a 3 b 3,. 4 A = [ a ij ] ms, B = [ b i j ] s, A B m C = [ c ij ], C = AB.

12 7, s c ij = a i b j + a i b j + + a is b sj = a ik b kj k = ( i =,,, m; j =,,, ), s s, b j [ ai, ai,, ais ] b j s = ai b j + ai b j + + ais bsj = aik bkj = cij k = b sj C ( i, j) ci j A i B j ( ).. A = , B = - 3, AB BA. A 3, B 3, A B, AB, AB = B A, BA. 3 A = AB = - 3 =, B = BA = 4 + ( - ) + ( - 3) 3 ( - ) + ( - ) ( - ) + 33 = - 9 =, AB BA. = 4 A = - - B = - - AB BA. AB = BA = = - - = - -

13 8, AB BA ; 3, ABBA ; 4, ABBA,.,,, ABBA.AB A B, BA A B.AB = BA, AB. 4, AO, BO, AB = O,,, AB = O A = O B = O ;, AO AB = AC, B = C. () : () ( AB) C = A( BC ) ; () ( AB) = (A) B = A(B), ( ) ; (3) A( B + C) = AB + AC, ( B + C) A = BA + CA ; (4) Im Am = Am = Am I. A( B + C) = AB + AC. A m s = [ a ij ], B s = [ b ij ], C s = [ c ij ], A( B + C) = M = [ m ij ] m, AB + AC = N = [ ij ] m A( B + C) = AB + AC.., s m ij = a ik ( b kj k = s s + c kj ) = a ik b kj + a ik c kj = ij k = k = 5. A = [ a ij ], B = [ b ij ], i < j a ij = b i j =. C = AB = [ cij ], c ij = k = a ik b kj = j - k = a ik b kj + i < j, b kj =, a ik =, c ij =, C = AB. k = j a ik b kj,.a, k, A k A, A k = AAA ( A, k l ) : () A k A l = A k + l ; () ( A k ) l = A kl. k k, A B,, ( AB) k = A k B k. 6 cos- si = si cos cos - si si cos. =,. = k cos- si = si cos k cos k- si k si k cos k

14 = k +,. cos- si si cos k +, = = = = cos - si si cos k cos- si si cos cos k - si kcos- si si k cos k si cos cos kcos- si k si- cos k si- si k cos si k cos+ cos k si- si k si+ cos k cos cos( k + ) - si( k + ) si( k + ) cos ( k + ) x y = cos- si si cos ( x, y) ( x, y ). 6,,. A T..3 5 A, A, A. x y 9 A = a a a a a a, A T = a a a m a a a m a m a m a m a a a m () () ( A T ) T = A ; () ( A + B) T = A T + B T ; (3) (A) T = A T, ; (4) ( AB) T = B T A T. (4). A = [ aij ] ms, B = [ bij ] s, ( AB ) T ( i, j) AB ( j, i), A j B i, B T i A T j, B T A T ( i, j), ( AB) T = B T A T 7 A = - 3 AB =, B = , ( AB) T. - = ,

15 ( AB) T = 4 ( AB ) T = B T A T = = A T = A, a ij = a ji ( i, j =,,, ), A. A T = - A, a ij = - a ji ( i, j =,,, ), A.. 8. A, B, C,, A = B + C, A T = B T + C T,, A T = B - C,, B = ( A + AT ), C = ( A - AT ), BC, A = ( A + AT ) + ( A - AT ).3.3. x, x,..., x y, y,..., y y = a x + a x + + a x y = a x + a x + + a x (.3.) y = a x + a x + + a x y = Ax y a a a x y = y, A = a a a, x = x y a a a x

16 (.3.) x = b y + b y + + b y x = b y + b y + + b y (.3.) x = b y + b y + + b y, y, y,, y x, x,, x, (.3.) (.3.), x = By, y = Ax = A( By) = ABy, AB, AB = I, I. x = By = B( Ax) = BAx BA, BA = I,. A, B, AB = BA = I, A, B A, A - = B. A =, A B, A - = B, B A, B - = A. - 3, A - = A, A. BC A, AB = BA = I AC = CA = I, A. B = BI = B ( AC) = ( BA) C = IC = C A, AB = I, BA = I. A, AA - = A - A = I, AB = I, BA = I( BA) = ( A - A ) ( BA) = A - ( AB ) A = A - IA = A - A = I 3 A, A -, ( A - ) - = A. 4 A,, A, (A) - = A -.,,. 5 AB, AB, ( AB) - = B - A -. A B, A - B -, ( AB ) ( B - A - ) = A( BB - ) A - = AIA - = AA - = I ( B - A - ) ( AB ) = B( A - A) B - = BIB - = BB - = I AB, ( AB) - = B - A -..

17 , 5, A, A,, A k ( A A A k ) - = A - k A - k - A - 6 A, A T, ( A T ) - = ( A - ) T., ( A T ) ( A - ) T = ( A - A) T = I T = I = ( AA - ) T = ( A - ) T ( A T ), ( A T ) - = ( A - ) T. A -. A A + A - I = O, A, A -. A = A + I + A - I = O A( A + I) = I, ( A + I) A = I,. A A + I = A + I A = I AB A + B, A - + B -,, A - + B - = A - I + IB - = A - BB - + A - AB - = A - ( B + A) B - = A - ( A + B) B - 5, A - + B -, ( A - + B - ) - = B( A + B) - A A - + B - = B - + A - = B - I + IA - = B - ( A + B) A -, ( A - + B - ) - = A( A + B) - B,,. I, I A, A, A, A = - 5-3, :

18 ( i) 3 -, (ii) 3 -, (iii) (i)a = A A A A, A = 6-4 3, A =, A = - 5, A =, : () A = A A r,, A = A s A sr A A r - 3 ; A s A sr () A B,, A A r A =, B = A s A sr A ij B ij, B B s B r B sr A = A + B A r + B r As + Bs Asr + Bsr (3) A ml, B l, A A t A =, B = A s A st B B r B t B tr A i, A i,, A it B j, B j,, B tj, AB = C C r C ij C s C sr t = A ik B kj ( i =,, s; j =,, r) k = A A r A T A T s (4) A =, A A s A sr T = A T r A T s r. A =, B = - -, AB A T. - -

19 4 A = -, B = A = - AB = A T = I A I A O I = I A I B =.4. O I T =,, AB O I, B = I + OB A I + IB = I T A T O T I T = I O = - = () A, A, Ai ( i =,,, s) ( ), A A I B A = A., A i ( i =,,, s), A - A s A - = A - A - s A A - s B = A, B - = A -. A s A - () m A m T A = T T m

20 5 T i = [ b i, b i,, b i ]A i. (3) m A A = [,,, ] a j j = a j A j. a mj A = [,,, ], AA T A T A. T AA T = [,,, ] T = T T T + T + + A T A = T T [,,, ] = T T T T T T T T T T A T A = I,,,,. 3 x + x + + x = b x + x + + x = b m x + m x + + m x Ax = b = b m x b A = [ a ij ] m, x = x, b = b A m, Ax = b Ax = T T x x = T m b b a i x + a i x + + a i x = b i b m b m

21 6 T i x = b i ( i =,,, m) A, Ax = b x [,,, ] x = b x x + x + + x = b Ax = b b a, a,, a. 4 A B, AB = BA = I.B I, A[ b, b,, b ] = [ e, e,, e ] Ab i = e i ( i =,,, ) e i I i, e i = [,,,,] T. A = O. A B b i Ax = e i ( i =,,, ). 5 : A m, x, Ax =.. x, x = ei = [,,,, ] T ( i =,,, ) ( ei i ), A A = [,, i,, ], Ax =, Ae i = [,, i,, ] = i = A i, i, A, A = O..5,.,. x + x - x 3 = 3 x + x = - - x - x - x 3 =

22 7 : () ; () ; (3) k ( k ).,,,.,,., A = ( A ).,..5. ( 3 ): () () (i, j ( ), r ij ( c ij ) ) ; () ( ) ( i () r i () ( c i ( ) ) ) ; (3) () k ( k ) () (i () k j (), r ij ( k) ( c ij ( k) ) ).. A B, A B, AB. : () AA ; () AB, BA ; (3) AB, BC, AC.. I, 3 ( )R ij ( C ij ), R i ( ) ( C i () ), R ij ( k) ( C ij ( k) ), R ij = C ij =

23 8 R i ( ) = C i () = R ij ( k) = C ji ( k) = k. i j m A, () B, m ( )()A.., A r i ()B, B m,, AB, A =, R i ( ) A = r ( ) i A B B = R i () A T T i T m, B = T T i T m T T i T m = T T i T m, A 3 c ij ( k)b, B m, c ij ( k) A B B = ACij ( k) = B, AB, A = [,,, ],

24 9, B = [,, i,, k i + j,, ] AC ij ( k) = [,,, ] k = [,, i,, k i + j,, ] = B,,, : r ij, R - i j r i ( ), R - i R ij ( - k).: = R ij ; r i () ( ) = R i ( ) ; r ij ( k)r ij ( - k), R - ij ( k) = C - ij = C ij, C - i () = C i ( ), C - ij ( k) = C ij ( - k).5.. m A,, m : N = Ir O O O, m A, R,, R l, C,, C s, R l R AC C s = I r O O O (.5.) r A mi ( m, ), r =, I. A. N = I r O O O.A = [ a ij ] m, A = O, (.5.),. AO, A., a. - a i a ; - a j a ; a ;,,, A

25 A ( m - )( - ). O T O A A, ( m - r)( - r)a r,. (.5.),, m A, m P Q, A = P I r O O O r A mi( m, ). (.5.) A. A = A = r ( 5 ) r ( ) r - 3 c 3 ( ),. r ( - ) c ( ) - 3 = [ I O] Q (.5.) R () R ( 5 ) R ( - ) R AC 3 () C 3 () = [ I O], A = R () R ( 5 ) R ( - ) R - [ I O] ( C 3 () C 3 () ) - = R - R - ( - ) R - ( 5 ) R - () [ I O] C - 3 () C - 3 (), = R R () R (5) R ( - ) [ I O] C 3 ( - ) C 3 ( - ) = P[ I O] Q P = R R () R (5) R ( - ) I = R R () R (5) = R R () - 5 Q = I 3 C 3 ( - ) C 3 ( - ) = = R - - = - C 3 ( - ) = () : [ I r O], I r O, I r, O. (),,,

26 . A = = - 4-3,. = [ I O] Q 3 A A. A, A, A, R,, R l, C,, C s, R l R AC C s = Ir O O O (.5.) A,, r =., (.5.) A. A = R - R - l C - s C - (.5.3) m AB : m P Q, PAQ = B. A A r, A AI. c, AI.. A, [ AI ] [ IB], B = A -., R, RA = I, R = A -, B = R = A -. [ IB] = R[ AI] = [ RAR] A = [ AI] = , A r 3 ( 5 ) r r ( - 3 ) r 3 ( ) r 3 ( - ) r 3 ( ) r 3 ( 3)

27 r ( - ) A - = Ax = b, A, A -, A -, x = A - b,.,, A, b = [, -,] T x = A - b = = ,, A - A - [ AB] = [ IA - B], [ AB], A I, B A - B. 3 AX = B, A = A, X = A - B. 3 5 [ AB] = r ( ) r 3 ( - ) r ( - ) r 3 ( - ) r ( - ), B = r 3 ( - 3 ) X = r 3 ( - ) r 3 ( - 5 ) B

28 3 X = A - B. YA = C Y = CA -, A C, c A C Y = CA -., YA = C, A T Y T = C T, [ A T C T ], Y T = ( A T ) - C T, Y. 4 AXB = D, A = I CA - [ A T C T r ][ I( A T ) - C T ], B =, D =, A B, AXB = D, R X = R - DC - 3 () = R DC 3 ( - ) = XC 3 () = D, C 3 ( - ) = 3-3. A = () [,,3] 3 x y u u z - 8, B = x ; () - 3 X + 3,x, y, z, u = O,X. [, - 3] ; (3) ; a a a 3 x (4) [ x, x, x 3 ].4 A = a a a 3 a 3 a 3 a 33 x. x 3, A A = 3, B = ;: () AB = BA? () ( A + B) = A + AB + B?

29 4 (3) ( A + B) ( A - B) = A - B?.7. () A = O,A = O ; () A = A,A = O A = I; (3) AX = AY,AO,X = Y..8. () A = (3) A = 3 [,, 3 ],A4 ; () A =,A ( ). 3-3,A 5 ;.9 f( x) = x 3-3 x + 3 x +,f( A),f( A) = A 3-3 A + 3 A + I, A = - -. A =, f( A) , B = - 6,AB, BA, B T A A.. A (trace), tr( A),tr( A) = a ii i =, : tr( AB) = tr( BA) ; tr( AA T ) =. A = I - T T, I,, A A = I. i = j = a ij.b.3 A, B, : () A ; () AB - BA ; (3) AB AB = BA..4 : () A m, m x, x T A = A = O; () A m,x m, y,x T Ay = A = O..5 A = ,A A - ( )..6 A A + A - 3 I = O,A -, ( A + I) -, ( A + 4 I) -..7 X. () A - XA = 6 A + XA, A = () AX - X + I = A, A = ;

30 5.8 AB A + B = AB. () A - I,AB = BA ; () B =.9 A = - 3 4,A -... () ; (). : () (3) X X = ; () = A = [ a ij ], P = ; (3) X, P = =.,A , P AP, P P A, AP P. r ij.3 A, AB.() B ;() AB - ; (3) A - i j B -..4 A = ;

31 . (Cramer)....,.,,,.,43 4,435 5,, ( - )( - ) =!,, ;.,,,,. 43, , j j j ( j j j ).,., 43 ; A = a a a a a a, a a a a a a a a a a a a A ( ), A ( determiat), A det A,, a j a j a j, j j j,,,,!, : j j j, ; j j j,.

32 7 A = a a a a a a = ( - ) ( j j j ) j j j a j a j a j (..) a a a. j j j a = a,.,,,,. A = a a a ( ) A. a a a a,, j =,, a, a,, j,. j =, j =.,,, a a a,.,,,,. a a a A = = a a a (..) A a a a,, = diag[,,, ], =., I, I =. A = a a, - a a -, a -, - a -, A., (..), A a a a, - a A a a, - a, = ( - ) ( ( - ) ) a a, - a = ( - ) ( - ) a a, - a = ( - ) ( - ) a a, - a 3 R i j (C ij )-. R ij a a a ij a ji a =, C ij = R ij = ( - ) ( ji) = ( - ) j - i - = - 3,

33 8 R ij = C ij = - ; R i ( ) = C i () = ; R ij ( k) = C ij ( k) = (..), a a a a = a a - a a (..3) a a a 3 a a a 3 = a a a 33 + a a 3 a 3 + a 3 a a 3 - a 3 a a 3 - a a a 33 - a a 3 a 3 a3 a3 a33 a a 3 a a 3 a a = a - a + a 3 (..4) a 3 a 33 a 3 a 33 a 3 a 3,,.!,,! (! = 3688),... a a j a a i a ij a i a a j a a ij i j, ( - ) - a ij, M ij.( - ) i + j M ij a i j, A ij, A ij = ( - ) i + j M ij, (..4) a a a 3 a a a 3 = a a 3 a 3 a 33 a a 3 a a 3 a a a 3 a 33 - a a 3 a 33 + a 3 a 3 a 3 : = a M - a M + a 3 M 3 = a A + a A + a 3 A 3, i, a a a a a a a a a = a i A i + a i A i + + a i A i = a ik A ik (..) k =

34 9 A ij a ij ( i, j =,,, )., j, a a a a a a = a j A j + a j A j + + a j A j = a kj A kj (..) k = a a a A ij a ij ( i, j =,,, )., (),. 4 A = (..) (..) A = 3( - ) A =. = 3( - ) = 35( - ) + ( - ) = (..) (..),, A = ( - ) ( - + ) - + 5( - ) A = ( - ) + + ( - ) ( - )( )3 A = ( - ) = = + + = = + + =,,..3,.. A A T, A T = A. A T, A,.

35 3,, ;. ( ),...3, -,, i j, k ( ki, j), A = a k A k + a k A k + + a k A k = a ks ( - ) k + s M k s s = M ks ( s =,,, ) -,, M ks, A =. 3 ( ),, ( )( ). a a j a a i + b i a ij + b ij a i + b i a a j a = a a j a a i a ij a i + a i a j a a a j a b i b ij b i a a j a, i,. 4 ( )( ) : () A ( ) B, B = A. () A ( )k ( ) B, B = A. (3) A ( ) B, B = - A. () B ( ) B = A. () A = [ a,, a i,, a j,, a ], a j k a i ( ij) B, 3 () B = a,, a i + ka j,, a j,, a = a,, a i,, a j,, a + a,, ka j,, a j,, a = A + = A. (), 3,. (3) i, j, ()() c ji ( ) A = a,, a i,, a j,, a a,, a i + a j,, a j,, a c ij ( - ) a,, ai + j,, - ai,, a c ji ( ) a,, aj,, - ai,, a = - a,, a j,, a i,, a = - B.

36 3 ( ). ( ),. 3 A, ka = k A. 4 A r - r 3 ( - 4) r 4 ( ) - A = = - ( - 8) 5 = r 3 ( - ) r 4 ( 5 ) - r 34 ( 3 8 ) A = a b c d a d c b c d a b c b a d 4 a b c d A r ( ) 3 r ( ) 4 a d c b a + c b + d a + c b + d a + c b + d a + c b + d = 4 a b c d a b c d A r ( - ) r ( - ) 34 d - b b - d c d a b = ( d - b) ( b - d) - c d a b = b - d d - b - 3. A,, A T = - A, A T = - A,, A T = A, 4 3, - A = ( - ) A, A = ( - ) A

37 3, A = - A, A =. 5 A, ik, jk, a ija kj = j = a i ja ik = i = k,, i, k, a kj = a ij ( j =,,, ki) a a a ija kj = j = a i a i a i a i k a a,,., ( ) ( ). (..)(..) A = a ija kj = j = a ij A ik = i = A i = k ik A j = k jk A 4 + A 4 + A 43 A 44 + A 45 (A 4 j ( j =,,5)a 4 j ). 4, = 7 5 A A A 43-4 A 44-4 A 45 = 7 4, 5, x = A 4 + A 4 + A 43, y = A 44 + A 45, 3 A A A 43 + A 44 + A 45 = 5 x - 4 y = 7 3 x + y = (.3.) (.3.)

38 33 A4 + A4 + A43 = x =, A44 + A45 = y = L ( + p): L = A O C B = a a O a a c c b b p c p c p b p b pp A r ij ( k), A, A = u L = A B (.3.3) O u u = u u u B c ij ( k), B, B = v O v p v pp = v v v pp, L r ij ( k), p c ij ( k), L L = A O C B = u u u c c v cp cp vp vpp L = u u u v v v pp = A B, A, B, A, B, U = A C O B = A B (.3.4) AB = A B (.3.5)

39 34 a a a b b b A = a a a, B = b b b, D = A O - I B, 4 a a a D = b b b A O - I B = O AB - I B = ( - ) - I B O AB = ( - ) ( - ) I B O AB = I AB = AB 6, D = A B, AB = A B. 5 4 D = a b c d - b a - d c - c d a - b - d - c b a A, D = A, A,. AA T = A = A A = A A T = AA T a b c d - b a - d c - c d a - b - d - c b a (, s = a + b + c + d ), A = AA T = s s s s a - b - c - d b a d - c c - d a b d c - b a. = s s s = s 4 = ( a + b + c + d ) 4 s D = A = ( a + b + c + d ),, a 4, a 4, D = ( a + b + c + d ).4,,,.. A = a a a ()

40 35 A ( - a) A = - a a a = a a a = - + ( - ) + = a - a - = a - ( a - ) a a - a a a = - a - ( - a ) = a - ( a - ) = a + ( - ) + ( - ) - + a 3,, A = a a a 3. a a = a a a a = ( a - ) a - a a b b b a b D = b b a,3,,, 4 a + ( - ) b b b b b a + ( - ) b a b a b D = = [ a + ( - ) b] a + ( - ) b b a b a b b a - b = [ a + ( - ) b] = [ a + ( - ) b] ( a - b) - a - b, ( - ), b b b b a b - a - b D = = b a + - a - b

41 36 ab a = b. ( + b a - b ) b b a - b. a - b 3, D = = b + ( a - b) b b b + a b = b + b a b b b a - b a - b b b b b a b + b b a = [ a + ( - ) b] ( a - b) - a - b b b a b b a + ( a - b) D - = b( a - b) - + ( a - b) D - D = ( a - b) D - + b( a - b) - D - = ( a - b) D - + b( a - b) - D = ( a - b) D + b( a - b) ( - ), ( a - b), ( a - b),, ( a - b) - D = ( a - b) - D + ( - ) b( a - b) - = ( a - b) - a + ( - ) b( a - b) - = [ a + ( - ) b] ( a - b) (Vadermode) V = x x x x x x x - x - x -.. V = x x = x - x = j < i ( x i - x j ) (.4.) =.(.4.) -,., V :, x,

42 37 V =, i ( x i x - x x3 - x x - x x ( x - x ) x 3 ( x 3 - x ) x ( x - x ) x - ( x - x ) x - 3 ( x 3 - x ) x - ( x - x ) V = ( x - x ) ( x 3 - x )( x - x ) - x ) ( i =,3,, ), x x 3 x x x 3 x x - x - 3 x - -,, ( x i j < i. 4 + V = ( x - x ) ( x 3 - x )( x - x ) j < i ( x i - x j ) = j < i ( x i - x j ) D + = a ( a - ) ( a - ) a - ( a - ) - ( a - ) - a ( a - ) ( a - ) - x j ), D + +,, ;, D + +,,, D + = a - a - + a =! ( - )!! = ( i - j) j < i + ( a - ) ( a - + ) a 5 x x 8 6 x 3, - x 4 - x 8-7 x 3 =

43 38, x - x - x + = 4 3 x 4 - x 4 x 8 6 x x x 3 = ( - ) ( - - ) ( - - ) ( x - ) ( x - ) ( x + ) = x = x = x = -.5, (Cramer)..5. A, AA * = A * A = A I (.5.) A A A A * = A A A = [ A ij ] T A A A A ( A ij a ij ), adj A. (.3.), (.3.) a a a A A A A a AA * a a A A A A = = a a a A A A A, A * A = A I. = A I A, A * = A -. (.5.) AA * = A I, A A * = AA * = A I = A I = A A, A * = A -, A =, A * = A - ( ). A -. A A, A - = A A * (.5.)

44 39 A, A -, AA - = I, A A - = I =, A. A - = A - (.5.), AA * = A * A = A I, A, A A A * =, (.5.). A A * A = I ( A * ) - = A A (.5.3) A, A ; A.. (.5.),. =, A =?, , B = A =, B =, A, B.A -. A = ( - ) =, A = ( - ) = - 3, A 3 = ( - ) = A = ( - ) = 6, A = ( - ) = - 6, A 3 = ( - ) = A 3 = ( - ) = - 4, A 3 = ( - ) = 5, A 33 = ( - ) = - A - = A A * = ( A * ) - = A A = T 3 = A 3, A = 3, (3 A) - - A - A *.

45 4 A = 3 A - = 3, A - = A A *, A * = A A -,, (3 A) - - A - A * = 3 A A * = 3 A A A - = ( 3 - ) A - = ( - 3 )3 A - = (.5.)(.3.). 3 A, Ax = b x = [ x, x,, x ] T, A j x j = A j A b A j. ( j =,,, ) Ax = b, x = A - b = A A * b = A A A A A A A A A A b b, b x j = A ( b A j + b A j + + b A j ) = A j A ( j =,,, ),., +. 3 : A x = b,. Ax =.3. Ax = A, ( ) x = x x =, x i, A. x Ax = A =. =.

46 4 3,? ( - ) x - x + x 3 = - x + ( - - ) x + 4 x 3 = x + 4 x + ( - - ) x 3 =,, D =. D = = = = ( - ) - 4 = ( - ) ( + 5-4) = ( - ) (- ) (+ 7) = = = a, b, c. A = a a b b = c c x + ax + a x 3 = x + bx + b x 3 = x + cx + c x 3 =, A = a a b b = A ; A = c c a b c = ( b - a) ( c - a) ( c - b) a b c a b = ; A 3 = c a b c = x = A A =, x = A A =, x3 = A3 A =. : () 43; () 3695; (3) ( - )3. x 3 x x. 4 x 4 x 3. x 3 x 3 x.3 :

47 4 () (4) : () (3) ; () x y x + y y x + y x x + y x y ; (5) + x.5 : () - x + y ; () - y a - b b - c c - a b - c c - a a - b c - a a - b b - c a + a b + b (3) abcd =, c + c d + d = ; () a b c d a b c d ; (3) a b c b c a c a b b + c c + a a + b. a + b x a x + b c a + b x a x + b c = ( - x ) a 3 + b 3 x a 3 x + b 3 c 3 =..6 AA T = I,A = -, A + I =. ; ; a b c a b c a 3 b 3 c 3 ;.7 A = [,, 3, 4 ] ; B = [,, 3, 4 ] 4, i = i ( i,,3), 4 = 7 4, A + B = 64,A..8 k,a k = O, A. : () ; () I - A, ( I - A) - = I + A + A + + A k -..9 cos cos cos cos = cos

48 43. : () (3) : () D = a c a c ; () a c b d ; + b d b d. ; () D = + a a a a + a a a a + a.. A 3,A = 3, ( 7 A) - - A *..3 3 A 3, A - A T O O A *..4 A = , A * A -..5 f( x), f( ) =, f() = 3, f( - 3) = 8. x + y + z =.6 ax + by + cz = d,a, b, c,,. a x + b y + c z = d.7 3 A = [ a ij ],A ij = a ij a = -,: ( ) A ; () Ax = e, e = [,,] T. x + x + x 3 =.8,,x + x + x 3 = x + x + x 3 =?

49 ,.??, ;. 3. k. 3.. m A, k k ( km, k),, A k, A k A r D, r + ( ), D A, r A, rak( A), r( A)., O r( O) =. B = b 34 =, b b 4 = - 5 b 3 b 34 3, r( B) ;, B 4, r( B) 3., B,, B 3 B,,., B 3, r( B ) < 3., r( B) =. : () A, r( A ) = ; () A k, r( A) k ; A k +, (3) m A, (4) A m, r( A) k r( A) = r( A T ) (3..) r( A )mi( m, ) (3..), A, r( A ); A r ( A ) =.

50 3 45 (),. m A C k mc k k., m A, m,., m : () ( k + ) ( ) k, ( k =,,, m - ). (),.,,. r( A). A = A, 3, A ,, r( A) = 3.,,,., : ( )? ( )?.. m A. m A = [ a ij ]. A,. A,, j j,, a j,, a i j,, A i, A, a j A, a, a,, a j -. j a j, A. A A, A, A., m, k ( km) A k.

51 46 A. AB, r( A) = r( B). : A B, r( A)r( B). r( A) = r, A r D r. r r ( k) ij i AB A B, B D r D r, D r = D r D r = - D r D r = kd r, D r, r( B)r. A r ji ( k) B, : Dr i ; Dr i j ; D r i j., B D r D r = D r, r( B) r;, ^D r, ^D r Dr = r i + kr j = r i + k r j = Dr + k^dr i A i r, r( B)r; ^D r =, D r = D r, r( B)r. A B, r ( A) r ( B). B A, r( B)r( A)., r( A ) = r( B ).,. A B, A T B T, r( A T ) = r( B T ), r( A) = r( A T ), r( B) = r( B T ), r( A) = r( B)., A B (AB), r( A) = r( B). A m, PQ m, r( PA ) = r( AQ) = r( PAQ) (3..3) () PQ,, PA( AQ PAQ) A ( ),.. : m A r( A) = r(i r ). A = PNQ = P I r O O O Q PQ, r( A) = r( N), r( N ) = r, r( A) = r,,,.

52 A = , r( A). A : A = r ( - 3 ) r 3 ( - ) , r( A ) =.. A = 4 A = r r ( ) r 3 ( - ) a b b b a b b b a ( - ),3,,, a b b r b a b b - a a - b b b a a b b b - a a - b ab a + ( - ) b, r( A) = ; ab a + ( - ) b =, r( A) = - ; a = b, r( A) = ; a = b =, r( A) =. c a + ( - ) b b b a - b a - b A =, B = 3, r( AB ). 5 3 AB = , r( AB) = A = 8, r( A) = 3; r( B) =, (3..3 ) r( AB) = r( B) =.

53 48 3. m a x + a x + + a x = a x + a x + + a x = (3..) a m x + a m x + + a m x = Ax = (3..) m A = [ a ij ], x = [ x, x,, x ] T, m. (3.. ) x =,.,, ( ),,. A, Ax =, : A m x = r( A) <, - r( A). Ax =, r ( A ) <., r( A) =, A D, D ( ),, r( A ) =, r( A) <. r( A) = r <, A r, - r.,,. - r c, c,, c - r, - r c, c,, c -,,. - r,. r( A ) < ( m = ),. A m x = r( A) =.,,. x - x - x 3 + x 4 = x - x + x3-3 x4 = x - x - x x 4 = A : r

54 3 49 r( ) x = c, x 4 = c, r ( - ) r 3 ( - ) x - x - x 4 = x 3 - x 4 = r 3 ( ) r ( ) x = c + c x = c x 3 = c x4 = c x x x 3 = c = c, ( c, c R) x 4 x + x - x 3 = 3 x + 7 x - 6 x 3 = 4 x + 8 x + x 3 =,., A r( A ) < 3. - A = r ( - 3 ) r 3 ( - 4 ) r( A) < 3, + 8 = = , x =., A = - 8. = 3.3 m Ax = b (3.3.) m A = [ a ij ], m( + ) A = [ Ab ], x = [ x, x,, x ] T, b = [ b, b,, b m ] T m.

55 5,, : A m x = b A A = [ Ab]. Ax = b, r ( A ) = r ( A )., r( A) < r( A ), A =,. r( A ) = r( A). r( A) = r ( A ),, A, r( A) = r( A ) = r ( r), A r, r, - r, - r,. AX = B, r( A) = r( AB).,, r( A ) = r( A) =,,.r( A) = r( A) <, - r, - r c, c,, c - r, - r c, c,, c - r,,, ( ).,, :. A m x = b : () r( A) = r( A ),., r( A) = r( A ) =,. r( A ) = r( A ) <,, - r ( A ) () r( A) < r( A ),. A, A = x - x + 3 x3 - x4 = 3 x - x + 5 x 3-3 x 4 = x + x + x 3 - x 4 = 3 r ( - 3 ) r 3 ( - ) r ( - ) r( A) =, r( A) = 3,

56 3 5, A = r ( - 3 ) r 3 ( - ) x - 5 x + 5 x3-3 x4 = x - x + 3 x 3 - x 4 = x - 3 x + x 3 - x 4 = r r ( ) r 3 ( - ) x - 5 x 3 - x 4 = - x - 4 x 3 = x 3 = c, x 4 = c, x = 5 c + c - x = 4 c - x 3 = c x4 = c x x x 3 x 4 = c c, x = c, x = c, x x x 3 x 4 = c + c , ( c, c R), ( c, c R),,.,, - r( A) ;,,.,.

57 5 3 x + x + kx 3 = 4 - x + kx + x 3 = k x - x + x3 = - 4 k, () ; ( ) ; ( 3)?., A = [ Ab], A = r 3 k - r ( - ) 3 k 4 - k k r 3 r ( ) r 3 ( - ) k - 3 k - 4 k k ( k - 4) ( k + ) k( k - 4 ) () k4 k-, r( A) = r( A ) = 3, ; () k = -, r( A) =, r( A ) = 3, ; (3) k = 4, r( A) = r( A) = < 3,., A x 3 = c, x = - 3 x 3 x = - x x x x 3-3 = c - + 4, ( cr ),. k k A = - k = k + k + = ( k + ) (4 - k) k (),.k4 k-, r ( A) = r( A ) = 3, ; () k = -, x + x - x 3 = 4 - x - x + x 3 = x - x + x 3 = - 4

58 3 53, A = r( A) = < r( A ) = 3,. (3) k = 4, x + x + 4 x 3 = 4 - x + 4 x + x 3 = 6 x - x + x 3 = - 4, A = x 3 = c, x = - 3 x3 x = - x x - 3 x = c - + 4, ( cr ) x 3 A, ( k - ) ( k - ), r ( -. )., ( k - ) = k -,,,, k4 k-, - x 3 = - x = ( k - 4) ( k + ) x 3 = k( k - 4), k k +, x + ( k - ) x 3 = 8, x = k + k + 4, x - x + x 3 = - 4, k + k( k + ) k r, r? 3., ( ) A = ; ( ) B =

59 54 3.3,. ( ) A = - 3 k - k - 3 k k - 6 ; () B = a b 3.4 A m, m,r( A) r( A )r( A) +, [ A ] A. a b a b a b. 3.5 A = a b a b a b a b a b a b,r( A) r( A ) A m, B m, ABx = Bx =., r( AB) = r( B) ( ) A = ; ( ) B = ; (3 ) C = 3 ; ( 4) D = ( )?,. x - x + x 3 = x - 3 x 3 = ( ) x + x + x x = ; ( ) x ; - x + 5 x 3 = - x + x 3 = x - 4 x + 5 x x 4 = x - x 3 = - x (3 ) 3 x - 6 x + 4 x 3 + x 4 = ; ( 4) x - 3 x 3 = - 5 x 4 x - 8 x + 7 x 3 + x 4 = x + 4 x = 3 x a a a 3,. x + a x = a x + a x = a x + a 3 x = a x - x + x 3 + x 4 = ( ) x - x + x 3 - x 4 = - ; ( ) x - x + x x 4 = 5 x - x + x 3 = - 5 (3 ) x + 5 x - 7 x 3 = 9 ; ( 4) 3 x + x - 6 x 3 = 3. : x + x + x 3 - x 4 = 3 x - x + 3 x 3 - x 4 = ; 3 x - x + 5 x 3-3 x 4 = x - x + 5 x 3 = 5 x + 3 x - x 3 = 4 x - 4 x + 6 x 3 =. 3 x + x + 4 x 3 = 9

60 3 55 a + a x - x = a x - x 3 = a x 3 - x 4 = a 3 x 4 - x 5 = a 4 x 5 - x = a 5 + a 3 + a 4 + a 5 =.,. 3.?. ( ) (3 ) x + x + x 3 = x + x + x 3 = ; ( ) x + x + x 3 = x + (4 - ) x + 7 = ( - ) x + x + 3 = ; ( 4) x + 5 x = x + x + x 3 + x 4 = x - x 3 + x 4 = ; x + 3 x + ( a + ) x x 4 = b x + 5 x + x 3 + ( a + 8 ) x 4 = 5 - x + x + x 3 = - x - x + x 3 = x + x - x 3 = B - x =, ( ) ; ( ) B ; (3) B. 3 -

61 .,,,.,. 4. ( ). m A = [ a ij ], m a j j = a j, ( j =,,, ),,, A. a mj m A m T i = [ a i, a i,, a i ], ( i =,,, m) T, T,, T m A.,. m Ax = b, A = [ Ab]. ( A ), A. x Ax = [,,, ] x = x + x + + x = b A,,,, b. x 4.. () :,,, m, m k, k,, k m, (), k, k,, k m k + k + + k m m. b,,,, m, b = m m, b ( ) ( ).

62 4 57 b = [, -, ] T, = [,,] T, = [,, ] T, 3 = [,, ] T, b = - + 3, b,, 3, b,, 3. = [ a, a,, a ] T e = [,,, ] T, e = [,,,] T,, e = [,,, ] T, = a e + a e + + a e., b (), x + x + + x m m = b.a m x = b.3.3, b ( ) :,,, m A = [,,, m ] A = [,,, m b]. () :,,, m () : b, b,, b s, ( ) (), ( ) ( ). () (),. b = - 3, = 3 - -, b =,, , [ x, x ] T, [, ] x x = b., [, b ]: b, b = r ( ) r 3 ( - 5 ) r 4 ( - ) x x = - 3 r 3 r 3 ( ) r 4 ( 3 ) r ( - ) r ( - ) = x + x = - 3. b,. 3 () :,,, m, k, k,, k m, k + k + + k m m = (4..) () ;, k = k = = km = ( 4.. ),

63 58 (). e = [,,] T, e = [,,] T, e3 = [,,] T, e, e, e3, = [ a, a, a 3 ] T, k e + k e + k 3 e 3 = [ k, k, k 3 ] T =, k = k = k 3 = e, e, e 3 ; a e + a e + a 3 e 3 - =, e, e, e 3,.,,.,,,,.3. () :,,, m A = [,,, m ], ( ),.3., x + x + + x m m =, Ax = () :,,, m A = [,,, m ] m; r( A) = m. m m,,, m,,, m = ;,,, m,,, m.. = [, -, 7] T, = [, 4,] T, 3 = [ 3, - 6, 3] T,, A = [,, 3 ],, A = r r ( ) r 3 ( 7 ) r ( - ) r ( 9 ) r ( 4 ) r 3 ( - 39 ) - (4..) r( A) = < 3( ),. k, k, k 3, k + k + k 3 3 =, Ak =, k 3 =, k = -, k =, k + k 3 = k - k 3 = k + k + k3 3 = =,, 3 =,, = , ( 4..),,,,. =

64 4 59., 3,, 3, 4, b b, b, b 3, b 4, = +, b = -, b3 = 3 + 4, b4 = 3-4. k b + k b + k 3 b 3 + k 4 b 4 = (4..3) k b + k b + k 3 b 3 + k 4 b 4 = k ( + ) + k ( - ) + k 3 ( ) + k 4 ( 3-4 ) = ( k + k ) + ( k - k ) + ( k 3 + k 4 ) 3 + ( k 3 - k 4 ) 4 =,, 3, 4, k + k = k - k = k 3 + k 4 = k 3 - k 4 =, (4..3), b, b, b3, b4 k = k = k 3 = k 4 =.,., B = [ b, b, b 3, b 4 [ b, b, b 3, b 4 ] = [,, 3, 4 ] - - ], A = [,, 3, 4 ], C = - -,,, 3, 4, r( A ) = 4; C = 4, C, r( C ) = 4; 3. r( B) = r( AC) = r( A) = 4, b, b, b 3, b m,,, m, m >,. m,,, m m A = [,,, m ], r( A) mi(, m) = < m,,,, m. 3,,, m,,,, m,,.,,, m, k, k,, k m, k + k + + k m m =,, k + k + + k m m + m =,

65 6,,,,.. 3 :,,, m,,,,,, m ( ). 4 j = [ a j, a j,, a rj ] T, j = [ a j, a j,, a rj, a r +, j ] T ( j =,,, m).,,, m,,,, m. A = [,,, m ], B = [,,, m ], r( A)r( B),,,, m, r( A) = m, r ( B ) r ( A) = m; B m, r ( B) m; r( B) = m,,,, m. 4 :,,, m,,,, m. 4 = [,,, 3, ] T, = [,,,, - ] T, 3 = [,,, - 3, ] T., k + k + k3 3 =, k k k 3 = 3 k + k - 3 k 3 k - k + k 3 k = k = k 3 =,,, 3. A = [,, 3 ] = r( A) = 3,,, ,, 3 45 e, e, e 3, e, e, e 3 =, e, e, e 3, 4,, = [,,,, ] T, 5 = [,,,,] T,,, 3, 4, 5 = 3-3 -,, 3, 4, 5, 3,, 3. =

66 4 6 5 a, a,, a k,,, k. k = +,, i = [, a i, a i,, a i ] T ( i =,,, k),,,, +,, a a a +,,, + = a a a + = j < i + ( a i - a j ),,, +. a a a + k > +,,,,, k. k < +,,,, k k,,, k,,,, k = ( a i j < ik - a j ),,, k, 4,,, k () :,,, m ( m ) m -. () m -, s,,, s -, s +,, m,,,, s -, s +,, m, m = + + s - s - + s + s s - s m m, + ( - ) s + s + s m m =,,, s -, ( - ), s +,, m m ( - ), ( ). (), k, k,, k m, k, k,, k m k + k + + k m m =, k s, s = - ( k k + + k s - s - + k s + s k m m ) s s,,, s -, s +,, m., ( ), () m -. = [, ] T, = [, - ] T,, 3 + =,,, =,. 3 :,,, m ( m),,, m i m -. 4 ( ) :,,, m, ( ) :,,, m, b

67 6 b () :,,, m,. A = [,,, m ], B = [,,, m, b], r( A) r( B). (), r ( A ) = m; ( ), r( B ) < m +. m = r( A) r( B) < m +, r( B) = m. r( A) = r( B) = m, 3.3 Ax = [,,, m ] x = b, b () :,,, m,. b,,, m, 3.3 [,,, m ] x = b, r[,,, m, b ] = r[,,, m ] = m,,,, m,,,, m, b. 6 b ( ) :,,, m, ( ) :,,, m -. ( ) :,,, m -, b. : m (), ( ). m ()., k, k,, k m -, m = k + + k m - m - (4..4) b ( ) :,,, m,,,, m, (4..4) (4..5), b = + + m - m - b = ( + m k ) + + ( m - b ( ) :,,, m -,. + m m (4..5) + m k m - ) m - m ()., ( 4..5 ) m ( b () :,,, m - ),, (4..5 ), m = [ b m m - m - ] 7,, 3,, 3, 4, ( ), 3?? () 4,, 3?? (), 3., 3, 4, 3, 3,,, 3, 4, 3. () 4,, 3., 4,, 3, ( ), 3, 4, 3, 3, 3, 4,, 3, 4, 4,, 3. 4.,,.. (), () r a, a,, ar,

68 4 63 () ( ) : a, a,, a r ; () () r + ( ( ) r + ), ( ) ( ) ( ). r ().,.., 4., ( ), ( ). A = [ a, a,, a m ]m, r( A ) = r, A r D r, A r. Dr r r B, Dr r( B) = r, 4. B r.a r + r( A ) < r +, 4. A r +. B r A, A r. D r r. r A, A. [ a, a, a 3 ] = r[ a, a ] =, a, a ; r[ a, a, a 3 ] = a, a, a 3, a, a a, a, a3., r[ a, a 3 ] = r[ a, a 3 ] = a, a 3 a, a 3 a, a, a 3., :..,,?,. [ a,, a ] = A = T, [ b,, b ] = B = T m T, r A B, A B, A T,, T m B T m

69 64 T,, T m, Ax = Bx =, x a + + x a = x b + + x b =, a,, a b,, b. B A, B b,, b, bi. a,, a b,, b, a,, a a i.,. = [, -, 5, - 3] T, = [4, -, -, 3] T, 3 = [5, 4, - 9, 5] T, 4 = [ -, -, 6, - 5] T,,. 4 A = [,, 3, 4 ] = A B r A r( A) = r( B) =,,., b, b B, AB,,,, 3, 4., 3 = - 3 +, 4 = - 3 ( ) ( ), ( ) (). () s,,,, s ; () t,,,, t, st.,,, s (), ( )( ), (),,, t,,,, s,,, t, [,,, t,,, s ] [,,, t,,] r[,,, t,,, s ] = r[,,, t ] = t,, st. r[,,, t,, s ]r[,,, s ] = s c = B

70 4 65. () ( ) s r,,, rs sr, s = r..,,. ( ) () (), ( ), () (), ( ) ( ). () r, r. ( ) (), ( ) r, () r +, (). 3 () ( ), () ( ), ( )..,, 3 AB m, k, A m, B p, (4..).A r D r, r( ka) = r( A). r( ka ) = r( A ) (4..) r( A + B)r( A) + r( B) (4..) r( AB)mi( r( A), r( B ) ) (4..3) r( A) + r( B) - r( AB) (4..4) ka r k r D r, k (4..).A = [ a,, a ], B = [ b,, b ], r( A ) = s, r ( B ) = t, AB, a i,, a is b j,, b jt. A + B a + b,, a + b, a i,, a is, b j,, b j t.,, s + t, r( A + B)s + t = r( A) + r( B) (4..3).C = AB, C A C = [ c, c,, c p ], A = [ a, a,, a ], B = [ b ij ], C = [ c, c,, c p ] = [ a, a,, a ] b b p b b p

71 66 c j = b j a + b j a + + b j a C A,, r( C )r( A). C T = B T A T, r( C T )r( B T ), r( C)r( B). (4..3). (4..4), [ 3 ]. A A - A - 3 I = O, r( A - 3 I) + r( A + I) =. A - A - 3 I = O ( A - 3 I) ( A + I) = O, (4..) (4..) (4..4) = r( 4 I) r( 3 I - A) + r( A + I ) = r( A - 3 I ) + r( A + I) + r[ ( A - 3 I ) ( A + I) ] = r( A - 3 I) + r( A + I ) =. 3 ()(),, () (). ()(). r, () () ( ) : a,, a r, ( ) : b,, b r Kr (Kr. ( ) ( ), ( ) ( ), r ) [ b,, b r ] = [ a,, a r ] K r ( ), r[ b,, b r ] = r. 3 (4..3), r( K r ) r, r( K r ) = r. K r r( K r )r[ b,, b r ] = r, [ a,, a r ] = [ b,, b r ] K - r ( )( ), ( )( ). ()() r. ( ) (), () () (, ) (), ( ) (, ), ( ) (, ), ( ) (, ), (, ) r.() r, () ( ) r,, ( )(, ), (, ) ( ). ( ) (, ), (, ) ( ), ( ) (), ( ) ().,,. ( ) ( ) ; ( ) (, ).,, () (),,. = [, ] T = [, ] T,. 4 = [,,, ] T, = [,,, ] T = [,, -, - ] T, = [, -,3, 3] T, 3 = [,,, ] T,,,, 3.,, 3 = +,, 3,,,, 3,,,, 3.

72 4 67 [,,, ]: [,,, ] = = -, = - + 3, [, ] = [, ],, ; [, ] = [, ] =, ,,,,,,,,,,, 3 A = [, ], B = [,, 3 ],, AX = B BY = A, 3.3, AX = B BY = A r( A) = r( AB )r( B) = r ( BA ), r( A) = r ( B ) = r( AB ). [,,,, 3 ] = r( A) = r( B) = r( AB) =,,,, V, V ( ), V., V V.,

73 68 : av, bv, a + bv; av, R, av. R 3,. 3 3, 3 3, R 3. 3, R 3., R,. V = x = [, x,, x ] T x,, x R.a = [, a,, a ] T V, b = [, b,, b ] T V, a + b = [, a + b,, a + b ] T V, a = [,a,,a ] T V V = x = [, x,, x ] T x,, x R.a = [, a,, a ] T V, b = [, b, b,, b ] T V, a + b = [, a + b,, a + b ] V 3,.x V = x = +, R = +, x = +, x + x = ( + )+ ( + ) V kx = ( k )+ ( k ) V.,,,, m V = spa(,,, m ). x = m m,,, m R (4.3.) V V, V V, V V. V, V R, R V,,, r, (),,, r ; () V,,, r,,,, r = r r (4.3.),,,, r V,,,, r, ( 4.3.),,, r, r V, dim V = r, V r. V, V,, V. 4 e = [,] T, e = [, ] T = [, ] T, = [, ] T

74 4 69 = [4,3] T.() e, e R ;, R ; () e, e,. () e, e, e, e, vr, 4. e, e, v, 4. 4 v e, e 3, e, e R., R. () = 4 3 = 4 e + 3 e e, e [ 4,3] T ;, = 4 3 = 3 +, [ 3,] T. R, ( ) ;. (), , V, V.,,,, m V = x = m m,,, m R,,, m,,,, m V,,,, m V., V, V,,, r ( V ), V V = spa(,,, r ),. 4,,,,,, V, [,,, ] = [,,, ] P (4.3.3) P, (4.3.3 ), P,,,,,,, P = [,,, ] - [,,, ]. V,,,,,, x = [ x, x,, x ] T, y = [ y, y,, y ] T,. x = Py y = P - = [,,, ] x = [,,, ] y, y = P - x. x = [,,, ] - [,,, ] y = Py 5 = [,, ] T, = [,, ] T, 3 = [,, ] T, x

75 7 = [6, 5,3] T, = [,, ] T, 3 = [,, ] T.( ),, 3,, 3 R 3 ; ( ),, 3,, 3 P ; (3),, 3 [, -,3],,, 3. (),, 3 =, 3,, 3 R 3.,, 3 = = 3 3 = -,,, 3, R 3 =,,, 3 R 3. () [,, 3 ] = [,, 3 ] P P = [,, 3 ] - [,, 3 ], [,, 3 ] =, [,, 3 ] - = - - P = [,, 3 ] - [,, 3 ] = (3) = [,, 3 ] - 3 y = P - - 3, - - = [,, 3 ] y, = = 3-7 = ,,, () A m x = r( A) <, - r( A). () A m x = b A A, r( A ) = r( A) = ; r( A ) = r ( A ) <, - r( A ).,

76 4 7 a x + a x + + a x = a x + a x + + a x = a m x + a m x + + a m x = (4.4.) Ax = (4.4.) m A = [ a ij ], x = [ x, x,, x ] T m.. A = A A = x 3 = c, x 4 = c,, Ax = x = c - + c - 4 -, ( c, c R) = [, -,, ] T, = [ - 4,,, ] T, spa(, ). :, - 4 -,,, t Ax =, c + c + + c t t Ax =, c, c,, c t. A ( c + c + + c t t ) = c A + c A + + c t A t =, c + c + + ct t Ax =., Ax = xy, x + y x ( R).Ax = N( A), ( 4.4.)., N ( A ),,, t, N( A ) = x x = c + + c t t, c, c,, c t R. Ax =, r( A) = r <, Ax = dim N( A) = - r. A r, A r, A

77 7 b b, - r B = B, b r b r, - r x = - b x r b, - r x (4.4.3) x r = - b r x r b r, - r x A B, ( 4.4. ) (4.4.3 ). (4.4.3), x r +,, x, x,, x r, ( ) ( ), ( 4.4.).x r +,, x (4.4.3) x x r = x r + x r + x = - b,, - b r - b,,,, - b r - r : - b, - r - b r, - r (4.4.3) ( ( 4.4. ) ) - r : - b - b - b, - r = - b r, = - b r,, - r =,,, - r N( A)., [ x r +, x r +,, x ] T - b r, - r - r - r,,,, r - r,,, - r., ( 4.4. ) = [,,, r, r +,, ] T,,, - r.,

78 4 73 = - r + - r r = [ d, d,, d r,,, ] T,,, - r, (4.4.), (4.4. ), (4.4.3) d = d = = d r =, =, = r + + r r (4.4.),,, - r. - r.,,, - r N( A). N( A),,, - r, ( ),,, - r ; ( ) Ax =,,, - r. (4.4.) k,, k - x = k + k + + k - r - r r.( 4.4. )., N( A) = x = k + k + + k - r - r k,, k - r R = spa(,,, - r ) (4.4.4) r( A ) =, = 4, - r( A) =. x 3 = x 4 x x =, x x 4 = - -, - x - ; x 3 x 4 = x = c x 3 = c ; x x - 4 x 4 = x = - 4, ( c, c R) x x 3 =, x = c c - ( c, c R) -,,, - r( A),. -, - 4

79 m Ax = b (4.4.5) Ax = ( ). Ax = b :,,, t Ax = b, = c + c + + c t t,, Ax = ;, Ax = b. c + c + + c t = c + c + + c t = A= A( c + c + + c t t ) = c = ( c + c + + c t ) b A + c A + + c t A t c + c + + c t =, A= ; c + c + + c t =, A= b., Ax = b, - Ax =, + Ax = b. 3 Ax =, Ax = b, x = + Ax = b. Ax = A(+ ) = A+ A= + b = b, x = + Ax = b. 3. = c + c + + c - r - r, Ax = b x = c + c + + c - r - r + ( c, c,, c - r R).,, { x x = c + c + + c - r - r +, c, c,, c - r R},,,, - r Ax =, Ax = b. x g = x h + x p x g Ax = b, x h Ax =, x p Ax = b. A = 3 4 -, b = Ax = x h = c c, Ax = b. - 4 A b Ax = b x p ( c, c R) = [,,, ] T, Ax = b

80 4 75 x g = x h + x p = c - + c ( c, c R),.. 3 3,,, 3, = [,,3, 4] T, + 3 = [,3, 4,5] T,. A, 4, Ax = dim N( A) = - r( A) = 4-3 = N( A ), x p, Ax = b., x p =., = N( A ), x = + c( ) = = + c ( cr) 3,, x = [ x, x,, x ] T, y = [ y, y,, y ] T, < x, y > = x y + x y + + x y x y.,, x y, < x, y > = x T y ( x, y, z, ) : () < x, y > = < y, x > = y T x ; () < x, y > = < x, y > ;

81 76 ). (3) < x + y, z > = < x, z > + < y, z >. x= < x, x > = x + x + + x, x x ( = ;, =, ( ). :. x, x> ; x =, x= ;. x= x; 3. x + yx+ y. < x, y > < x, x > < y, y >,, < x, y > = x ycos. x, y, x y. = arccos < x, y >, [, ] x y < x, y > =, x y, x y, xy., x =, x. r( A T A ) = r( A). A m, x. Ax = A T Ax =. x Ax =, A T A T ( Ax) =, ( A T A) x = ; x ( A T A) x =, x T Ax=,, Ax =. x T ( A T A ) x = x T =, ( Ax) T Ax =,, Ax = A T Ax =,, r( A T A) = r( A) ,,, r,,,, r,,, r r r = T i ( i =,,, r),, T i i = i, T i i = i, i = ( i =,,, r).,,, r.

82 R 3 = [,, ] T, = [, -, ] T, 3,,, 3. A = Ax =, T T = -, 3 T 3 =, T 3 =, - x x x3 =, A = - - x - = - x, x 3 = x. 3 = -., :,,. 3,,, r V( V R ),,,, r,,,,, r V., R, e, e,, e R, ei I i ( i =,,, ), e, e,, e R. R 3, = 3., = -, 3 = 6,,, r V, V.,,, r,,,, r,,, r.,,,, r.. ( Schmidt ),,, r V.,,, r V. i =,,, r = - = <, > <, > r - < j, r > r = r - j = < j, j > j -

83 78 i =,,, r V., i i [,,, r ] = [,,, r ] P P, P,,,, r,,, r,,, r. 3 A = [,, 3 ] =, R 4. = = [,,, ] T, = - <, > <, > =, 3 = 3 - <, 3 > <, > - = R 4 =, = <, 3 > <, > = =, A - -, 3 = = = 3 = , 4 T 4 =, T 4 =, T 3 4 =, - - x + x 3 = - x + x + x 3 + x 4 = x + x - x 3 + x 4 = = [, -,, ] T, 4 = [, -,, ] T, R 4

84 4 79,, 3, 4. 4 = [,,] T,, 3,,, 3.,, 3 T x =, =,. x + x + x 3 = - <, > =, <, > =, =, = - =, 3 = - <, > <, > -, 3 = = - - -, =,, :, x + x + x 3 = x = c + c = [ c, c, - c - c ] T ( c, c R). =. 3, x,, T x = x 3. T x =, c = - c, c =, x = [, -,] T, 3 = [, -,] T,.. 4 A A. A, T T - - A T A = I ( A - = A T ) [,,, ] = I = T T i j = i = j ij = - - ( i, j =,,, ) : A A.

85 8 A T A = I AA T = I, A., A () R 3 R 4. :.,, 3, 4 Q = [,, 3, 4 ] = () ; () ; (3) ; (4). - -, () = 3, = ; () = -, = 3, 3 = - ; (3) =, = 3 4, 3 = = [,,] T,,. () = [,3,] T, = [, -,] T, 3 = [7,5,] T ; () = [,,] T, = [,3, ] T, 3 = [,,] T. 4.3 = [,,3] T, = [,,] T, 3 = [3,4, a] T, a,,, 3 ; a,,, 3? 4.4 4,, 3, 4,3, : 4 k, k, k 3, k 4, k + k + k k 4 4 =. 4.5 = [,,] T, = [,-,] T, 3 = [,3,+ 3] T,= [,,- ] T, : (),, 3,? (),, 3,? (3),, 3? 4.6.

86 4 8 (),,, m,, 3, m. ( ) +, +,, m + m,,, m,,,, m. (3 ),,, m,,, m, +, +,, m + m. (4),,, m,,,, m+. 4.7, :,,, m i,,, i ,. () =, =, 3 = 4.9., 4 = ; () = () = [, -,4] T, = [,,3 ] T, 3 = [,, - ] T ; () = [,,] T, = [, -,] T. 4. e, e,, e,,,,,,,, e i ( i =,,, ) i. 4.,, = -, = +, 3 = - + 3, :,, A m, B m, () m >, AB = ; ( ) m < AB = I, r( B) = m. 4.3,,,, = +, = + 3,, = +,,,,. -, = -, 3 = 4.4 A m, r( A) =,A = T. 4.5 A = I A, A, r( A - I ) + r( A + I) = 4.6 V = { x = [ x, x, x 3 ] T x + x + x 3 = }, V = { x = [ x, x, x 3 ] T x + x + x 3 = }, R 3, R 3,? 4.7,, 3 V = spa (,, 3 ) V dim V, = [, -,4] T, = [,,3] T, 3 = [,, - ] T. 4.8 = [,, 3, - ] T, = [ 3, -,, ] T, 3 = [, 3, 4, - ] T, 4 = [4, - 3,,] T, (),, 3, 4 ; () ; (3 )R = [5,,,] T, = [,,,] T, 3 = [,,8,5 ] T, 4 = [,, 3,] T R 4, = [,,,] T, = [,,,] T, 3 = [,,,] T, 4 = [,,, ] T, (),, 3, 4,, 3, 4 P ; () = ,, 3,

87 8 ( ) x + x + x 3 - x 4 = x + x + x 3 - x 4 = x + x + x 3 + x 4 = ; () x + ( - ) x + + x - + x =. 4. AB,AB = O, r( A) + r( B). 4. A lm, B m, r( AB) = r( B). 4.3 A, r( A ) = r( A + ). 4.4, = [,,,4 ] T, = [ -,,, - 3] T. 4.5 A -, A * A, A * x =, x + 3 x 3 + x 4 = x - 3 x + x 4 = - x + x + 7 x 3 + x 4 = 5 4 x + x + 4 x 3 = , x, x, x 3,, x = [4,3,,,] T, x = [,,,4,] T, x 3 = [,8,,,] T. 4.8 Ax = b r ( A 53 ) =,, + = [,3,] T, + 3 = [,5,] T,. 4.9,,, - r A m x = b - r +, r ( A ) = r, -, -,, - r - Ax =. 4.3 = [,,] T, = [,,] T, 3 = [,,] T. 4.3,, 3, = + + 3, = + + 3, :,,?,,. 4.3 AB,A = - B, : A + B =.

88 ;., ; ,. A, x, Ax = x (5..), A ; x A ( ). (5..) ( A - I ) x = (5..), (5..) A - I = A - I f( ) = A - I = a - a a a a - a a a a - = ( - ) + a a - + a A.A - I = A. A, A. :, A : () A - I,, A () i,, A - i I x =

89 84,, t, i x = c + c + + c t t ( c, c,, c t ) A i,, i, ( A - i I) x =, N( A - i I).N( A - i I) dim N( A - i I) = - r( A - i I) i, i, i f() =, m i. A, A - I =, A - I = A = A =, =, 3 = 3. =, ( A - I) x =, ( A - I) = - x = c[,,] T, ( c ). =, ( A - I ) = x = c[,,] T, ( c ). 3 = 3, ( A - 3 I) = x = c[,,] T, ( c ). = ( - ) ( - ) (3 - ) = ,,. A =

90 5 85 A - I = = -,, 3 = = ( - - ) = - ( + ) ( - + ) = - ( + ) (- ) = = -, ( A - ( - ) I) x =, ( A + I ) = = [,,] T, m - = - =.,3 =, ( A - I) x =, ( A - I) = x = c[,, ] T, ( c) = [ -,, ] T ; m =, = ( > ). x = c[ -,,] T, ( c) 3 A = A - I = = 5,, 3 = -. x = (5 - ) - ( + ). = - ( + ) = 5, ( A - 5 I) x =, ( A - 5 I) = = c, ,3 = -, ( A + I) x =, = (5 - ) = (5 - ) ( + ) = x = c[,, ] T, ( c)

91 86 ( A + I ) = x = c, x 3 = c, m 5 - x = c - + c, ( c, c ) = 5 = ; m - = - = ( = ) A = [ a ij ],,,, i = i = i =, i = A (5..3) i = aii = tr( A) (5..4) A - I = f() = ( - ) ( - ) ( - ) =, ( 5..3). (5..4), ( - ) -., ( - ) ;, ( - ) - A - I ( a - ) ( a - )( a - ) ( - ) - a i i,, tr ( A)., i = i = i = i = a ii = tr( A) A x, A x., Ax = x, A A x = Ax, A x = x A x. A x, A ( A) = a I + a A + + a m A m ( ) = a + a + + a m m, x. 3 A A T A T. A =,. - I = A T - ( I) T = ( A - I ) T = A - I =, A A T,, =, x = c[,] T, ( c ) ; A T =,

92 5 87, =, x = c[, ] T, ( c),. 4 A, x A -, x., Ax = x, A - A - x = x ; A - x = x, A Ax = x. A, x A * A, x. 5,,, m A m, x, x,, x m,,, m, x, x,, x m..,. k, k +,,, k, k + x, x,, x k, x k +.. k +, t x + t x + + t k x k + t k + x k + = (5..5) t k + x + t k + x + + t k k + x k + t k + k + x k + = (5..6) (5..5) A A( t x + t x + + t k + x k + ) = (5..7) (5..6), t x + t x + + t k k x k + t k + k + x k + = (5..7) t ( - k + ) x, x, x,, x k + t ( - k + ) x, ti ( i - k + ) = ( i =,,, k) + + t k ( k - k + ) x k = ( i - k + ) ( i =,,, k), t i =., ( 5..5 ) t k + x k + =, x k +, t k + =. x, x,, x k, x k A, B, P, B = P - AP (5..) A B, A B. : A A (). A B, B A (). 3 A B, B C, A C ( ).

93 88 4 A B, A T B T, A m B m ( m ). 5 A B, A - B -. 6 A B, A B,. 5, 6,. A B, P, P - P - A - P = B -, A - B -.5, 6. A B, P, B = P - AP, AP = B, B - I = P - AP - I = P - ( A - I) P = P - A - I P = A - I A B, A B. ( 5..3) ( 5..4) :.,. A B, A B. : A =, B =, A, P. A B (- ), P - AP = P - IP = P - P = I B A, B.B. A, A. A A. A, P, P - AP = P, P = [ x, x,, x ], x i AP = P A[ x, x,, x ] = [ x, x,, x ] Ax i = i x i ( i =,,, ) AP =,, i A, i x i. P, x, x,, x. A x, x,, x,,,, Ax i = i x i ( i =,,, ).P = [ x, x,, x ], AP = A[ x, x,, x ] = [ x, x,, x ] = [ x, x,, x ] P - AP = 5. 5 : = P

94 5 89 A, A. A m,,, m, r, r,, r m, r + r + + r m =, r i i r i ( i =,,, m), A. A,,, : A, m =., m, m =,, < m, A.,,,, P, P - () A = AP =. - 3 ; ( ) B = ; (3) C =,, 3, : () A,. 3 x = [,, ] T, x [,, ] T, x 3 [,, ] T P = [ x, x, x 3 ] = = 3, P - AP =.. () B, = < = m, B. (3) C 3 5, -, -, 3 x = [,, ] T, x = [ -,, ] T, x 3 = [ -,,],, C, P - CP =. P = [ x, x, x 3 ] = A B, A = - - a 3, B =, = b () a, b ; () P, P - AP = B ; (3) A ( ). () (5..3) (5..4 ) tr ( A ) = tr ( B ) + a + 3 = + + b; A = B (3 a - 4 ) = b; b = a +, b = 3 a - 4 a = 3, b = 5.,

95 9 () =, ( A - I ) x = [,, ] T ; =, ( A - I ) x = [, -,] T ; = 5, ( A - 5 I ) x = [,, ] T. P = -, P - AP = B. (3) A = PBP -, A = PB P -, P = A = =, P - = , 5.3,,,. A. A,, m =. 3, A, x, x, x x. 3. x = Ax, x = Ax,. A, x, x T = ( x ) T = ( Ax ) T = x T A T = x T A x T x = x T Ax = x T ( x ) = x T x ( - ) x T x =, ( - ), x T x =, x x., P P -, Q, Q -. AP =., AQ = Q T AQ =? A, 3,,, Q. A,,

96 5 9, Q.. A, Q, Q - () A = () A - I = ; ( ) A = = ( - ) (- 3)= AQ.. = ( - ) ( - 3+ ) - ( - ) =, =, 3 = 3. : x = [ -, -, ] T, x = [ -,, ] T, x 3 = [,, ] T,, = x x = [ - 3, - 3, 3 ] T, = Q = [,, 3 ] = () A - I =, =, 3 = - 7., =, A - I = - x = c + c = Q x x = [ -,,] T, 3 = AQ = Q T AQ = = = ( - ) (- ) (+ 7) = , 3 = - - x 3 x 3 = [ 6, 6, 6 ] T, = ( - ) ( + 5-4) -, ( c, c ), =, x = c, x 3 = c, -, =,

97 9 = = = 5 x - 3 = - 7, ( A + 7 I) = -, = 45 = c, x = c, = - <, > <, > = , ( c). 3 = - ( - 4) = , 3 =, 3 3-3, Q = [,, 3 ] = , Q - AQ = = A, 6 = [, -, ] T, ( ) 3 ; ( ) A. A,, 3 x T x = [, -, ] x =, x - x + x 3 =, x = c, x 3 = c, 3 x = c - + c, ( c, c ) () () = [,,] T, = [ -,, ] T, = = = [,, ] T, = -, = , 3 = <, > <, > = -, T, = 3-3 3, Q = [,, 3 ],

98 5 93 Q - AQ = A = QQ - = QQ T = - 6, P = [,, ] = A = PP - = P - AP = = , P - = = = , ,, ax + bxy + cy = (5.4.) x = x cos- y si y = x si+ y cos mx + y = m,. (5.4.),,,.,.,. 5.4.

99 94 f( x, x,, x ) = a x + a x x + + a x x + a x + + a x x + + a x (5.4.),. aij, f ; aij, f..,. a ij = a ji, a ij x i x j = a ij x i x j + a ji x j x i, (5.4.) f = a x + a x x + + a x x + a x x + a x + + a x x + + a x x + a x x + + a x f = a ij x i x j = a ij x i x j (5.4.3) i = j = i, j = f = x T Ax (5.4.4) a a a x = [ x, x,, x ] T, A = a a a, A T = A. a a a, f A, A f, f A.,. f( x, x, x 3 ) = x + x x + x x 3 + x - x 3. 3, a ii x i,, aij = aji, xi xj. A = - f( x, x, x 3, x 4 ) = x + x x + x x 3 + x - x 3, A = -

100 5 95,., ( 5.4.4), P, f y, y,, y (5.4.5) (5.4.4), P, D, (5.4.6). x = Py (5.4.5) f = d i y j = y T Dy (5.4.6) i = D = diag[ d, d,, d ] f( x) = x T Ax = ( Py) T A( Py) = y T ( P T AP) y P T AP = D (5.4.7) A B, P, A B. B = P T AP P, A B, B A. P, Q, A B, B C, B = P T AP, C = Q T BQ, C = ( PQ) T A( PQ), A C.., ( ),.5.3 ( ), f( x) = x T Ax x = Qy, f = y T y. 5.3 A, Q Q T AQ =, x = Qy f( x)f( x) = x T Ax = ( Qy) T A( Qy) = y T Q T AQy = y T y = i y i, i =,,, A., ( ), P.,,. (5.4.6), r( D), r( A), f A,. f = x T Ax x = Qy A. x = Qy, ( ), x = Qy, x = y

101 96, x = < x, x > = < Qy, Qy > = ( Qy) T ( Qy) = y T Q T Qy = y T y = < y, y > = y,,,.,, f =. f f( x, x, x 3 ) = x - 4 x x + x - 4 x x 3 A = A - I =, A, A - I = = ( - ) ( - ) + + ( ) ( ) - + ( - ) ( ) - = 4- ( - ) ( - )+ 4(- ) = (- )(8 + - ) = (- ) (4 - )( + ) = = 4, =, 3 = -. = 4, ( A - 4 I) x =, ( A - 4 I) = = [, -, ] T, = 3 [, -, ] T. - =, ( A - I) x =, ( A - I) = = [,, - ] T, = 3 [,, - ] T. 3 = -, ( A + I) x =, ( A + I ) = = [,, ] T, 3 = 3 [,, ] T.

102 Q = [,, 3 ] = , x = Qy f =,. f = x - 4 x x + x - 4 x x 3 = 4 y + y - y 3 3 f = ax + x x + x x 3 + bx x 3 x = Qy, f = y + y - y 3, a, b Q. f A = a b b f = y + y - y 3,,, - A, a + a + a 33 = A = 3 a = + - b( - ab) = - a =, b = -, =, ( A - I) x =, ( A - I) = = [,, ] T, = [,, ] T, = = [,, ] T, = - = [,,] T, = [, -, ] <, > <, > =, - T, 3 = -, ( A + I) x =, ( A + I) = = [ -,, ] T, 3 = 3 [ -,,] T, 6-3 Q = [,, 3 ] = ,.

103 () f = y + y + + y,,, f A,, x = y.,,. x = Py. f f = x - 4 x x + x - 4 x x 3 y = x - x f = ( x - x ) - x - 4 x x 3 = ( x - x ) - ( x + x 3 ) + 4 x 3 x = y + y - y 3 y = x + x 3, x = y - y 3, y 3 = x 3 x 3 = y 3 x x x 3 = - - y y y 3 f = y - y + 4 y 3 y = ( x - x ) x = y - y + y 3 y = x 3, x = - y + y 3 y 3 = x + x 3 x 3 = y, x x x 3 = - - y y y 3 f = y + y - y 3 x = Py f = x x + x x 3 - x x 3.,.

104 5 99 x = y + y x = y - y, x = x 3 = y 3 x 3 x - y y y 3 = P y y y 3, f = x x + x x 3 - x x 3 = y - y + ( y + y ) y 3 - ( y - y ) y 3 = y + y y 3 - y + 3 y y 3 = ( y + y 3 ) - y + 3 y y 3-4 y 3 = ( y + y 3 ) - ( y - 3 y 3 ) + y 3 z = y + y 3 z = y - 3 y 3 y = z - z 3, y = z + 3 z 3 z 3 = y 3 y 3 = z 3 y y y 3 =, f - 3 z z z 3 = P z z z 3 f = z - z + z 3 x y z - z z x x 3 = P y y 3 = P P z z 3 = - 3 z z 3 = - - z z ,, x = Py,,,. f = x T Ax, r, x = Py x = Qz f = t y + t y + + t r y r ( t i ) f = k z + k z + + k r z r ( k i ) ( i =,,, r) t, t,, t r k, k,, k r.,,,,, r.

105 f f = y + y + + y p - y p y r,,,,, -, -,, -,,,, (),.. ( ), ( ) f = x T Ax, x, x,, x, f >,,, A > ; f <,,, A < ; f,,, A ; f,,, A ; f,, f.,. f( x, x, x 3 ) = x + x + x 3, f( x, x, x 3 ) = x + x, f( x, x, x 3 ) = x - x.,,.,? : f = x T Ax. x = Py f = d i y i i = d, d,, d >, x, y = P - x (x ), f( x). f( x) = d i y i > i =. <, x = Py f = x T Ax f = d i y i, d s, y = e s i = x = Pe s, = [,,,,] T, f = x T Ax = e T s P T APe s = d s f = x T Ax. =. A A., : 3 f = x T Ax, : () x, f = x T Ax > ; () f ;

106 5 (3) P, A = P T P ; (4) f = x T ( - A) x. x = Py x = Qy,, A, ()() ; x = Cy,, A y T Iy, C T AC = I ( ), P = C - ()(3) ; f = x T ( - A) x = - x T Ax, x, x T Ax > x T ( - A) x <, ( )( 4)... A, a i i > ; A, a ii <. A, x = e i = [,,,,,] T, f = x T Ax = e T i Ae i = a ii > ( i =,,, ), A, x = e i = [,,,, ] T f = x T Ax = e T i Ae i = a ii < ( i =,,, ) A, 3 P A = P T P, P, P p, p,, p, A, a ii, A, A -, A * a ii = p T i p i = p i > <.. A -, A *. A, 3 A, A - A,, A -. A, A * = A A -, A *., A -, A *. A, P, A = P T P, A - = P - ( P T ) - = Q T Q Q = ( P T ) -, A -. A * = A A - = R T R, R = A ( P T ) -, A *. 3 t,. f = x - 4 x x - x + 4 x x3 - x x x3 + t( x + x + x 3 ) q = x - 4 x x - x + 4 x x 3 - x x x 3 A = A - I =,,, - 7, q q = y + y - 7 y 3

107 ,, f f = y + y - 7 y 3 + t( y + y + y 3 ) = ( + t) y + ( + t) y + ( t - 7) y 3 f, + t >, + t >, t - 7 >, t > 7 A k a, a,, a kk k A k (),,. 4 A A, D >, D >,, D > (5.5.) a a a k D k = a a a k, ( k =,,, ) a k a k a kk, ( 5.5.) D <, D <,, D < A., 4 : A ( - ) k D k > ( k =,,, ) (5.5.) D k A k. ( - f) = x T ( - A ) x, - A k ( k =,,, ) - a - a - a k - a - a - a k = ( - ) k D k > - a k - a k - a kk - A, A. 4 f = - 5 x - 6 y - 6 z + 4 xy + 4 xz. f a = - 5 <, f. a a a a = A = A. = 6 >, A = - 8 <, 4

108 5 3 5,. f( x, x, x3 ) = x + x + 3 x 3 + x x - x x3 + x x3 f A = f,, - ( + >, ) < < -,. >, A > 5.. () A = 3 ; () A = ; (3) A = A,,, u, u,, u, A *. 5.3 = [,,, ] T, A = a T,A. 5.4 :B = P - AP, A, g A, P - g B. 5.5 A = a - c 5 b 3 - c - a [ -, -,] T, a, b, c. 5.6 A A = A( A ), : () A ; () A + I. 5.7 A, -,,: () B = A - 4 A + 3 I ; () B ; (3) A - 5 I. -, A A * 5.8 A,, 3.( ) B = ( A) - A * ; ( ) B ; (3) A + A + A , A,, + A. 5.?? ( ) ; () ; (3)

109 4 5. A = - a 3, B = - b A B, a b P, P - AP = B. 5. A = x, x A. 5.3 = [ a, a,, a ] T,= [ b, b,, b ] T, T =, A = T, : A. 5.4 AB, A,AB = BA. : ( ) A B ; ( ) B. 5.5 A =, =, 3 = 3, = [,,] T, = [,,4] T, 3 = [, 3, 9 ] T, = [,, 3 ] T.( ),, 3 ; ( ) A ; (3) A. 5.6 A, A. 5.7 Q,. ( ) A = 3 3 ; () B = A 3. = [,, - ] T = 4. ( ) = -?,. ( ) A?,. 5.9,. ( ) f = x + x - 4 x x - 4 x x 3 ; ( ) f = x x - x 3 x f = 5 x - xy + 6 xz + 5 y - 6 yz + cz, c ;f =. 5. ( ) P, ;? ( ) f = x + x + x 3 + x x + x x x x 3 ; ( ) f = x x - x x x x f = x T Ax, A, : x, x T xx T Ax x T x. 5.3 A, B, A + B ( ) f = 3 x + x + 6 x x + 6 x x 3 + x x 3 ; ( ) f = x + 3 x + 9 x x 4 - x x + 4 x x 3 + x x 4-6 x x 4 - x 3 x 4 ; (3 ) f = - 5 x - 6 x - 4 x x x + 4 x x 3 ; (4 ) f = ( x + x 3 - x 4 ) + ( x + x 3 - x 4 ) + ( x 3 + x 4 ). 5.5 A A( A - 6 A + I) = 6 I, A. 5.6 A, A + I >. 5.7 t,f. ( ) f = x + x + 5 x 3 + tx x - x x x x 3 ;.

110 5 5 ( ) f = x + x + ( - t) x 3 + tx x + x x f = 9 x + 3 x x x + 6 x x 3 - x x 3 + k( x + x + x 3 ) x = Qy - 7 y + y + y 3 ( ) k Q ; ( ) f k. 5.9 A m, B m, : B T AB r( B) =.

111 6. x = - 4, y = -, z =, u = -.. X = () ; () ; ( 3) (4) a x + a x + a 33 x 3 + ( a + a ) x x + ( a 3 + a 3 ) x x 3 + ( a 3 + a 3 ) x x 3. a b c a b a.6 () () (3)..7 () A =.8 () I. ( a, b, c ). - - ; () A = ; (3) A =, X =, Y = ; (). AB =, BA = 5 - -, BT A = 3-3 ; (3 ) ; - ( - ) , A = , A = I ;, A = - A. A - = 4 A..6 A - = 3 ( A + I), ( A + I) - = 3 A, ( A + 4 I) - = -.7 ( ) X =.8 ( ) A =.9 A - = 3-3 ; () X = ( A - I).

112 7. ( ) ( ) X = ; ( ) - ; () X = a + a 3 a a ; (3) ; (3) X = a a a P AP = AP P =.3 ( ) R ij. a 3 + a 33 a 3 a 33, P P A = a + a 3 a a 3 a + a a 3 a a + a a 3 a a 3 + a 3 a 33 a 3. a + a 3 a + a 3 a 3 + a 33 a a a 3,.4 A = I O O O () (43) = 4; () (3695) = 4; (3) ( ( - ) 3) =. x 4 6, x () - 3; () ; (3) - 8 ; (4) ; (5 )..4 () - ( x 3 + y 3 ) ; () a 3 + b 3 + c 3-3 abc; (3) x y..7 A = 7. ( - 3) ( - ) +.. ( ) 3! ( - 3)!; () ( - ) ( - ) ( - ) ( - ) - ; (3 ) ( ) ( ad - bc) ; ( ) + a i. i = A * = f( x) = x - 3 x +., A - = a, b, c,, x = ( b - d) ( c - d) ( d - a) ( c - d), y = ( b - a) ( c - a) ( b - a) ( c - b).7 ( ) A = ; ( ) x = [ -,,] T..8 = = () r( A) = ; () r( B) =., z = ( d - a) ( d - b) ( c - a) ( c - b).

113 8 3.3 () k =, r( A) = ;k = -, r( A) = ;k k-, r( A) = 3; () a = - 8 b = -, r( B) = ; a- 8 b = -, r( B) = 3; a = - 8 b-, r( B) = 3; a- 8 b-, r( B) = = [ a, a,, a ] T,= [ b, b,, b ] T,r( A) = r( A ) =, T =,. 3.7 () A - (3) C 3.8 () c - ; () B 3 ; (4) D + c - () ; (3) c (4) c -, ( cr) c - + c , ;, ; 5 ;, ( c, c,, c - R) ;, ( c, c R) ; x - 3. ( ) x x 3 = + c + c, ( c, c R) ; x 4 x x 7 - () ; (3) x = x ; (4) x = x c, ( cr). x a + a + a 3 + a 4 x a + a 3 + a 4 3. x 3 = a 3 + a 4 + c, ( cr). x 4 a 4 x 5

114 9 3. ( ), ; =,, x x x 3 = c c, ( c, c R ) ; = -,, x x = c, ( cr). x 3 () a-,;a = - b, ;a = - b =,, x x x 3 x 4 = c - + c - +, ( c, c R) (3), ; = -, x x x = - 5 ; =, x x =. x = (4) -,; =,, x x x 3 = = -,, 3.3 ( ) = ; () B = ; (3) B = 4. () ; () ; (3). 4. () ; (),= a = 5, a5. x x = x 3 + c + c., ( cr) ;, ( cr). ; =, 4.5 () -,,, 3 ; ( ) =,,, 3,; (3) = = -,,, () = [,,] T, = [,,] T, ; ( ) = [,] T, = [,] T, = [ -,] T, = [, - ] T +, +,,,,. (3) = [, ] T, = [, ] T, = [, ] T, = [, ] T, +, +.

115 (4) = [,] T, = [,] T,, 3 = [3,886] T. 4.8 (),, 4, 3; ( ),, 3, V, V. 4.7,, dim V =. 4.8 ( ),, ; () 3 = -, 4 = - + ; (3 ) e 3, e 4,, e 3, e 4 R ( ) P = ( ) = [ 4 3, - 3, 4 3,] T ; () = 4.4 x + x 3 = - 3 x - 4 x + x 4 = ; ( ) [ - 7,9,4, - ]. -, = - +,, - =. 4.5 A = [,,, ],dim N( A * ) = -, A - A *. 4.6 = [,,,] T, = [ - 3, -,,] T. 4.7 [4,3,,,] T + c [,,, - 4,] T + c [, - 5,, -,] T, ( c, c R). 4.8, 3, T + c -, -, 5 T, ( cr). 4.3 = [,,] T, = 6 [, -,] T, 3 = 3 [,, - ] T = - ; = 6, = () =, = c[ -, ] T, = 4, = c[,3 ] T, ( c) ; (),,3 =, = c [ -,,] T + c [,, ] T, ( c c ) ; (3),,3 =, = c [,,,] T + c [,,,] T + c 3 [ -,,,] T, ( c c c 3 ), 4 = - 3, = c[, -, -,] T, ( c). 5. A * A, u i ( i =,,, ). i ( + ) ( + ). 5.5 a =, b = - 3, c =. 5.7 (),8, - ; () B = ; (3) A - 5 I = , - 3,, -,

116 5.8 (), 4, 6,6,3, ; () B = 3 4 ; (3) A + A + A 33 =. 5. ( ) 5. a =, b = -, P = 5. x = -, A = ; () ( ) = ; () A = 5.7 ( ) Q = ( ) Q = , Q T AQ = = 6 6 -, P - AP = B. 6-6, Q T AQ = = 5.8 ( ) = [,,] T, 3 = [ -,,] T, () A = 5.9 ( ) f = - y + y + 4 y 3, Q = ( ) f = - y - y + y 3 + y 4, Q = 5. c = 3, f =. 5. ( ) f = y + y, P = ; (3). ; (3) A = ; ; , r( f) =,, ;.

117 ( ) f = z - z + 6 z 3, P = ( ) ; () ; (3) ; ( 4). 5.7 ( ) ( ) k = 4, Q = 4 5 < t < ; () - < t < , r( f) = 3,,. ; () k >.

118 ...,,,.,, ;, ;, ;, 3, ;., :..,.,, ;, ; 5 3,,,, 3. :,,.,,.,,,,.,,,.,,,,,.,,.,,.,,,. :,,. ( 43,97 )..,...,,. :

119 6 ; () ; (),,, (3),,,,... E.. E,,,.. E,, E :, : i = i ( i =,,,6), = {,, 3, 4, 5, 6 }. E :. : (),, : = {,, 3, 4 }. = {, }, = {, }, 3 = {, }, 4 = {, }, (),, : = { }, = {, }, 3 = { }, = {,, 3 }. :,,,. 3 E :,,.: = {,, }. i = {i } ( i =,, 3, ), 4 E :,, t, = { t t > },. 5 E : x y, -, 5.( x, y), - x y 5, = { ( x, y) : - x y 5},.

120 7 () ; () (, ) ; (3) ().,., : A = { 4 }, B = {}, C = { 4}., A, B C (, B = {, 4, 6 }, C = {,, 3, 4 }),,.,,,,., A, B, C., A, A, A.,.,.,.,,.,,,,,, ;,,,,,,,.,,.,,,..,.,.,,.,. ) B A AB, B A, A B, B A A B. ( Joh Ve, 83493),.,,, A B, A B( ). : () A, A A; () A, B, C,, A B, B C, A C; (3) A, A ;

121 8 (4) A, A. ) A B A B, B A, A B, A = B., : () A = A; () A = B, B = C, A = C; (3) A = B, B = A. 3) A B A B, A B, A + B A B. A B A + B., A B, A + B = A; A B, A + B = B ; A = B, A = A + B = B. A, A,, A A, A,, A, A + A + + A ( A A A ), Ai i = A i. i = A, A,, A,, A, A,, A,, A + A + + A + ( A A A ), A i i = A i. i = 4) A B A B, A B, AB A B. 3 A B AB., A B, AB = B; B A, AB = A. A = B, A = AB = B. A, A,, A A, A,, A, A A A A A A, A i i = i = Ai. A, A,, A, A, A,, A,, A A A (A A A ), A i i = A i. i = 5) A B A B, A B

122 9. AB =. A, A,, A, A i A j = ( i < j ),. E.,. 6) A B A B A + B =, AB =, BA., B A, A B,, A B (), B = A A = B. 4 A A. 3 4, =, =, A = = A, A B A B. :,.,, A = {, 3}, B = {, 6}, A B, A B. 7) A B AB AB, A - B. 5, AB ; 6, A B.A - B. 5 6 : A - B = A - AB A - B = AB. 4.A i i ( i =,,3, 4), A i :

123 () ; ( ) ; (3) ; ( 4). ( ) A A A 3 A 4 ; ( ) A A A 3 A 4 ; ( 3) A A A 3 A 4 + A A A 3 A 4 + A A A 3 A 4 + A A A 3 A 4 ; ( 4) A A A 3 A 4 + A A A 3 A 4 + A A A 3 A 4 + A A A 3 A 4 + A A A 3 A 4. : () A + B = B + A, AB = BA ; () ( A + B) + C = A + ( B + C), ( AB) C = A ( BC) ; (3) ( A + B) C = AC + BC, AB + C = ( A + C) ( B + C) ; (4) ( ) A + A + + A = A A A (..) A A A = A + A + + A. (..), (..) Ai ( i =,,, ), A i ( i =,,, ).., (..).,,.,., : ( A + B) C = AC + BC. ( A + B) C, ( A + B) C, A + B C, A B, C. A C, B C, AC + BC. ( A + B) C AC + BC., AC + BC, AC + BC, AC BC. A C, B C, A B, C, ( A + B) C. AC + BC ( A + B) C., : ( A + B) C = AC + BC., A, B, C. () ABC ; () ABC = C; (3) C A. ( ) ABC,. ( ) ABC = C C AB,, ABC = C. ( 3), C A.

124 3 A, B, AB + AB + A B = AB. : AB = A + B = A+ B = A( B + B) + ( A + A) B = AB + A B + A B + A B = AB + AB + A B, AB + AB + A B = AB. 4 A, B, C, ( A + B) - AB = AB + AB. : ( A + B) - AB = ( A + B) AB = ( A + B) ( A + B) = AA + AB + BA + BB = + AB + BA + = AB + BA, ( A + B) - AB = AB + AB..3,.,., ( ),.,,,,..,. A,, A, f ( A),. f ( A) : () f ( A) ; () f ( ) = ; f ( A) =, (.3.) (3) A, B (AB = ), f ( A + B) = f ( A) + f ( B). : E. ( A), =, =., A : f ( A) = =.55;

125 4, A, A 4. f 4 ( A) = 4 =.5,,, A,,,, A,.5,,.,, : ,,,,, : E T O A N I R S H D L C F U M P Y W G B V K X J Q Z ,,..,,, A,., A.,,,,., :,, A f ( A) =. A, A, P( A).,,.,,.

126 3,,,,.,,,,., A, A,., : (),,,,, ; (),. P( ) = P( ) = = P( ),, E, :,,,,. A, A k, A k, k, A P( A) = k = A = (.4.). A. (.4.) 3, 9, 3. A = {3 }., 3 3, 3.3, 3, P 3, A P 3., P( A) = P3 3 = 9 8 = 8 5.,. i = { i } ( i =,,,6 ), = {,,, 6 }.

127 4 6, A = {}, A = {, 4, 6 }. A 3, P( A) = 3 6 =. :,. = {, },, P( ) = P( ) =, P( A) = P( ) =.,,, A , 5, 5 5,,. 5 C 5, C C 3 C , P = C C 3 C 5 C 5 = 3 5 = 5. 4 ( ), N ( N), : () ; (), ; (3) k(). N, N. (),!, P =! N ; () N, C N,,!, : P = C N! N = N! N ( N - )! ; (3) k, C k, - k, ( N - ) - k, k : P 3 = Ck ( N - ) - k N.,,,.

128 5, (Bose Eistei) ;,, ( Fermi. Dirac) 5 ( ) 4 4, 3,,,,3. 4 4, 4 4,, 4 4., 4,4 4 4!, P{ } = 4! 4 4 = 3 3 ;, 4,,. 4, C 4, 3, C 3., 4, C 4,, : P{ } = C 4 C 3 C 4! 4 4 = 9 6 ;,,! () 3,. 4 3, C 4, 3, C 3., 4 3, C 3 4,,. (). 4, C 4., 4, C 4,,. P{ } = C 4 C 3 C C 4 C = 64 ; C 4 3, ,, 4,, P{ 3} = C = ( ),, k. A = {k }. :,. (+ ), (+ )!.k, + - (+ - )!.,

129 6 P( A) = (+ - )! (+ )! = +. :,, C +.k, - C - + -,, P( A) = C- + - C + = +..4.,, ()., E,,. : A, S A, AS A, A., A: P = S A S.,,, ( ), A A A, A P( A) = A = A, (.4.) (.4.).. 7 ( ) 7 8,,,. 7 x, 7 y. x 6, y 6.,, { ( x, y) x 6, y 6}. 6, S = 6. A, x - y, A = { ( x, y) x - y, x 6, y 6}. 7 A 7, S A = 6-4., P( A) = S A S = = 5 9.

130 7 8 { ( x, y) < y < 4 x - x },, x 6. {( x, y) < y < 4 x - x }, S =. A = ( x, y) 3 3 x < y < 4 x - x 8, SA = 3 4 x - x x d x = 4-3, 3 P( A) = S A S = = ,,.,,, :.. :, A, P( A), : () P( A) ; () P( ) = ; 3 () A, A,, A,, P( A) A. P A i i = = P( A i ). i =,., = , P() =. (.5.) P() = P( ) + P() + + P() +,

131 8 P() =. () A, A,, A, P A i i = = P( A i ). (.5.) i = Ai = A + A + + A + + +, P() =, i = 3 A, P A i i = A + A =, AA =,, = P( A i ). i = P( A) = - P( A). (.5.3) P( A) + P( A) = P( ) =, 4 A, B, P( A) = - P( A). AB A B, P( A - B) = P( A) - P( AB). (.5.4) A = A= A( B + B) = AB + A B,, P( A) = P( AB + A B ) = P( AB) + P( A B). P( A - B) = P( A B) = P( A) - P( AB). 5 A, B, A B, P( A - B ) = P( A) - P( B). (.5.5) A B, AB = B, P( A - B) = P( A) - P( AB) = P( A) - P( B). A B, P( A ) P( B). (.5.6) 6 A, B, P( A + B) = P( A) + P( B) - P( AB). (.5.7) A + B = A( B + B) + B = A B + AB + B = A B B, A B + ( A + ) B = A B + B, P( A + B) = P( A B + B) = P( A B) + P( B) = P( A - B) + P( B),

132 9 4 P( A - B) = P( A) - P( AB), P( A + B) = P( A - B) + P( B) = P( A) + P( B) - P( AB)., A, B P( A + B) = P( A) + P( B)., 6. A, A,, A, P( A + A + + A ) = i = P( A i ) - i < j, = 3 P( A i A j ) + + ( - ) - P i = A i. (.5.8) P( A + B + C) = P( A) + P( B) + P( C) - P( AB) - P( BC) - P( AC) + P( ABC).,., 5.5,. A = { 5 }. A i = {5 i } ( i =,,,5 ), A = A + A + A 3 + A 4 + A 5, A, A,, A 5, P( A i ) = Ci 5 C 5 - i 95 C 5 ( i =,,3,4, 5), 5 P( A) = i = C i 5 C 5 - i 95 C 5 = A 5, P( A) = C5 95 C 5 = , P( A) = - P( A).34.,,.,,. 99, () 3 ; ( ) 3. A = { }, B = {3 }, A - B 3 ; AB 3, 6. 9, 45, 3 3, 6 5,

133 3 P( A) = , P( B) =, P( AB) = , (.5.4) (.5.7) P( A - B) = P( A) - P( AB) = = 3. P( A + B) = P( A) + P( B) - P( AB) = = ,.6,, 45,5.4,, 35, 5., A = { }, A = { }, B = { }, B = { }. B B A AB = { } A B = { } A = { } A AB = { } A B = { } A = { } 5 5 B = { } 6 B = { } 4 : () ;,, = 8,, A = { } 8, () ; P( A) = 8 =.8.,, 6,, B = { } 6, P( B) = 6 =.6.

134 3 (3) ;,, 45,, AB = {} 45, P( AB) = 45 =.45. (4),. B, A,, P( A B). P( A B), P( A) P( B), P( AB).,, 6,,,, B 6,, A 8 AB 45., P( A B) = AB B = 45 6 =.75. : P( A B) = 45 6 = 45 6 = P( AB) P( B).,.,, A m, B m, AB m, P( A) = m, P( B) = m, P( AB) = m. B, B (B ), B m, A AB, m,, : P( A B) = m m = m m = P( AB) P( B). B P( B ) >, B, A P( A B) : P( A B) = P( AB) P( B). (.6.), P( A) >, A, B P( B A) : P( B A) = P( AB) P( A). (.6.), P( A B) :

135 3 () A, P( A B) ; () P( B) = ; (3) A, A,, A,, B =, P( A ) =. P A i B = i = i = P( A) P( ) P( A i B). = P( A),, 5 3,,,,,? A i = {i } ( i =,), P( A A ). : P( A ) = 5 8, P( A A ) = C 5 C 8 = 8, P( A A ) = P( A A ) P( A ) = = 4 7. :. A, 4 3, A 7, A 4, P( A A ) = , : A, B.P( B) >, P( A) >, P( AB) = P( B) P( A B ). (.6.3) P( AB) = P( A) P( B A ). (.6.4), P( B) = ( P( A) = ), (.6.3) ( (.6.4) ). P( AB) P( B) = ( P( AB) P( A) =.. A, A,, A, P Ai = P( A ) P( A A ) P( A A A - ). (.6.5) i = 5, 5. 3,?

136 33 A i A A A 3. = {i } ( i =,,3 )., P( A ) = 45 5, P( A A ) = 44 49, P( A 3 A A ) = 5 48, P( A A A 3 ) = P( A ) P( A A ) P( A 3 A A ) = , P 3 5, A A A 3 P 45 P 5, P( A A A 3 ) = P 45 P 5 P ,, %, 3%,5%,.5,.,.5.,? : B = {}, B = {}, B 3 = {}, A = { }, A = AB + AB + AB 3, AB, AB, AB 3, : P( B ) = %, P( B ) = 3%, P( B 3 ) = 5%, P( A B ) =.5, P( A B ) =., P( A B 3 ) =.5, : P( A) = P( AB ) + P( AB ) + P( AB 3 ) = %.5 + 3%. + 5%.5 =.5.,,,,,,,,,.,,. 3 B, B,, B, Bi Bj = ( i < j ),

137 34 P( B i ) > ( i =,,, ), A B i, A B, B,, B i =, A P( A) = P( B i ) P( A B i ). (.6.6) i = B, B,, B, AB, AB,, AB, A B i, A = i = AB i, i = P( A) = P AB i i = = P( A( B + B + + B ) ) = P( AB + AB + + AB ) = P( AB i ) = i = i = P( B i ) P( A B i ). 4 :.,,.5, %, 55%, 3%, : : ()? (),? () : A = {}, B B B 3 = { }, = { }, = { }. P( B ) = 5%, P( B ) = 55%, P( B 3 ) = 3%, P( A B ) =.5, P( A B ) =.5, P( A B 3 ) =.3, P( A) = P( B ) P( A B ) + P( B ) P( A B ) + P( B 3 ) P( A B 3 ) = 5% %.5 + 3%.3 = 8%. (), P( B A) = P( AB ) P( A) = = 5%.5 8% 3 P( B ) P( A B ) i = P( B i ) P( A B i ).4. : A, B i.

138 35, (Bayes). 4 B, B,, B, Bi Bj = ( i < j ), P( B i ) > ( i =,,, ), A B i, A B, B,, B i =, A B i, P( B i A) =, B i P( B i ) P( A B i ) P( B j ) P( A B j ) j = ( i =,,, ). (.6.7) P( B i ) ( i =,,, ),., A, Bi, A, B i P( B i A),, A,. 5 4,4 5%, %,3%,35%, 4.5,.4,.3..,? A = {, }; Bi = {, i } ( i =,, 3,4) ; 4 P( A) = P( B i ) P( A B i ) i = = =.35 = 3.5%.,, P( A B i ).5,.4,.3..,. 6 5,,.,,,?4 (,, 3 )?, P( B i. 4, A) i ( i =,, 3,4) P( B 4 A) =, 4 P( B 4 ) P( A B 4 ) i = P( B i ) P( A B i ).

139 36 4 P( A) = P( B i ) P( A B i ) =.35, i = P( B 4 ) P( A B 4 ) =.35. =.7,, 4.%. P( B 4 A) = , 3 3.8%,5.4%,8.6%. 7.4%., 95%, 4%.,,. A = {}, : B = {}, B = {}, P( B) =.4%, P( B) = 99.6%, P( A B) = 95%, P( A B) = 4%, P( B A) = = P( B) P( A B) P( B) P( A B) + P( B) P( A B).4% 95%.4% 95% % 4% 8.7%.,,. :, B, B,, B, A B, B,, B, A ; A, B i ( i =,,, ), P( B i A) ( i =,,, ),..7 :.7..A, B P( B A) = P( AB) P( A)., P( B A) P( B),, P( B A) = P( B), B, A, A., : A, B, P( B A) = P( B), (.7.)

140 37 B A. (.7.), P( AB) = P( A) P( B A) = P( B) P( A B) P( A B) = P( A). (.7.) A B.. (.7.) (.7.), A B., : A B, P( AB) = P( A ) P( B). (.7.3), (.7.3) A B. (.7.3 ), A. A B, A B, A B, A B. P( AB) = P( A - AB) = P( A) - P( AB) = P( A) - P( A) P( B) A B. = P( A) [ - P( B) ] = P( A) P( B), A B, A B A B. :. A B A B, P( A + B) = P( A) + P( B) ; A B A B, P( AB) = P( A) P( B)..7. A, A,, A, m( m ) A i, A i,, A im. P( A i A i A im ) = P( A i ) P( A i ) P( A im ), (.7.4),, m, m =,,.,,,. :, = {,, 3, 4 }, P( i ) = 4 ( i 4), A = {, }, B = {, 3 }, C = {, 4 }, P( A) = P( B) = P( C) =,

141 38 P( AB) = P( BC) = P( AC) = 4, A, B, C.ABC = { }, P( ABC) = 4 P( A) P( B) P( C), A, B, C. 3 A, A,, A, P( A A A ) = P( A ) P( A ) P( A ). (.7.5) 4 A, A,, A, m( m ).., A, A,, A A, A,, A, :, P( A + A + + A ) = - P( A A A ) = - P( A ) P( A ) P( A ). = - ( - P( A ) ) ( - P( A ) )( - P( A ) ). (.7.6), 5, 3, 4,. A, A, A 3 B = {}, A i = {i } ( i =,, 3),, P( A ) = 5, P( A ) = 3, P( A 3 ) = 4, P( B) = P( A + A + A 3 ) = - P( A A A 3 ) = - P( A ) P( A ) P( A 3 ) = = 3 5.,,.3,., : (), 9 ; (), 3, 3,. A i = {i }, A i = {i }, P( A i ) =.3, P( A i ) = - P( A i ) = -.3 =.7, ( i =,, 3,4, 5,6), S = {}. () 9,,, S = A + A + A3 + A4 + A5 + A6 = A A A3 A4 A5 A6,

142 39 P( S) = - P( A A A 3 A 4 A 5 6 A 6 ) = - i = P( A i ) = = = (), 3, 3,,,,, S = ( A + A ) ( A 3 + A 4 ) ( A 5 + A 6 ), P( A + A ) = P( A A ) = - P( A A ) = - P( A ) P( A ) = -.7 = -.49 =.5;, : P( A3 + A4 ) =.5, P( A5 + A6 ) =.5, P( S) = P( ( A + A ) ( A 3 + A 4 ) ( A 5 + A 6 ) ) = P( A + A ) P( A 3 + A 4 ) P( A 5 + A 6 ) =.5 3 = ,,,.., : () : A A; (), P( A) = p, P( A) = - p ( < p < ) ; (3).,, (Beroulli).( 5,, (), A., 5,,.),,,

143 4 P( A) = p =, P( A) = - p =,,,,,.,.., A p( < p < ), A m P ( m) = C m p m ( - p) - m ( m =,,,, ). (.8.) B m = { A m }, A i = {i A }, A i = {i A }, B m = A A m A m+ A + A A m - A m A m + A m + A + + A A - m A - m + A, B m C m,,, p m ( - p) - m., : P ( m) = P( B m ) = C m p m ( - p) - m., p =.7, 6.,,,,, =, A = { }, p = P( A) =.7, m = 6, (.8.) P{ 6 } = P (6 ) = C ( -.7 ) ,,,,, = 4, A = { }, p = P( A) =., - p = P( A) =.9. m, (.8.) : P{3 m 5} = P{ m = 3} + P{ m = 4} + P{ m = 5} = P 4 (3 ) + P 4 (4) + P 4 (5) = C C C %. :,.95?,,,, A = { }, p = P( A) = % =., (.8.) P{ } = - P{ } = - P () = - C p ( - p) - = - ( - p)..95,

144 4 P{ } = - ( - p).95, ( - p) = ( -.) log( -.95) -.95, log( -.) 98.7, = 99, ,., A p( < p < ),, A r,, :, A r,?, - A r -, A ( A r ). (.8.) P{ - A r - } = P - ( r - ) = C r -, P{ A } = p. P{ A r, } - p r - ( - p) ( - ) - ( r - ), = P{ - A r - } P{ A } = C r - - p r - ( - p) ( - ) - ( r - ) p = C r - - p r ( - p) - r.,, A r, P = C r - - p r ( - p) - r ( = r, r +, ). (.8.). : () ( ),8 ; (),.A:5, B :8 ; (3) 5,,,3,4,5,3, ; (4), 5 ; (5),.. 8,A 4, B, C 3. () ; (),? (a) AB ; ( b) A + B; ( c) B ; ( d) A - B ; ( e) BC; ( f) B + C..3 ABC : () A ; ( ) A, BC ; (3) ; (4 ) ; (5) ; (6 ) ; (7).

145 4.4 = {,,,}, A = {,3,5}, B = {3,5,7}, C = {,3,4,7}, : () A B; () A( BC)..5 A, B, ( A - B) + ( B - A) = AB + A B.6 Probability, 7,ability..7 5,5 3,()? () 3?.8,,,,9,: A = {,5}; A = { 5} ; A 3 = {,5}..9,,,., (a) ; (b)... 6, : () 6? () 6? (3) 6?. 3 4,,,3?. a,..3 ( p, q) :[ -,] [ -,], x + px + q = : () ; ()..4 ABC,P( A) = P( B) = P( C) = 4, P( BC) = P( AC) = 8, P( AB) =. A BC..5 ( N) (), K?.6 AB P( AB) = P( A B),P( A) = p,p( B)..7 P( A) = a, P( B) = b, P( AB) = c. () P( A + B), () P( A B), (3) P( AB), (4) P( A + B)..8, ( ),?,?.9 4,,, :?.,,,?. 96 %, 4%.,.98,.5...,,,,.8,..,,, 4,,,. () ; (),..3,,, -.7

146 43..; -, ,,?..4 5, 5,.,,,,?.5 P( B A) = P( B A) A B..6 A, B, C, A - B C..7 P( A) >, P( B) >, AB AB..8,a, b, c.?.9 AB, P( A) =.4 P( A + B) =.7,P( B) = p. (a) p A B? (b) p A B?.3,.,,..3,88,?.3, p =.4,,.: () 5,; (),99 %?.33,.7,.3,.8 ;., ( ) ( ). () ; () ; (3).

147 .,,,,.,,., A, { A k } {=,,,,. k }, A,, {= k} { k },,,,. 3, (), [, + ). {= t} = {t },,.. 4. = { }, = { }, = { } = {= }, { } = {= }. 5,. = {, }, = {, }, 3 = {, }, 4 = {, }., {= } = { 4 }, {= } = { 3 }, {= } = { }, {= k} = { k }, k =,,.

148 45, =, A = { }, A.,,.,,,,.,.,,,, :,, (), ().., f( x) x, f( x),, () ; ( ),.,,,.,.,,,,,,,.,,,. 5,,,,,., :.,,...,,, x, x,, x,,., x, x,, x,, p, p,, p,, P{= x i } = p i ( i =,, ), (..) : x x P{= x i } p p

149 46,., (..) ( )., { p i } : () p i ( i =,, ) ; () p i =. i =,. ABC, P( A) =.5, P( B) =.8, P( C) =.. A, B, C..,,,,3.A, B, C : : p = P{= } = P( ABC) = ( -.5 ) ( -.8 ) ( -. ) =.8, p = P{= } = P( ABC + ABC + ABC) =.4, p = P{= } = P( ABC + ABC + ABC) =.4, p 3 = P{= 3} = P( ABC) =.8. 3 P{= x i } :,. :

150 47 P{< } = P{= k} = P{= } + P{= } =.5, k = P{ < < 3} = P{= } + P{= } =.84.., 5. 4,. 4,,,,3, 4 5. : P{= k} = Ck 5 C 4 - k 5 C 4 ( k =,,, 3,4). : 3 4 P{= x i } , N A, B, M( M N) A.,, A, : P{= k} = Ck M C - k N - M C N ( k =,,,, l). N >, M >, N - M, l = mi( M, ).. ) :. P{= k} = C k p k q - k, k =,,,, q = - p, < p <. (..) p, b(, p)., b(, p). : k = P{= k} = k = C k p k q - k = ( p + q) =.. C m p m q - m ( p + q) m +,,, A p,, A, p,

151 48 b(, p). 3, 3 4., 4, 4, 3.,,,,,, = 4, A = { }, p = P( A) = = 4 p = 3 4, b(4, 3 4 )., 3 : P{= 3} = C = 7 64 = ,, h h?.,,,, h h,,,, = 5, A = {h }, p = P( A) =.. 5, h. = 5 p =., b(5,.). : P{> } = - P{= } - P{= } = - C 5 (.) (.9) 5 - C 5 (. ) (.9) 4.85., =,, P{= x i } q p :, b(, p).,, P{= k} = p k q - k, k =,, q = - p, < p <. C k p k q - k = C - k q - k p k, p =.5,, p.5,,,,. ) ( Poisso) : P{= k} = k k! e-, k =,,, >. (..3)

152 49 (Poisso)., P(). P(), P(). : P{= k = k} = e - k = k k! = e- e =. 3 =.5,= 5 =.,,. 3,, : ; ; ;.

153 5 5. ( Rutherford) (Geige)., 7.5. N = 68, 68 94, k N k, N k N p k = k, 3.87 k. k N k N k N p k,,,..,,. 6, =.95%,?, a, a, : P{ a}.95. P( ), a k = k k! e-.95,, : 4 k = 5 k = k k! e- k k! e-.966 <.95,.953 >.95.

154 5, 5 ( ) 95%., P{= k} = C k p k ( - p) - k,,,, p. () A, P( A) = p ( < p < ), P{= k} = C k p k ( - p) - k k k! e- ( k =,,,, ) (..4) = p. = p, p =, P{= k} = C k p k ( - p) - k = =! k!( - ( - )( - k + ) k! = k k! - - k)! pk ( - p) - k k - k k - - k. lim k - - k =, lim - = e -,,>, p =, lim Ck p k ( - p) - k = lim k k! k k = k k! e-., p = p, P{= k} = C k p k ( - p) - k k k! e-. 7.5, 3., = 3, p =.5, q = - p =.995, k =, P{= } = C ,= p = 3.5 = 6, k =, P{= } 6! e , p,. 8 3,,..

155 5.( ),.?( ),.( 3) 3 8, ()?, b(,.). () x, P{> x}.. b(3,. ). = 3 p =., = b(3,.)., P{> x} = P{> x}., 3 C k 3 (.) k (.99) 3 - k k = x+ 3 k = x+ 3 k e - 3 k!. p = 3 3 k = x + 3 k e - 3 k!.. x = 8. () 8. () P{ }, b(,.), = p =., (..4),. P{ } = - P{ } - k e -. k = k!.76. (3) P{ 4}. b(8,.), = p =.8, 3.8 k e -.8 P{ 4} = - P{ 3} - k = k!.9. () ( 3), (3) ( (), (3) 7 ),,.,.,. 3), A p( < p < ), A,, A,. A. = k, k - A, k A,, P{= k} = ( - p) k - p ( k =,, ). (..5) (..5). p, g( p). p, g( p)., P{= k} = k = p( - p) k - = k = p - ( - p) =. 9 3.,,

156 53,,., : () ; (). A = { },,, A, p = P( A) = 5 =.4. p =.4, g(.4), P{= k} = ( - p) k - p =.6 k -.4 ( k =,, ). P{ } = k = P{= = ( - p) - p - ( - p) k} = ( - p) k - p = ( - p) - p k = k = = ( - p) - =.6 -. ( - p) k P{ }.,, -, P( A) = - p =.6,,, -, ( - p) - =.6 -.,, P{ } = ( - p) -..3,,., [ a, b] ( -, + ),.,,.,., ;, (mm mm ).,,. { a} { a < b}, P{ b} = P{ a} + P{ a < b}, P{ a < b} = P{ b} - P{ a}.,, x, P{ x}, x,

157 54 F( x) P{ x}, (.3.) F( x) ( -, + ) [,],.,... : 3 P{= x i } , F( x) = P{ x} = P{= x i } = x x i 4. x <.8 x <.5 x <.9 x < 3 x 3 4 F( x), x i F( x), p i.,,, F( x) : () x < x, F( x ) F( x ) ; () F( x), F( + ) = lim F( x) =, F( - ) = lim x+ (3) F( x + ) = F( x). F( x) = ; x -,.,

158 55,. F( x),, F( x). : P{> x} = - P{ x} = - F( x), P{ a < b} = P{ b} - P{ a} = F( b) - F( a), P{< x} = lim + P{ x - } = lim + F( x - ) = F( x - ), P{= x} = P{ x} - P{< x} = F( x) - F( x - )., { a < b}, { a b}, { a < b}, F( x). F( x).,,,,,.,. x,,,,,,.,,..4, F( x), ( x), x, F( x) = x ( t) d t (.4.) -, ( x) (F( x) ),. :, ( x) () ( x) ; () + ( x)d x =. (.4.) - : P{ a < b} = F( b) - F( a) = b ( x) d x. (.4.3) a :( a, b] (.4.), y = ( x) x. 5.

159 56 5 a = b, P{= a} =,,.,,.,,,., P{ a b} = P{ a < < b} = P{ a < b} = P{ a < b} = b ( x) d x, a., ( x),, ( x), ( x),. ( : ), : ( x) = x e - x > x () 5 5?? () F( x). () = + - e - ( x) d x = + =. x d x =, 5 5 : P{5 < < 5} = 5 x 5 e- d x = e - - e

160 57 P{< } = x e- d x = - e -.63., 63.%. () F( x) = x - = - e- ( x) d x = x x e- d x x > x x x > x F( x) ( x) : F( x) = x ( t)d t. -,, ( x) x >, : d F( x) = ( x). (.4.4) d x ( x) = lim x F( x + x) - F( x) x = lim x P{ x < x + x} x, x, P{ x < x + x} ( x)x., x ( x),, ( x) x,.,. R, O. O x P{ x} x, > R ( 6). () F( x) ( x)., (), O i R i + R, - i. - i ( i =,,,, 9). (), [, R], x <, F( x) = P{ x} = ; x R, F( x) = P{ x} = ; x < R, F( x) = P{ x} = kx.(, P{ x} x, k.) F( x) : lim F( x) = F( R), lim kx = kr =, xr - xr -

161 k = R, F( x) = x < x R x < R x R 7. ( x) = x R x < R, ( x). () - i i R < < i + R, P i R < < i + R = F i + R - F i R = i + - i = i + ( i =,,,, 9)..; 9.3;.. ) [ a, b],, ( x) = c x [ a, b]

162 59., b cd x = c( b - a ) =, a c = b - a, [ a, b] : ( x) = b - a x [ a, b] (.4.5) F( x) = x a x - a b - a a < x b x > b (.4.6), [ a, b] [ x, x ] : P{ x x } = x ( x) d x = x - x x b - a, [ x, x ] [ x, x ], [ x, x ]..4.. [ a, b], U( a, b) , 7:,7: 5, 7:3.7: 7:3. () 5, (). 7, (,3), 5, 7: 7: 5 7: 5 7:3, () : P{ < < 5} + P{5 < < 3} = 5 3 d x d x = 3., 7: 7:5 7:5 7:,, ( ) : 3. ) P{ < < 5} + P{5 < < } = 3.

163 6 ( x) = e - x x > x (.4.7),, E(), F( x) = - e - x x > x (.4.8) =..,,,, P{= k} = k k! e- ( k =,,, )., t, (, t) P( t).(, t),,,., ( ).,,,,. ( ),,. 4, (, t) P( t).. t =, F( t) = P{ t}., t, P{ t} =. t >, {> t} {(, t) } {= }, P{> t} = P{= } = ( t) t! e- = e - t. P{ t} = - P{> t} = - e - t, F( t) = - e - t t > t., ( ),.

164 6., s >, t >, P{> s + t > s} = P{> s + t} P{> s} ( s+ t) = e- e - s = e - t = P{> t}. (.4.9),, s, t s, s, (, s ),. 5 =, 5km,??, =, P{> 5} = - F( 5 ) = e - 5 = e , F( x), P{> t + 5 > t} = - F( t + 5) - F( t), t.,, t. 3) : ( x) = ( x - ) e- ( - < x < + ), (.4.) >,, : F( x) = x e - ( t- ) - d t. (.4.), N (, ).,,,,.,,,,,, 5.,,,,,,.. ( x) 8 (= ).

165 6 8, ( x) x =, x = ( x),.,, (4 ). =,= N (,), ( x) = ( x) : e- x ( - < x < + ). (.4.) ( x) = x - e - t d t. (.4.3) N(, ).x, ( x), x <, x, ( x),. ( - x) = - ( x) (.4.4) N(, ), F( x)?, x - = t, F( x) = = x - - x - e - ( x - ) e - t d t d x = x -. (.4.5)

166 63 N(, ), = N(, ) N(, ). - N(,),. 6, A,,,, ( ) N(7, 5 ).B,,,, N( 3, ).() 3, ( ) 34,?,., (.4.5) 3 - P{> 3} = - P{ 3} = - F(3 ) = -. () A: P{> 3} = - (.6).743, B: P{> 3} = - ( ) =.5. A. () A: P{> 34} = - (.4).88, B: P{> 34} = - ( ).8. B. 7, (, ), ( )., + A, + B, - C, - - D, - E ( ). + - P{> + } = - = - ().587, + - P{< < + } = - - = ( ) - ().343, - - P{- < < } = - - = () - ( - ).343, P{- < < - } = ( - ) - ( - ).359, P{< - } = ( - ).8., A 6%,B 34%, C 34%, D 4%, %., N(, ), P{ - < } = () - ( - ) = () -.686, P{ - < } = ( ) - ( - ) = () , P{ - < 3} = ( 3) - ( - 3) = (3) , 3, 95%,

167 %, 3.3%.,, N (, ),,3 - < 3, 3., 3, ,,.,,.,..,,. f( x) x,, x, y = f( x),, = f( )..,, = f( ). - - P{= x i } =. -, -,,,, =,,4.

168 65 = : P{= } = P{= } =., P{= } = P{= } + P{= - } =.3, P{= 4} = P{= } + P{= - } =.6, 4 P{= y j }..3.6 P{= x i } = si. si = - = 4 k - = 4 k - 3 = k ( k =,, ) = si : -,,; : P{= - } = P{= 4 k - } = = k = 3 7 P{= } = P{= k} = = k = 4 6 P{= } = P{= 4 k - 3} = k = = 5 9, = 3, = 5, = 8 5, P{= y i } , = f()?. 3 [ a, b],

169 66 ( x) = b - a x [ a, b], < a < b, = ( y). ( y),, F ( y) = P{ y}. ( y) = y <, = d F( y) = ; d y y,, y, F ( y) = P{ y} = P() =, F ( y) = P{ y} = P{ y} = = y - a - a - P{y } = y - d x = y < a y d x + a b - a d x a y b b y d x + a b - a d x + d x y > b b ( x) d x = y < a y - a a y b b - a y > b, ( y) = d d y F ( y) = y - a b - a = y < a = y b - a a y b = y > b y <, ( y) = y b - a a y b 3,., ( x) = ( x) x ( a, b) f( x) ( a, b), x ( a, b), f( x) (,),, f( x) f - ( y), y (, )., = f()

170 67 ( y) = f - ( y) d d y f - ( y) y (, ) (.5.),, F ( y). y, F ( y) = P{ y} = P() =, ( y) = y, F ( y) = P{ y} = P( ) =, ( y) = < y <, F ( y) = P{ y} = P{ f() y}., f( x) ( a, b), d d y F ( y) =. d d y F ( y) =. F ( y) = P{ y} = P{ f() y} = P{ f - ( y) } = F ( f - ( y) ), ( y) = d d y F ( y) = d d y F ( f - ( y) ) = ( f - ( y) ) d d y f - f( x) ( a, b), f - ( y) (,), d d y f - ( y), ( y) = ( f - ( y) ) f( x) ( a, b), d d y f - ( y). ( y), F ( y) = P{ y} = P{ f() y} = P{ f - ( y) } = - F ( f - ( y) ), ( y) = d d y F ( y) = d d y - F ( f - ( y) ) = - f - ( y) d d y f - ( y), f( x) ( a, b), f - ( y) (,), d d y f - ( y), ( y) = f - ( y) y,< d d y f - ( y). y <, y, = f() ( y) = 3. f - ( y) d d y f - ( y) y (, ) f( x) = x x [ a, b], d d y f - f - ( y) = y, y [ a, b]. ( y) = d d y (y ) = y, (.5.) ( y) = f - ( y) d d y f - ( y) y [ a, b ]

171 68 = b - a y = y b - a a y b 4 N (, ), = ( x) = a+ b ( a ) ( y). ( x - ) e-, x ( -, + )., f( x) = ax + b, x ( -, + ), f - ( y) = y - b, y ( -, + ). a d d y f - ( y) = d d y y - b a =, (.5.) a ( y) = f - ( y) = = e- d d y f - ( y) y - b - a a ( y - a- b ) a e- a, y ( -, + )., N (, ) a, = a+ b N( a+ b, ( a ) ), (.5.)., (.5.) a =, b = - - N(, ).., ( x) = ( x) x ( a, b) f( x) ( a, b), ( a, b) : ( a, b ), ( a, b ),, ( a, b ), ( a i, b i ), f( x), f( x) f - i ( y), y ( i, i ), ( i =,,, )., = f(), ( y) = i = i ( y), i ( y) = f - i ( y) d d y f - i ( y) y ( i, i ) (.5.3)

172 69 ( i =,,, ). F i ( y) = P{ y, ( a i, b i ) },, i ( y) = d d y F i ( y) = f - i ( y) ( i =,,, ). d d y f - i ( y) y ( i, i ) F ( y), F ( y) =,= f( ) P{ y} = P{ y, ( a i, b i ) } = i = i = F i ( y), ( y) = d d y F ( y) = d F d y i ( y) = d y F i ( y) = i = i = d i = i ( y)., f( x),,, = f( ). 5, ( x), x ( -, + ), = ( y). f( x) = x ( -, + ). x ( -,), f( x) = x, f - ( y) = - y, y (, + ). d d y f - ( y) = d d y ( - y) = -, (.5.3) y ( y) = = f - ( y) ( - y) d d y f - ( y) y (, + ) - y = ( - y) y y (, + ) x (, + ), f( x) = x, f - ( y) = y, y (, + ). d d y f - ( y) = d d y y =, (.5.3) y ( y) = f - ( y) d d y f - ( y) y (, + )

173 7 = ( y) y = ( y) y y (, + ), = ( y) = ( y) + ( y) = ( - y) + ( y) y y (, + ) (.5.4), N (,), ( x) = e- x (.5.4), = ( y) = y - e - y y > y, ().. 4 8,,,,.,.. 5,,,3,4,5, 3. (),, 4 ; (),, > 3..3, 3,,, =.,,,, 3,.,,,, =..4 5,,, 3 3.: ( ) 5 3 ; ( ) 3 3 ; (3) 3 ; (4) , 3,, 4?.6, b(, p), b(3, p), P{ } = 5 9, P{ }..7 9,., 3,

174 7,. : () 3, 3, 3, ; () 3 9, 9 3,;,?.8 = 3. () 3. (),( )..9,, P(3)., 99%..,3,,,,,.,,,?.,.4,, 9%?. ( ) A p,, A r,.r =,?.3 ( x) = C( x - x 3 ) < x < 5?,C..4 ( x) = Ae - x ( - < x < + ). () A ; () (,) ; (3)..5, ( ), ( x) = 5( - x) 4 < x <.,?.6 + x, () ( -, + ) ; () ( -,)..7 : F( x) = x < Ax x < x : () A; () ( x) ; (3) P{ -.3 < <.7}..8 ( ) F( x) = A + B arcta x ( - < x < + ), () A B ; () ( -, ) ; (3).

175 7.9 [,5], x x, x < < x < 5, [ x, x ].. ( : ) =. : ()? () 9,?. t N( t) t. () T ; () 8, 8.. N(, ), () P{<.} ; () P{ -.6 < 5.8}; (3 ) P{ 3.5} ; (4) P{ 4.56}..3 U(,5 ), 4 x + 4 x + + =..4, () N(7, ),96.3 %, V, 4 V 4V,.,... N(,5 ), () ; (), 4 V..6 N(, ),, P{= x i } 3 a 6 3 a a 3 ) a; ) =..8 E( ),= l...9, : ( x) = 4 x a 3 e - x a x > x ( a > ) = m ( m )..3 N(, ), () = + ; () =..3 = si ( y). ( x) = x < x <

176 3.,,.,. (),. 4, ; ;. (), R (3), R (, ) ; (4), R (,, ),,,,, (,,, ). i,.,,,,.,.,,,. 3..,,. (,), ( x i, y j ) ( i, j =,, ), (,), : () pij ( i, j =,, ) ; () i p ij =. j p ij = P{= x i,= y j } ( i, j =,, ) (3..)

177 74, 3 (, ). 3 y y x p p x p p,, 5% A, % B, 3%., A, B, (, ).,,. = i, = j, i A j B, - i - j. P{= i, = j} = C i C j - i.5 i. j.3 - i - j ( i, j =,,). i, j =,,, m > C m =, P{=, = } = C C.5..3 =.9, P{=, = } = C C.5..3 =., P{=, = } = C C.5..3 =.4, P{=, = } = C C.5..3 =.3, P{=, = } = C C.5..3 =., P{=, = } = C C =, P{=, = } = C C.5..3 =.5, P{=, = } = C C =, P{=, = } = C C = ,3.,

178 3 75 i = i i ( i =,). (, ).. () P{ =, = } = =.3, P{ =, = } = =.3, P{ =, = } = =.3, () P{ =, = } = 5 4 =.. P{ =, = } = =.36, P{ =, = } = =.4, P{ =, = } = =.4, P{ =, = } = 5 5 =.6. () () (, ) (, ), F( x, y) = P{ x, y} ( 3.. ) (,). (, ) xoy, F( x, y) ( x, y) (,) 3., F( x, y) : () x y ; 3

179 76 () x y, F( x, y) = F( x +, y), F( x, y) = F( x, y + ) ; (3) x y, F( x, y), F( -, y) = lim F( x, y) =, (3..3) x - F( x, - ) = lim F( x, y) =, (3..4) y - F( +, + ) = lim x+ y+ F( x, y) =. (3..5), F( x, y) = P{ x, y} = (,) xi x y y j P{= x i, = y j } = x x y y i j p ij (3..6) (,) F( x, y), ( x, y) ( x, y) R, F( x, y) = x y - - ( u, v) d vd u (3..7) (,), ( x, y) (, ). : () ( x, y), ( x, y) R ; () + + ( x, y) d xd y =. (3..8) - - : P{ x < x, y < y } = x y, D, D R, x y ( x, y)d yd x (3..9) P{(, ) D} = ( x, y) d xd y. (3..) D, z = ( x, y), (, ) D, z = ( x, y), D. ( x, y), ( x, y) = 4 (, ) F( x, y). (3..) x y ( x, y) = ce - ( x+ y ) < x < +, < y < +

180 3 77 : ( ) c; ( ) F( x, y) ; ( 3) (, ) 3 D. () ( x, y)d xd y =, + ( x, y) d xd y = + ce - ( x + y) d xd y = c + e - x d x + e - y d y 3 = 4 c, c = 4. () (3..7) F( x, y) = x y - - x = y ( u, v) d vd u 4e - ( u + v ) d vd u < x < +, < y < + = ( - e - x ) ( - e - y ) < x < +, < y < + (3) (3..) P{(, ) D} = ( x, y) d xd y D = d x - x = - 3e -. 4e - ( x + y ) d y (, ),, (, ).. c ( x, y) ( x, y) = ( x, y) d xd y =,

181 78 ( x, y) = cd xd y =, c = S =. ( x, y) P{ (, ) A} = A ( x, y) d xd y = d xd y A = S A = = , (,) ( x, y) S,, ( x, y) = S ( x, y) (,). (,), S A P{ (,) A} = S A A, A. S. A, 3. (, ),, (,, ), (, ) ( ). (,). (,) 3 p ( x i ) = P{= x i } =,, P{= x i, } = j P{= x i, = y j } = j p i j ( i =,, ) (3..) p ( y j ) = P{= y j } = P{,= y j }

182 3 79 = i P{= x i, = y j } = i p ij ( j =,, ) (3..) 3 5, p ij., p ( x i ) = P{= x i } ( i =,, ), p ( y j ) = P{= y j } ( j =,, )., (, ),,,. 3 5 y y P{= x i } x p p p j j x p p p j j P{= y j } i p i i p i 3., A B,. 3 (, ), A B,. P{= x i } P{= y j } , (, ). ( 3..) (3..), P{ = x i } P{ = y j }.6.4 P{ = x i } P{ = y j }.6.4,,. (,) (), ( ).,,,. (, ),.,

183 8 (, ).,,. : : F( x) = P{ x} = P{ x, < + } = F( x, + ) = x ( u, y) d yd u, (3..3) F ( x) = P{ y} = P{< +, y} = F( +, y) = y ( x, v) d xd v. (3..4) ( x) = ( y) = d + d x F ( x) = - d + d y F ( y) = - ( x, y) d y, (3..5) ( x, y)d x. (3..6), : F( x, y) F ( x) = F( x, + ) F ( x) ( x, y) = F( x, y) x y F( x, y) = x y ( x) = - - ( u, v)d vd u d d x F ( x) F ( x) = x - ( u) d u ( x, y) ( x) = + - ( x, y)d y ( x) (,). ( 3..5) ( x) = + ( x, y)d y = + - 4e - ( x + y) d y x > x = e - x x > x ( y) = + - ( x, y) d x = e - y y > y

184 ,,. P{= x i }, P{= y j } P{= x i,= y j }, = y j = x i P{= x i = y j } = p ( x i y j ), : P{= x i,= y j } = P{= x i = y j } P{= y j },, P{= y j }, p ( x i y j ) = P{= x i = y j } = P{= x i,= y j } P{= y j }. (3.3.) j i, p ( x i y j ) ( i =,, ). : () p ( xi yj ) ( i =,, ) ; () i p ( x i y j ) =., p ( x i y j ) ( i =,, ).,, = y j,.,, p ( x i ), = y j., P{= x i }, p ( y j x i ) = P{= y j = x i } = P{= y j,= x i } P{= x i } ( j =,, ) (3.3.) ( 3.3.) 3. =,, 3 8 P{= y j = } P{= y j = }.6.4 P{= y j = }. 39, 6 8, 5% 7% 9%, :

185 % 7 % 9 % %,?, (,) 3 : P{= x i } P{= y j } , P{.7} = - P{=.5} = , 3 P{= x i = y j } = P{= x i,= y j } ( x P{= y j } i = 6, 8,,; j =,,3) P{= y j = 6} P{= y j = 8} P{= y j = } P{= y j = } P{.7 = x i } = - P{=.5 = x i }, = 8, -.83 =.97,.,, 8,. 8,,.,,

186 3 83, P{ x = y} P{= x i = y j }, P{ x = y} = y, F ( x y), F ( x y) = P{ x = y}. (3.3.3) P{= y} =, F ( x y). ( ( x, y) ( y), ( y) > ). F ( x y) = P{ x = y} = lim P{ x y < y + y} y = lim y = lim y = F( x, y) y x - = P{ x, y < y + y} P{ y < y + y} [ F( x, y + y) - F( x, y) ]y [ F ( y + y) - F ( y) ]y d F ( y) d y ( u, y) d u ( y) = y :. ( x y) = ( x, y) ( y) ; (3.3.4), = x : ( y x) = ( x, y) ( x). (3.3.5) (3.3.) (3.3.). 3 ( ),,,, r, >, >, r <. ( x, y) = - - r e ( - r ) x - - r x - y - + y -,, N(,,,, r).. : ( x) = + - ( x, y) d y, x - = u, y - = v, ( x) = + - e - - r ( - r ) u - r u+ v d v

187 84 = = e - e - u = + e ( - r - ) u - e ( x - ) - ( v - ru ) ( - r ) d v, (3.3.6) ( y) = + - ( x, y) d x = - e ( y - ). (3.3.7), N (,,,, r), N(, ) N(, ),, r r, N(,,,, r ) N(,,,, r ).( ) ( 3.3.7).,,,. : ( x y) = = ( x, y) ( y) - r e - ( - r ) x - - r y - = - r e - ( - r ) x - + r ( y - ), ( y x) = - r e - ( - r ) y - + r ( x - )., N(,,,, r), N + r ( y - ), ( - r ) N + r ( x - ), ( - r ). 3 (, ). 3, x = x ( x, y), ( x, y).( x, y) = x ( y x ) = ( x, y) ( x ),,.,,, x = x.y = y,,.

188 ,, = x, : ( y x) = ( x y)e - xy ( < y < + ), x., x,,,,.,,.5.35,. 9. [.5,.35], ( x) =.5 x.35 = x.(, ) : ( x, y) = ( x) ( y x) = x ye - xy.5 x.35, < y < + : ( y) = + x, y) d x - ( =.35.5 x ye - xy d x ( < y < + ). P{> 9} = + ( y) d y 9 = + d 9 y.35.5 x ye - xy d x =.35 x d.5 x + ye - x y d y = = - 9 ( 9 x + )e- 9 x (9 x + )e - 9 x d x = 9 (4.5e e )

189 86 3.4,, (, ),.. (,). 3. ( x i, y j ) P{ =, = } =.36 = P{ = } P{ = }, P{ =, = } =.4 = P{ = } P{ = }, P{ =, = } =.4 = P{ = } P{ = }, P{ =, = } =.6 = P{ = } P{ = }. P{ = x i, = y j } = P{ = x i } P{ = y j },,,..,, P{ x, y} = P{ x} P{ y}, F( x, y) = F ( x) F ( y), (3.4.).,. : P{= x i, = y j } = P{= x i } P{= y j } ( i, j =,, ), (3.4.) p ij = p ( x i ) p ( y j ) ( i, j =,, ),. 3. (), ().. ( x, y) = ( x) ( y), (3.4.3),,,..,,.

190 3 87,,, [,6] ( 3 P{+ < } + P{+ < }. 4).: P{+ < } = x + < y ( x, y) d xd y = ( x) ( y)d xd y x+ < y = 6 d y y - 6 = 6 6 ( y - ) d y = d x 3 4,,,,.,,, ;,.,,,. 3.5., (,), = f(, ),,(,,, ) = f(,,, ). 3.5.,., b(, p),. = +. i P{ i = x j } q p ( i =,) = +,,. P{= } = P{= } = P{ =, = } = P{ = } P{ = } = q, P{ =, = } + P{ =, = } = pq,

191 88 P{= } = P{ =, = } = P{ = } P{ = } = p. P{= y j } q pq p b(, p)., = + + +, i, i b(, p) ( i =,,, ), b(, p).,.,, P{= x i } = p ( x i ) ( i =,, ) P{= y j } = p ( y j ) ( j =,, ), = + P{= zk } = P{= z k } = i P{= x i } P{= z k - x i } ( k =,, ). (3.5.) P{+ = zk } = i P{= xi,= zk - xi },,, P{= x i,= y j } = P{= x i } P{= y j }, P{= zk } = i P{= xi,= zk - xi } = i P{= xi } P{= zk - xi }.,. (, ) ( x, y), = + ( z) = + - ( x, z - x) d x ( z) = + - ( z - y, y)d y (3.5.),, ( x) ( y), = + ( z) = + - ( x) ( z - x) d x ( z) = + - = +. = + F ( z) : F( z) = P{ z} = P{+ z} = ( x, y) d xd y x+ y z = + - z - x - x + y = u, y = u - x, ( x, y)d y d x F ( z) = + - z ( x, u - x) d ud x - = z ( z - y) ( y) d y (3.5.3) ( x, u - x) d x d u.

192 3 89, ( z) = d d z F ( z) = + ( x, z - x) d x. -,, ( x, y) = ( x) ( y), ( z) = + x, z - x)d x - ( = + - ( x) ( z - x)d x., = +, ( z) = + z - y, y)d y - ( = + - ( z - y) ( y)d y.,, N(, ). = +., N(, ), ( x) = x e- ( y) = e- y, (3.5.3), = + ( z) = + - = + - ( x) ( z - e- x - xz+ z d x x)d x = + x ( z - x ) - e- e- d x = = = z 4 e- z 4 e- - e + - e - z ( ). N(, ), N (, ). x - z d x - z,. () i N(, ) ( i =,,, ), i N (, ). i = () i N( i, i ) ( i =,,, ), i i = N i = i,,,. i = i., :

193 9 (3) i N( i, i ) ( i =,,, ), c, c,, c, i = c i i N i = c i i, i = i. c i,. 3, ( x) = xe - x x > x,. ( ),, ( x), ( x) x = ( ),.,.,,., 4, 4 ( x) d x = 4, 9%.. xe - x d x = - 5e i ( i =, ) i,, = +., ( x), ( ) : ( y) = + - = ( x) y 3 e - y 3! ( y - x)d x = y y > y 4,, : xe - x ( y - x)e - ( y - x) d x y > y ( x) = x ( y) = e - y y > y = + ( z). F ( z). (, )

194 3 9 ( x, y) = ( x) ( y) = e - y x, y >, ( x, y) 3 6, ( x, y) = ( x) ( y)., F ( z) =, x + y z 3 P{+ z} = ( x) ( y) d xd y x+ y z z, ( 3 6). 6, () z, F ( z) = ; 3 6 () < z, F ( z) = z - x e - y d yd x = (3) z >, F ( z) = z - x e - y d yd x = F ( z) = ( z) =. z - d d z F ( z) = ( e - ) e - z ( z + e- z z ( - e x - z ) d x = ( z + e- z - ) ; ( - e x - z ) d x = - z > - ) < z z (e - )e - z z > ( - e- z ) < z z (e - )e - z.

195 9, (.5.) ( x) = x, ( 3.5.3) = + ( z) = + - = = ( x) ( z - x) d x = ( z z - e - y d y z > d y z - + z z - (e - )e - z z d y e - y d y) z > ( - e- z ) < z z ( z - x)d x = z z - < z z ( y) d y,.,, a+ b,. 5, N(,),, = +. F ( z) = P{ z} = z, F ( z) = ; z >, F ( z) = x + y z P{ + z}, ( x) ( y) d xd y = x + y z e- x + y d xd y = z d r - e rd r = z - r e rd r, ( z) = d d z F ( z) = e- z z > z, ()

196 3 93 ( x, y) (, ), = ( z) = + -, ( z) = + - ( x) ( y). 6 y ( yz, y) d y (3.5.4) y ( yz) ( y) d y (3.5.5) ( x, y) = e - ( x+ y ) < x < +, < y < + =. =. z >, F ( z) = P z z, F ( z) =. - ( x + y) e x y z = = + ( - e - yz )e - y d y = - d xd y = + yz e - ( x + y) d xd y z +, ( z) = d d z F ( z) = ( z + ) z > z (3.5.4), z >, ( z) = + y ( yz, y) d y - = + ye z, ( z) =, - ( yz+ y ) d y = ( + z) ( z) = ( z + ) z > z 3.5.3,,,, Fi ( x) ( i =,,, ). = max(,,, ) = mi(,,, )

197 94 F ( x) = F max ( x) F ( x) = F mi ( x). {max(,,, ) x} { x, x,, x}, F max ( x) = P{max(,,, ) x} = = P{ x} P{ x}p{ P{ x, x,, x} x} = F ( x) F ( x)f ( x) = i = F i ( x) (3.5.6) {mi(,,, ) x} {mi(,,, ) > x} { > x, > x,, > x}, F mi ( x) = P{mi(,,, ) x} = - P{mi (,,, ) > x} = - P{ > x, > x,, > x} = - P{ > x} P{ > x}p{ > x} = - ( - Fi ( x) ) (3.5.7) i =,,,,. F( x), (3.5.6 ) (3.5.7) : F max ( x) = F mi ( x) = P{max(,,, ) x} = [ F( x) ], (3.5.8) P{mi(,,, ) x} = - [ - F( x) ]. (3.5.9). 7 L ij ( i =,; j =,,3), 3 7. ij E(),. L L L 3 L L L (.4.8), ij ( i =,; j =,, 3) F ij ( x) = - e - x x > x i ( i =, ). Li, Li, Li3, i, (3.5.9) i i = mi( i, i, i3 ), F i ( y) = - e- 3y y > y.,

198 3 95, = max(, ), (3.5.8) F ( z) = - e - 3 z z > z ( z) = d d z F ( z) = 3 z 6e- - e - 3z z > z 3. 3,5,.,. () ; () ; (3) a, b,,..,(, ). () ( ) ; () ( ). 3.3,7, 3,,.(, ). 3.4,,,.(,) ,,,3,4,5, 3,,,(,). 3.6, p,,.(, ). 3.7 (,) F( x, y) = A B + arcta x C + arcta y 3.

199 96 : ( ) A, B, C; () (,) ; (3). 3.8, (,), (,), (,) (,),.: () ; () (,) (,) : ( x, y) = Ae - ( x+3 y) x >, y > : ( ) A; ( ) (, ) ; (3) ; (4 ) (, ) D = ( x, y) x >, y >, x + 3 y < (, ) ( x, y) = ab e - x a + y b (,) x a + y b = k , = =,? 3. b(,.6 ), = = : 3 3 P ( y j ) 4 4 P ( y j ) 6 3 (, ),. 3.3 (,) ( x, y) = xe - x( y+ ) x >, y > = y, = x,? 3.4 : ( x y) = ( y) = 3 x y 3 < x < y 5 y 4 < y < P{> }. 3.5, ( ),(, ) F( x, y) = - e -. x - e -. y + e -. ( x+ y ) x, y ()? ().

200 , [,],E().: () (,) ; ( ) P{ } ,5, 3,.? P{ = x i } 4 4 P{ = y j } P{ = } =. ) ; ) ; 3 ) = max(, ). 3.9, () b(, p), b(, p) = + b( +, p) ; () P( ), P( ) = + P( + ). 3., U(,), [,], ( y) = = +. y y - y < y 3.,,,. E( ) E(). + ( = ). 3. (,) ( x, y) = = (,) e - ( x+ y) x >, y > ( x, y) = e - ( x + y ). : () = + () =. 3.4., >.,,. T. 3.5 L ij ( i =,; j =,,3),. ij E().. L L L 3 L L L 3

201 ,,,.,,.,,.. 4. : 4..,, ?., N, : : 8.3 N + 9. N +.5 N = 9. N, : 8. N N +.3 N = 9. N., 9., 9.,.,., P{= x k } = p( x k ) ( k =,, ). k = x k p( x k ), k = x k p( x k ), E, x k k = p( x k ),. E= x k p( x k ). (4..) k =

202 4 99.,, x k, x k,. P{= x k } q p E= x k p( x k ) = p + q = p. k = E ,,,,,3..4 5,,,,3 3 3, 9 6, , 3 P{= x k } (4..) : E= = x k. = ( - ) k k k ( k =,, ), p( x k ) = k, p( x k ), k = p( x k ) = k = =,, k k = x k p( x k ) = k = ( - ) k k k = k k = ( - ) k k = - l, x k k = p( x k ) = k = k. =,. 4.., ( x), x < x < < x +, [ x i, x i + )

203 x i = x i + - x i, P{ xi < xi + } = x i + ( x)d x, x i P{ x i < x i + } ( x i )x i, x x x P{= x i } ( x )x ( x )x ( x )x, x k ( x k )x k. (4..) k =., (4..) + x( x)d x -.: ( x), + x( x)d x, - + x( x) d x, E, - E= + x( x) d x. (4..3) - + x ( x) d x,. - x. 5 ( x) = + x - x - x < x E. (4..3) E= + x - ( x) d x = - x( + x) d x + 6. x( - x) d x =.

204 4 ( x) = x - e - x x > x E= + x) d x - x( = + x x - e - x d x = x e - x d x = + = =. ( k) (Gamma), : ( k) = + x k - e - x d x. : () k, ( k) = ( k - )!; () ( k + ) = k( k) ; (3) ( ) =. 7 ( x) = ( + x ) ( - < x < + ),. + - x ( x) d x = - - x ( + x ) d x + + x ( + x ) d x, + x ( + x ) d x = l( + x ) + =,. 4..3,.,., = f( ).. 8 [,],

205 ( x) = x = e, E. ( x), = e,.5 (.5.), = e ( y) = ( f - ( y) ) d d y f - ( y) = y y e E= + y - ( y) d y = e y y d y = e -.,,,.., = f( ), f. (), P{= x k } = p( x k ) ( k =,, ). f( x k ) p( x k ), k = E= Ef() = f( x k ) p( x k ). (4..4) k = (), ( x). + f( x) ( x)d x, - E= Ef() = + f( x)( x)d x. (4..5) -,. 8 E( 4..5), 9 E= Ee = e x d x = e P{= x k } = E.

206 P{= y k } (4..) E= =.3. ( 4..4) E = ( - ). + ( - ) =.3., ( : ), [,4],, 3,,,,., ( x) = x 4 y ( y 4 ), = f( ) = 3 y y 3- ( y - ) < y E= + f( x) - = ( x) d x = y ( 3 x - ( y - x) ) ( - y + 7 y ), d x + 4 y 3 y d x ( E ) y = -. y =, y = 35, ( E) y = <, y = 35, E 4..4., B, C, E ( B+ C) = BE+ C. (4..6), P{= x k } = p( x k ), E( B+ C) = ( Bx k + C) p( x k ) = k Bx k p( x k ) + k C k p( x k ) = BE+ C.

207 4, ( x), E( B+ C) = + Bx + C)( x) d x = - ( B + x) d x + - x( C + - C,, C, 3, C, f( ) g( ), ( x) d x = BE+ C. EC = C. (4..7) E(+ C) = E+ C. (4..8) E( C) = CE. (4..9) E[ f( ) g() ] = Ef( ) Eg( ). (4..), P{= x k } = p( x k ), E[ f( ) g() ] = [ f( x k ) g( x k ) ] p( x k ) k = f( x k ) p( x k ) k g( x k ) p( x k ) = Ef( ) Eg (). k, ( x), = + E[ f( ) g() ] = + - x)( x)d x - f( + - [ f( x) g( x) ]( x) d x g( x)( x) d x = Ef() Eg(). 3 a, b, a, b, a b a E b. (4..), P{= x k } = p( x k ) ( k =,, ), a b, x k a x k b ( k =,, ), a = k = ap( x k ) k = x k p( x k ) k = bp( x k ) = b., ( x), a b, x a x b, a = + a - ( x) d x + - x( x) d x + -, a b, a E b. b( x) d x = b.,,,,, A ( A > ), A,?,

208 4 5 - A P{= x k }..998 E= ( - A) = -. A, E>, < A <,. A = 5, E= -. 5 =.,. 4.,,. 4.., mm,.mm, 9.8mm.mm,. 6, ( : mm) : : : mm,,,.,, ;,. - E,, E - E,.,,. E(- E ),, D Var(), D= E(- E). (4..) D,,.,. p( x k ) = P{= x k } ( k =,, ). D= ( x k - E) p( x k ), (4..) k = D= + ( x - E ) ( x) d x, (4..3) -

209 6 ( x).,, ;,.. : D= E - ( E) (4..4) ( 4..6) (4..) D= E(- E) = E[ - E+ ( E ) ] = E - EE+ ( E ) = E - ( E)., P{= x k } q p ( p + q = ), D. E= q + p = p, E = q + p = p, D= E - ( E) = p - p = p( - p) = pq D. (4..3)(4..5) (4..4) E= - E x( + x)d x + = x ( + x) d x - + D= E - ( E ) = 6. x( - x) d x =, 4 [, ], = si, D. x ( - x) d x = 6, ( x) = x (4..5) E= + E si x)( x) d x - ( = = + - ( si x) ( x) d x = si x d x =, si x d x =,

210 4 7 (4..4) D= E - ( E) = ,,., B, C, ( 4..6) (4..) D( B+ C) = B D. (4..5) D( B+ C) = E[ B+ C - E( B+ C) ] C, = E[ B(- E ) ] = E[ B (- E ) ] = B E(- E ) = B D., C, 3, C,, C, C = E, E (- C) D. DC = (4..6) D(+ C) = D. (4..7) D( C) = C D. (4..8) D E (- C). (4..9) ( 4..8)(4..)(4..4) ( 4..7) D= D(- C) = E(- C) - [ E(- C) ] = E(- C) - ( E- C) E(- C). E= C, E(- C) D. 3 (),. P{ - E } D (4..) ( x),. P{ - E } = x - E ( x)d x x - E x - E ( x) d x

211 8 + x - E) - ( ( x) d x = D. - E - E<, : P{ - E } + P{ - E< } =, P{ - E< } - D, (4..) P - E D. (4..),,.,. 4, E. ( 4..) D= P{= E} = (4..3) P{ - E } D =. >, P{ - E } =, P{ E } =, P{= E } =. 5 N(, ), 4.3,, E=, D=, P{ - 3 }. E=, D=, P{ - 3 } D = = (3 ) 9 9..,.4, 3,, N(, ) P{ - < 3 }.9974, P{ - 3 }.6. P{ - 3}.,,. E, D, * = - E D.

212 4 9 E * = E D * = - E D = E(- E) =, D D(- E ) D = D D = b(, p) E p k = C k p k q - k ( k =,,,, ) p + q =. E= kp k k = = k = = kc k p k q - k = k = k =! ( k - )!( - k)! pk q - k = pc - p k - q - k k = = p( p + q) - = p. (4.3.) = k = k C k p k q - k = k =! ( - k)!( k - )! pk q - k k( k - ) C k p k q - k + kc k = ( - ) p ( p + q) - + p = p - p + p, k = p k q - k + p = ( - ) p C k - - p k - q - k + p k = D= E - ( E ) = p + p( - p) - p = pq. (4.3.). P() p k = k k! e - E= kp k E k = = ( k =,,, ), k = k - = k p k = k( k - ) p k + kp k k = k = k = = k = D=. g( p) k - ( k - )! e- =. (4.3.3) ( k - )! e - + = +, E - ( E ) =. (4.3.4) p k = q k - p ( k =,, ), E= kp k k = = kq k - k = p = pkq k -, k =

213 k = kx k - = k = x k x = - x = ( - x) x <, E= p ( - q) = p. (4.3.5) E = k k = q k - p = k = = pqk( k - ) q k - + k = p, k( k - ) q k - p + k = kq k - p k = k( k - ) x k - = k = x k = x - x = ( - x) 3 x <, E = pq + ( - q) 3 p = q + p p, D= E - ( E ) = q + p p - p p, q p N(, ) = q + p - p = q p. (4.3.6) E= + x)d x - x( = + x - ( x - ) e- d x, x - = z, E= = + - D= + ( x z + ) e - z ( ze - z d z d z + + e - z - d z = + =. (4.3.7) ) ( x) d x = + - ( x - ) ( x - ) e- d x, x - = z, D= = z ze- z e - z d z = e- + z zde - z = + =. (4.3.8),,. d z

214 4 U( a, b) E= + x ( x)d x - = b x a b - a d x = b - a x b a = ( a + b). (4.3.9) = + E - x ( x) d x = b a D= x b - a d x = b - a x3 3 E - ( E ) = ( b - a) b a = a + b + ab, 3. (4.3.),. E() E= + E x) d x - x( = + = - xe - x = x e - x d x = - - x ( x)d x = + = - x e - x e - x d x e - x d x = + x x e- + - xde - x e - x d x = - + = + xe - x d x = + x e - x d x = =, D= E - ( E ) = -. =. (4.3.) x de - x =. (4.3.) (, ) E E, D D. :, E= i x i p( x i, y j ) (4.4.) j E= yj i p( xi, yj ) (4.4.) j D= i ( x i - E) p( x i, y j ) (4.4.3) j

215 D= i ( y j - E) p( x i, y j ) (4.4.4) j p( x i, y j ) ( i =,,, j =,, ) (,)., E= E= D= D= ( x, y) (, ). x( x, y)d xd y (4.4.5) y( x, y) d xd y (4.4.6) ( x - E) ( x, y)d xd y (4.4.7) ( y - E) ( x, y)d xd y (4.4.8) (3..) ( 3.. ), (4.4.) (4.4.4) E= i x i p ( x i ) (4.4.9) E= j yjp( yj ) (4.4.) D= ( x i - E ) p ( x i ) (4.4.) i D= j ( y j - E) p ( y j ) (4.4.) p ( x i ) ( i =,, ), p ( y j ) ( j =,, ). (3..5) ( 3..6 ), (4.4.5) (4.4.8) ( x), ( y). E= + x ( x) d x (4.4.3) - E= + y ( y) d y (4.4.4) - D= + ( x - E) ( x) d x (4.4.5) - D= + ( y - E ) ( y) d y (4.4.6) -,.,,, = =

216 4 3 (,) E, D. (, ) (4.4.) E= i y j p( x i, y j ) j = = 5. (4.4.) P{= y j } E= = 5, E = = 5, D= E - ( E) = 5-5 = = 6 5. (, ) (4 ) ( x, y) = x >, y >, x + y < E, E, D, D. E= x, y) d xd y - x( = d x - x = x( - x) d x = x - x3 3 xd y = 3 ; 4 E + + = - - x ( x, y) d xd y = = x ( - x) d x = x x4 4 d x - x x d y = 6 ; D= E - ( E ) = 6-3 = 8 ;

217 4 E= 3 ; D= 8. ( x) = + - ( x, y) d y = - x d y = ( - x) < x < + d y = - E= + - ( x) d x = x x( - x) d x = x - x3 3 = 3 ; = + - E ( x) d x = x x ( - x)d x = x x4 4 = 6 ; D= E - ( E ) = 6-3 = 8 ; E= 3 ; D= , = f(,),. f, E f(, ) = i f( x i, y j ) p( x i, y j ), (4.4.7) j p( x i, y j ) ( i =,,, j =,, ). f, E f(, ( x, y). + + ) = - - f(, ) =, (4.4.), ( ) ; f(, ) =, (4.4.), ( ) ; f(, ) = (- E ), (4.4.3), (4.4.7) ; f(, ) = (- E ), (4.4.4), (4.4.8). f( x, y)( x, y) d xd y, (4.4.8) = si (- ). (, )

218 4 5 E= si ( - ) 9 5 ( - ) + si 6 5 ( - ) + si 6 5 ( - ) + si 4 5 =. = si (- ) : - P = z k E= =. E = ( - ) = 5 D= E - ( E ) = 5. 4,, N(, ), = +. (, ) (4.4.8) E= = - + x ( x, y) = x + y e- + y ( x, y) d xd y = d + rde - r = - re - r e - r d r = + r e- r d r = ,, E(+ ) = i E(+ ) = E+ E (4.4.9) ( x i + y j ) p( x i, y j ) = j i x i p( x i, y j ) + j i = E+ E.,, E(+ ) = ( x + y)( x, y)d xd y - = x, y) d xd y - x( y( x, y) d xd y j y j p( x i, y j )

219 6 = E+ E. E k k = = 5. p k = Ck M C - k N - M C N N >, M >, N - M, l = mi( M, )., i = k = E k. (4.4.) ( k =,,,, l). i i P{ i = } = M N, E i = M N + N - M N = M N = + +,, ( i =,,, ), E= E( + + ) = i = E i = M N = M N. (4.4.) M N. 6,,,.,,,,,.,,,.,,, : i = i ( i =,,, ) A i i, P( A i ) = ( - )!! =, P{ i = } =, P{ i = } = - = + + +, = -. E= E + E + + E = E i = + - D= E - ( E ) = E( ) - = E i + E i j - i = i < j =,

220 4 7 = i = E i + i < j E i j -, E i = + - =, E i j = P{ i j = } + P{ i j = } = P{ i =, j = } = ( - )!! = ( - ), D= E - ( E) = + C ( - ) - =..,, -,,, k k,, k k. k k,,, v k = E k, (4.5.) v = E. (4.5.) Ek k,, =, k = E(- E. k = E(- E) k, (4.5.3) k ) k = i = = D. (4.5.4) C i k ( - E ) k - i E i k = i = C i k ( - v ) k - i v i (4.5.5) v k = E k = E(- E+ E k ) k = i = C i k E(- E ) i ( E k ) k - i = i = C i k i v k - i (4.5.6)

221 8,., ( x) = e- ( x - ) k. E=, D=. k = + - ( x - ) k ( x) d x = + - ( x - ) k e - ( x - ) d x, x - = z, k = k + - z k e - z d z. k,,,. k,,, k = k + z k e - z d z. z = x k = k k + = k k k + = k k k - = k k k - x k - e - x d x = k k k + = k k k - k - 3 k - 3 k - 3 = k ( k - )!! = k ( k - )!!. + k - = x k + - e - x k + d x, k = k ( k - )!! k k 4 = 3 4. (4.5.7) ( x) = e- x x > x k,.

222 4 9 k vk = E k E=, D=. = + x k e - x d x = - + x k de - x = - x k e - x = = k! v k = k!. k e - x d x k = k + x k - e - x d x = k v k - 3 = + = v 3-4 = + = v 4 - x - 4 v 3 x - 3 v v - 4 e v - e - x d x = + = 3! x d x = + x 4 x 3-3 x + 3 x - 3 e - x d x 3! = x3 + 6 x - 4 x e - x d x 4 v 3 + = 4! - 4! + 6! = (, ),,. 3 E(- E ) (- E), Cov(, ), :,, Cov(, ) = E(- E) (- E ). (4.5.8) Cov(,) = D, Cov(, ) = D. (4.5.9) Cov(,) = E() - EE. (4.5.) Cov(, ) = E(- E) (- E) = E( - E- E+ EE) = E( ) - EE- EE+ E= E( ) - EE.,,,, E() = E+ Cov(, ). (4.5.)

223 D( ) = D+ D Cov(, ). (4.5.) D( ) = E[ ( ) - E( ) ] = E[ (- E ) E(- E ) ] = E(- E ) + E(- E) E(- E ) (- E) = D+ D Cov(,). 3,, 4,, a, b, c, d, Cov(,) = Cov(, ). (4.5.3) Cov( a+ b, c+ d) = accov(,). (4.5.4) Cov( a+ b, c+ d) = E[ ( a+ b) - E( a+ b) ] [ ( c+ d) - E( c+ d ) ] = E[ a (- E ) ] [ c(- E) ] = ace(- E ) (- E) = accov(,). 5,,, Cov( +,) = Cov(, ) + Cov(,). (4.5.5) ( 4.4.9), (4.5.) Cov ( +, ) = E[ ( + ) ] - E( + ) E = E( ) + E( ) - ( E + E ) E = E( ) - E E+ E( ) - E E = Cov(, ) + Cov(, ).,,,, Cov(, + ) = Cov(, ) + Cov(, ). (4.5.6), Cov( +, + ) = Cov(, ) + Cov(, ) + Cov(, ) + Cov(, ). (4.5.7),,. 4 Cov(, ),, DD = Cov(, ) DD (4.5.8). * = - E D, * = - E D,, (4.5.4), Cov( *, * ) = Cov - E - E, D D = Cov(,) DD = (4.5.9),, *, *. (4.5.8)

224 4 Cov(,) = DD, (4.5.) (4.5.), D( ) = D+ D DD. (4.5.) 3 N (,9), N(,6 ), = - : ( ) E, D; ()., = 3 +. () E= E 3 + = 3 E+ E= 3 + = 3 ; D= D 3 + = D + D + Cov 3 3, = 9 D+ 4 D+ 3 DD = = = 3. () Cov(, ) = Cov, 3 + = 3 Cov(, ) + Cov(, ) = 3 D+ DD= , = Cov(,) DD =. :, 9 6 = 3-3 =. (4.5.) * = - E D, * = - E, = * *, D D= D( * * ) = D * + D * Cov( *, * ) = + =,,,. = a+ b,,, = a > - a < (4.5.3) = a+ b, E= ae+ b, D= a D Cov(, ) = E(- E ) (- E) = E(- E) ( a+ b - ae- b) = ae(- E ) = ad,

225 = Cov(, ) DD = a >, = ; a <, = -. =, = * *, ad Da D = a a, D= D( * * ) = D * + D * Cov( *, * ) = = ( ), =, D=, ( 4..3 ), P{= E} =,, E= P E( * * ) = E * E * =, - E E - D D = =, P = a+ b =, a = D D, b = D E D E.,. 5,.,,.,. () Cov(, ) = () (3) E= EE (4) D(+ ) = D+ D 5 (4.5.8) ( ) ( ) ; (4.5.) Cov(, ) = E- EE( ) (3) ; (4.5.) D(+ ) = D+ D+ Cov(, ) () (4).,.,., ( x, y) = ( x) ( y), Cov(, ) = ( x - E) ( y - E)( x, y) d xd y

226 4 3 = + - ( x - E ) ( x) d x + - = E(- E ) E(- E) =, ( y - E ) ( y) d y., E= EE (4.5.4), D(+ ) = D+ D (4.5.5), :,,, E( ) =,,, D( ) = 4 - E E E (4.5.6) D + D + + D (4.5.7) P{= x i } =, :?? E= =, E= E= E = ( - ) = 3, E 3 = ( - ) =,, Cov(, ) = E - E E= - 3 =,. : =,. 5 ( x, y) = () ; (). r x + y r () (4.4.5) (4.4.6) E= E= + + ( x, y) d xd y = r - r x x, y) d xd y - - y( = - r r xd x r - x - r - x r d y =. yd y r - y - r - y r d x =.

227 4 (4.5.) Cov(, ) = E(- E ) (- E) = E, =. () ( x) = + - ( y) = + -. ( x, y) d y = r - x ( x, y) d x = r - y,. = r - r xd x r - x d y = r - x - r - x r r - r - x y r d y =. - r x r,, d x = r - y - r - y r r ( x, y) ( x) ( y), - r y r. 6,,. ( x, y) = - r exp - ( - r ) ( x - ) - r( x - ) ( y - ) + ( y - ) (4.5.) Cov(, ) = ( x - ) ( y - ) ( x, y) d xd y = + - r - z = Cov(, ) = e - ( y - ) d y + ( x - ) ( y - )exp - - ( - r ) x - - r y - x - - r y -, t = y -, d xd y = - r - r d zd t, te - z ( - r z + rt) - t d zd t d x = - r + + zte - z - - ze - z = - r t d z + - d zd t + r + te - t + t e - z - - d t + r + t e - t - + t d zd t d t + e - z - d z = - r + tde - t = - r - tet + - e - t + r + - d t

228 4 5 = + r = r. = Cov(, ) DD = r = r. r. =, r =, ( x, y) = exp - x - + y - = exp - ( x - ) exp - ( y - ). = ( x) ( y),. 3, =,,, (,),. 4. 5,,,3,4,5.3, 3. 4., ( ) E 4..,,,,,3.? 4.3, ( x) = e - x x x > x ( > ) E E. 4.4, ( x) = 4 x a 3 e - x a x > x ( a > ) E E m. 4.5, [ a, b],. 4.6,3,55,,

229 6,. 4.7, [,3],[,3], 5., ;,, 3,: ( ) 98,? ( ),. 4.8, k ( k =,) E().(),,, ; (),,,. 4.9,,,. : () ; (). 4. E, D, E( + ). - P{= x i } ( x) = e- x,e, D. 4. F( x) = x < - a + b arcsi x - x < x : ) a, b; ) E, D. 4.3 ( x) = x x P{ - E< D}. 4.4 ( ), 5, P{45 55} =.9, D. 4.5 ( x) = x e - x x > x P{ < < 6}. 4.6, E, a >,P{< a} - E a. 4.7 a b, D b - a.

230 ,, 5,,. 4.9, E(+ e - ) D(3- ). 4. (, ) E, E, E si (+ ), E max(,), D max(, ). 4. ( x, y) = 8 ( x + y) < x <, < y < E, E, E. 4.,N(, ), E max(, ) = ,,,,E (, ). 4.4 [ a, b],. 4.5 D= 4, D= 9, () D(- ) =, ; () =.4,D(+ ). 4.6,, E= E=, E= -, D= D= D=, =, =, = - E(+ + ) D(+ + )., 4.7 ( X, Y) XY, = ax + b,= cy + d, a, b, c, d, ac >, = XY. 4.8 (,) ( x, y) = y < x, < x < () E, E, Cov(, ) ; ()?

231 8 4.9 (,) () E, E, Cov( ), ; ()? 4.3,E(- ) (- ) =,. 4.3,, 3, U(,6), N(,4), 3 E(3), = , N,,E -, D -.

232 ,,,.,.,,,.,. 5., () ;, ()., (- 3, + 3 ) 99.7% (,) ,,,,,,,,., (),.,,,, = + + +, a, a,, a,, >, { }. (5..) ( ). lim P{ - a < } =, (5..) lim P{ - a } =. (5..),,,,,, D i ( ), (5..3) i =

233 3 >, lim P i = i - i = E i < =. (5..4) (5..3). ( 4..) P i = i - i = E i < - D i i = = - D i i =, (5..3),, lim P i = i - ( ) i = E i < =.,,,,,, C, >, D i C ( i =,, ). (5..5) lim P i = i -,,,, D i =, lim P i = i = i = i - i = D i C i =. 3 ( Poisso) E i < =. (5..6) ( ), E i < =., A k p k, A, >, lim P - p + p + + p < =. (5..7) k = k A k A ( k =,, ) = + + +,

234 5 3 E k = p k, D k = p k ( - p k ) 4., > lim P i - i = i = E i < =, lim P - p + p + + p < =.. 4 (Beroulli) ( ) A, p A, >, lim P - p < =. (5..8) P - p < - D = - pq = - pq - ( ). 4 lim P - p < =., p = p = = p = p.,,,, : 5 ( ),,,,, E i = ( i =,, ), >, lim P i =,. i - < =. (5..9),, -,,, E.,, x, x,, x,,,,,,, x, x,, x

235 3, x i, (5..) i =... { }. - l l ( =,, ) E = - l + l = D = l + l = l ( =,, ). D i i = l = l ( ),. { }, { }. P = k = ( k =,, ). k k { }, E = k = k k k = k = k <,,. 3,,,,,,,, lim P i = i - < =.,, x, x,, x, x i i =.,,,,.

236 ,,.,,,9 (Gauss),,,,..,,?. ( ),,,,, E i = i, D i = i ( i =,, ). B = i i =, >, B + i = + E i - i ( ), (5..) x, E B i i = N (, ).. lim P (- B i i = (- i ) < x = i ) =, D B i - ( Lideberg i = (- i ) x - e - t d t = ( x). (5..) =, B i,,,,, i = (- i ) Levy) () E i =, D i = > ( i =,, ). lim P i = i - < x = e - t x - d t = ( x). (5..3) E i = i - i - =, D i = =.

237 34,, i i = N (, ),,,. - N (, ), i i = i i = N(, ). -,,,,., ( ),, ( -.5,.5). () 5, 5? ().9? i ( i =,,, ), i ( x) = E i =, D i = -.5 < x <.5 ( i =,,, ). 5 () = i, i = P{ > 5} = - P{ - 5 5} = - P = - P [ (.34) - ( -.34) ] = [ - (.34) ] = [ -.999] =.8, 5.8. (),, P i = i < =.9, P - < i i = < - =.9.

238 5 35 =.95, =.6449, 443.5, = P{ - p }, p.., P{ = k},, - p pq,,. 3 (De Moirve Laplace) ( ) ( ) A, < p < A, () [ a, b], a x k = - p pq b, P{ = k} pq e - x k. (5..4) () x,, lim P - p pq x = e - t x - d t = ( x). (5..5) ( ). ( ) -. = i = (5..3) i, i b(, p), = E i = p, = D i = pq, lim P (5..4),, - p pq x = e - t x - d t = ( x). C k p k q - k - p k pq pq. (5..5),, P - p pq x - x,, - p pq e - t d t = ( x). N(, ), N ( p, pq).

239 36 b(, p),,,, b(, p) N( p, pq). 5..,, b(, p),,, N( p, pq).,, : P{ a b} b - p pq - a - p pq (5..6),., p,, ;, p,. 4,.,,.,, b(, p), P{ }. (). P{= k} = C k p k q - k ( k =,,,, ), = 4, p =., q =.98. P{ } = P{= } + P{= } + P{= } = C C C (). P{= k} k k! e- ( k =,,, ), = p = 4. = 8, P{ } = P{= } + P{= } + P{= } 8! e- 8 (3). + 8! e ! e P{ a b} b - p pq - a - p pq, p = 4. = 8, pq = =.8, P{ } ( -.4) - ( -.86) ( ) - ( ).4.

240 5 37, P{ }, P{ } ( -.4) ,,,.,,,,,.,,. p =., p,. 3,.6,., 99.9%., (), =, p =.6, q =.4, p =.6 =, pq =.4 = 48., b(, p), N( p, pq). b, (5..6) P{ b} b - 48 b ( - 7.3) b = b - 48,, P{ b} b - 48 =.999, b - 48 = 3.9, b = = b = 4, 4, 99.9%., 8.%,.48. 4, A.75,, A A, A. (), b(, p), p =.75, p =.75, pq =.75.5 =.875, (5..6) P{.74.76} = P{ } =

241 38 P{.74.76} , , , = ,, 574, A (), b(, p), p =.75,, E = p =.75, D = pq =.875, (4..) P{.74.76} = - D = - (. ) P{ } = P{ - E. }.875 = (. ),, P{ }.9, -.9, = 875.,, 875, A P, p, - p >,,. P - p = P - p pq pq - pq pq e - x d x = pq - - pq = pq -. (5..7) P - p > - pq. (5..8) (5..7)., p,, P - p <, (5..7), ( x).. p,? pq - (5..9)

242 5 39. = pq x pq. = +, pq = x, = x pq,.. ( x) - = x, = x pq. p, pq 4 5, 6. x. () 6, 6 6 %? () 6,,.99?,? (3), () 6 6 (), = 6, p = 6, =.. (5..7) P <. = =.964, - = (.785) () (5..7) P - p < = pq - =.99, pq =.995, x= pq = = pq x = P 6-6 <.4 =.99,

243 4 P{95.6 < < 74.4} = (3) (5..9) pq - =., p = 6, q = 5 6, = , , =.975,.6 5 =.96, 5. { k } ks - k S, S,,,,. 5. { }, P = log k = ( =,,), k, { } P(.), D, D,, D 3,: D, D ; D, D 3,.D i ( i =,,,3) =.( ) -, T 3.T 35( )? 5.5,,,.,,88%. () 9,( ). ().99,? 5.6., 5, 5? ( ) ; () ; (3) (, ). 5.7,.5%,., 9%? 5.8,,,,,, 8., :

244 5 4 () ; () ,,,, %.9? 5. :,,.4.6 9%.

245 6.,.,,.,,,,.,,,,,,,,,,.,,,,,,,.,,,,,,., :,?,?,,?,.,,,,,,.,., ( ),,,., ( X, X,, X ), ( Y, Y,, Y ),,,. ( X, X,, X ) X i.,, ( x, x,, x ). ( X, X,, X ),, : (), X, X,, X. (), X i.,.,,,.,, :

246 6 43, P{= k}, ( X, X,, X ) P{ X = x, X = x,, X = x } = P{ X i = x i } = i = i = P{= x i }., ( x), ( X, X,, X ) * F * ( x, x,, x ) = i = X i ( x i ) = i = ( x i ). F( x), ( X, X,, X ) ( x, x,, x ) = F X i ( x i ) = i = i = F( x i ). 6.,., ( x) : a = a < a < a < < a r = b, [ a, b] r., ( a k -, a k ] k, k, k. ( a k -, a k ], k a k - a k -,., ( 6 )., 6 ( a k -, a k ] = k a k - a k - ( a k - a k - ) = k = ( ak -, ak ] ( ak -, ak ] = a k ( x) d x = ( a k -, a k ] ( x), a k -, ( x).,

247 44 P{= x k } ( k =,, ) :, x k ( ) k. k = x k x k = P{= x k },, k ( k =,, ) P{= x k } ( k =,, ). 6.3 ( X, X,, X ),.,,. : X = X i ; i = S = ( X i - X) = i = i = X i - X ; S = S = i = ( X i - X) ; S * = - i = ( Xi - X) = - S ; S * = S * = - i = ( X i - X) = - S; k ( ) X k = X k i. i =. SHARP ( ) TRULY( ) CASIO().SHARP TRULY, CASIO, CASIO,.SHARP TRULY,. (CASIO,.) () df, SD.) ONC,, STAT.(CASIO, () ( x, x,, x ) x DATA x DATA x DATA. x i, x i df CD. ( CASIO, x i Shift DATA x i Iv DATA.)

248 6 45 (3) x, X ; (CASIO, Shift x Iv x, X ) S, S * (CASIO, Shift ; - Iv -, S * ) S x, S * ; df, S ; (CASIO, Shift Iv, S ) df x, S ; (4), df ONC, STAT,,., df ONC. (CASIO, Shift AC Iv AC,.) 6.4,,.,,..,,,,..,,.,,,, m, ( X, X,, X ), m,,,, m., T = T ,, ( ) E( k ) ( k =,, ).,, X k ( k =,, ).,, X k E( k ).,,.

249 46,,,?,,,,,,.,,.. : () E( k ) = f k (,,, m ), k =,,, m, m ( m ). () f (,,, m ) = E = X ) f (,,, m ) = E(= X f m (,,, m ) = m ) E(= X m,,, m,,,, m. N(, ),,>, ( X, X,, X ),,., E=, E( ) = D+ ( E ) = +. () = X, () = E = X () + = E( ) = X () = X - ( X) = X i i = - X = S, = X = S = S, >,, = S, = S. [, ],. ( x) = x >, ( X, X,, X ),. E= + = E x) d x - x( = xd x =. = X.,,, : (). = X.

250 6 47,, ( X, X,, X 5 ) = (,, 3,5, 9), = X = , X 5 9,, = 8. (),. = 8,, [,8 ] = 9,,,,.,,,,.,,, E( ) = - x ( x) d x = x d x = 3. 3 = E( ) = X, = 3 X,, = X. (3),., ( x) = x x > x,>. E= x - ( x) d x = xd x = ( lim l x - l ). x,,.,,. 6.4.,.,,,.,, ;,., P{= } = - p, P{= } = p, < p <., P{= k} = p k ( - p) - k, k =,. 5,, 3.p.., p.( X, X,, X 5 ) = (,,,, ). p. p, (, ),, p ().:

251 48 p p, (,,,, ) P{ X =, X =, X 3 =, X 4 =, X 5 = } = p ( - p) = = = =.5, p, (,,,,), p =.4.,, p ( - p) 3 p =.4.,, p.4 ( ), p.4.., P{= k}, ( x, x,, x ) P{ X = x, X = x,, X = x } = P{= x i }. i =, ( x), ( x, x,, x ) * ( x, x,, x ) = ( x i ). i =, L.,,, L., m,,, m,. : () L., P{= k}, L = P{= x i }; i =, ( x), L = ( x i ). i = (),,, m, L,,, m,., l L(l L, L ).l L,,, m, l L = l L = m,,,,.,,,, m, L (

252 6 49 l L ),,,.,,,,, m,,, m. 3, P{= k} = p k ( - p) - k, k =,, < p <, ( X, X,, X ), p. L = P{= i = x i } = p x i ( - p) - x i i = = p x i i = ( - p) - x i i =,, l L = xi l p + i = d p = p i = dl L xi - - xi l( - p), i = - p - xi =. i = p = x i, l L, p i = p = i = X i = X. 4 N(, ),,>.( X, X,, X ).,. L = ( x i ) = i = i = - e ( x i - ) = - e () x - ) i i = (, l L = - l() - l- i = ( xi - ),, l L = - - i = ( x i - ) = i = x i - = () l L = i = ( x i - ) = () () = x i = x, () i = = i = ( x i - x) = s, = s = s, >,, = s.

253 5 = X l L,, = S. = s L, = s L,, = S. 5 [, ], ( x) = x >, ( X, X,, X ),. : L = ( x i ) = i = i = x i ( i =,,, ) = mi x i max x i i i dl L d = - L, L, l L = l( ) = - l., =, L,.,, L.L =,, L.,, max i x i, L =,, = max i x i, L.,, = max X i. i, ( X, X,, X5 ) = (,, 3,5,9 ), = max Xi = 9, i = 8, = 8, = ,. E D, ( X, X,, X ), X, S,

254 6 5 ( ) EX = E; ( ) DX = D ; ( 3) E( S ) = - D. () EX = E i = () DX = D i = X i = i = X i = i = EX i = i = DX i = (3) E( S ) = E X i - X = i = i = i = E= E ; D= D ; E( X i ) - E( X ) = E( ) - E( X ) = [ D+ ( E ) ] - [ DX + ( EX) ] = [ D+ ( E ) ] - = - D. D + ( E ),.,?: ().,,,,,., ( ).., E=,. N (, ), = X, = S, = X = S? S S *, E= EX = E=, = X. E() = E( S ) = - D= -, = S.,, S * =, E( S * ) = E. (). - S = - i = - E( S ) = ( X i - X) = - - S - =,,,.,, D( ) D( ),. N (, ), ( X, X ), = 3 X + 3 X,

255 5 = X + X.. E = 3 EX + 3 EX = 3 E+ 3 E = EX + EX = E+ E= E=, E= E=,,,. D = 4 9 DX + 9 DX = 4 9 D+ 9 D= 5 9 D= 5 9, D = 4 DX + 4 DX = 4 D+ 4 D= D=, < 5 9, D < D,. (3) (). 3,, >, lim ( ). P{ - < } =,,., X, X,, X, X i N(, ), i =,,,, X i i =, ( ). ( x) = x - e - x x > 6 6. x ( x) : () x, ( x) =.

256 6 53 () x +, ( x). (3) x = -, ( x). ( m), ( ),, + ( m + ).. ( m), ( ),,, X, X,, Xm, X i Y j m N(,), i =,,, m, = X i ; Y, Y,, Y, i = N(, ), j =,,,, = Y j. j =, X, X,, X m, Y, Y,, Y + = X i m i = + Y j,, + ( m + ) t j =. m + N(, ), ( ),, t, t( ). t ( x) = + + x - +. t t t ( x) : () ( x) x =. () x, ( x). (3) x =, ( x). (4), ( x) e - x,,, t( ) N (,)

257 F 3 ( m), ( ),, m ( m, ) F, F( m, ). F ( x) = m + m m m x m - ( mx + ) m + x > F 6 4. x F ( x) : () x, ( x) =. () x +, ( x). (3) x = - m F, ( x). F F( m, ), F F(, m). F F( m, ), F, ( m), ( ),, F = m. F = m, F, F F(, m) ( X, X,, X ) N (, ), X,

258 6 55 X N (, - ), X N(, ). X X, X,, X, X, X,, X Xi, 3.5 :,, X. 6.5, EX = E=, DX = D =, X N (, - ), X = X - N (,). ( X, X,, X ) N (, ), X, S, : () X S ; ( ) S ( - ).,. 3 ( X, X,, X ) N (, ), X, S, S + ( X - ) ( ). X - N(,), ( X - ) = X - (). S ( - ), X S,, ( X - ) S, S + ( X - ) ( ). 4 ( X, X,, X ) N (, ), X, S *, X - S * t( - ). X - N(, ), S ( - ), X S, X - S,, t, X - S * = X - S ( - ) t( - ). 5 ( X, X,, X m ) N(, ), ( Y, Y,, Y ) N (, ),, X, Y,,

259 56 ( X - Y ) - ( - ) m + N (, ). X N (, m ), Y N(, ), ( ), X - Y N( -, m + ), ( X - Y) - ( - ) m + N(, ). 6 ( X, X,, Xm ) N(, ), ( Y, Y,, Y ) N (, ), =,, X, Y,, S x, S y,, ( X - Y ) - ( - ) S w m + t( m + - ),, S w = ms x + S y m + -. = =.5 ( X - Y) - ( - ) m + N(, ). ms x ( m - ), S y ( - ), ( ),, ms x + S y ( m + - ). X S x, Y S y,, ( X - Y) - ( - ) m + ms x + S y., t, ( X - Y) - ( - ) S w m + = ( X - Y) - ( - ) m + ms x + S y ( m + - ) t( m + - ). 7 ( X, X,, X m ) N(, ), ( Y, Y,, Y ) N (, ),, X, Y,, S x, S y,, [ S x + ( X - ) ] F( m, ). [ S y + ( Y - ) ]

260 ms x + m( X - ) ( m), ( ),, F, S y + ( Y - ) ( ), [ S x + ( X - ) ] = [ S y + ( Y - ) ] ms x S y + m( X - ) + ( Y - ) m F( m, ). 8 ( X, X,, X m ) N(, ), ( Y, Y,, Y ) N (, ),, S * x S * x S * y F( m -, - ). ms x ( m - ), S y ),, F,, S * y,, ( - ), ( S * x S * y = ms x ( m - ) S y ( - ) F( m -, - ) , 53, 4, 6, 35, 74, 6, 96. X,S, S,S * S *. 6. ( X, X,, X ), ( Y, Y,, Y ),: Y i = X i - a b ( i =,,, ) a, b.: () X = X i Y = i = Y i ; () S x = ( X i - X) S y = i = i = 6.3 [ a, b], i = ( Y i - Y). ( x) = ( b - a) a x b, a < b, ( X, X,, X ),a, b. 6.4, P{= k} = k k! e -, k =,,,,,>,

261 58 ( X, X,, X ),: () ; (). 6.5 ( x) =,: x - < x < () ; ().,,>, ( X, X,, X ) 6.6,P{= k} = ( - p) k - p, k =,,,, < p <, ( X, X,, X ),p. 6.7 Laplace, ( x) = e -,. 6.8 Maxwell, x,,>, ( X, X,, X ) ( x) = 4 x a 3 e - x a x > x, a >, ( X, X,, X ), a. 6.9, ( x) = e - x x > x, >, ( X, X,, X ), X, S, EX, DX E( S ). 6. P(), P{= k} = k k! e -, k =,,,,, >, ( X, X,, X ), X, S *, :, = X + ( - ) S *. 6. N(,), ( X, X, X 3 ), () : = X + 4 X + 4 X 3, = 3 X + 3 X + 3 X 3 3 = 5 X + 5 X + 5 X 3 ;,, () 3 3,? 6. ( X, X,, X m ) N(, ), ( Y, Y,, Y ) N(, ),, : () m X i + Y j i = j = ( m + ) ; ( ) m X i i = Y j j = m t( ). 6.3 : T t( ), T F(, ). 6.4 ( X, X,, X m, X m+,, X m+ ) N(, ), :

262 6 59 m m i = m+ i = m+ ( X i - ) ( X i - ) F( m, ). 6.5 ( X, X,, X m ) N(, ), X, S, X m+ N(, ), X m+ X, X,, X m, : X - X m+ S m - m + t( m - ).

263 7.,,.,,,.,., ( : kgmm ) N(, ),, = 48.5, 46.8, 45., 48.3, 45., 44.7,.? H : = 48,,, 9,,., ( : ) : 3, 46, 5, 34, 48, 45, 39, 6, 58, 67 8,, 55, 45, 39, 35, 4, 38,?, N(, ) N(, ),,,,, H : =.? 3 x, x,, x,, H : N(, ),,.,,..

264 7 6,,, H,,,, H.,, H : = 48, H : 48., H : =, H :. 3, H : N(, ), H :.,.,,.?,,. N(, ), =, ( X, X,, X ), H : = ( ). N(, ), 6.7, X - N(,). U = X - = X H : =, =, U = -. X - = X - N (,). H : =,, >, <. 7 - >, >, U N(,),,, U, N(, ), ;

265 6 - <, <, U N(,),,, U, N(, ),. U, U,, H,, H ; U,, H,, H., u -,, U : W = { U U u - } W = { U U > u - }.W, W.U, U W, H ; U W, H. H,?.: U W, H U W, H H, H H, H,.,,,,;,.,,,, ( =.,=.5 =.),.,, W W.,.,, = P{ U W H } = P{ U > u - H }. H, U = X - N(, ), = P{ U > u - } = [ - P{ U u - }] = [ - ( u - ) ], ( u - ) = -,, ( u) = u x - e- d x N(, ). u - U =, :, N(, ),, ( u - ) = -, P{ U > u - } =. X -,, U > u - H ; U u -

266 7 63 H., : () H ; () U, U ; (3), U,, W W, H, P{ U W } = ; (4) U W, H ; U W, H ,, N(, ), =, ( X, X,, X ), H : =.,, :, N(, ), = 4.( mm),, X = 6.56(mm)., 7(mm), :?( =.5) H : = 7. U = X - = = =.5, N(,), u - =.96, U = -.48 =.48 <.96, H : = 7,. 7..,, N(, ), >, ( X, X,, X ), H : =. N(, ), 6.7 4, X - t( - ). S * T = X - S * - H : =, S * = X - + S * =, -. S * T = X - = X - t( - ). S * S *

267 64 - H : =,, T t( - ) S *,, T, t( - ),. :, ( - ), t t - ( - ), P{ T > t - ( - )} =, T = X -, T > t S * - ( - ) H, H. k = t, t, t., - p = - t - ( - )., N(, ), 5, 46.8, 45., 48.3, 45., 44.7, H : = 48.( =.5 ) = 5, X = 45.98, S * =.535, T = X - = S * = =.5, - = 4, t, t - ( - ) =.7764, T = -.94 =.94 >.7764, H : = ,, N(, ),, ( X, X,, X ), H : = ( = ). N(, ), 6.7, S ( - ). = S = S. H : =, =, = S = S ( - ) ; H : =,, >, <. >, >, ( - ),,, ( - ), ;

268 7 65 <, <, ( - ),,, ( - ),. :, -, ( - ) - ( - ), P{ < ( - ) } = P{ > - ( - )} =, = S, < ( - ) > - ( - ) H, H.,,., () k = - p = ( - ) ; () k = - p = - - ( - ). 3 N(, ), =.48. 5,.3,.55,.36,.4,.44, :?( =.5) H :=.48. = 5, S =.64, = S = = =.5, - = 4,, ( - ) = ( - ) =.43, = 3.57 >.43, H :=.48, :. 7..4,, N(, ), N(, ),,, =, ( X, X,, X m ), ( Y, Y,, Y ),,, H : =. N(, ), N(, ), =, 6.7 6, ( X - Y) - ( - ) S w m + t( m + - ),, S w = ms x + S y m + -. T = X - Y S w m + = ( X - Y ) - ( - ) S w m S w m +.

269 66 H : =, - S w m + T = X - Y S w m + H : =, - S w m + =, = ( X - Y ) - ( - ) S w m + t( m + - )., T t( m + - ),, T, t( m + - ),. t - :, m + -, t ( m + - ), P{ T > t - ( m + - )} =, T =, T > t - ( m + - ) H, H. X - Y S w m + 4 9,, ( : ) : 3, 46, 5, 34, 48, 45, 39, 6, 58, 67 8,, 55, 45, 39, 35, 4, 38, :?( =.5 ) N(, ) N(, ),, =. H : =. m =, X = 47.9, S x = 5.9, = 9, Y = 36., S y =. S w = ms x + S y m + - = =.533, T = X - Y S w m + = =.458. =.5, m + - = 7, t, t - ( m + - ) =.98, T =.458 =.458 >.98, H : =, :. 7..5,,, =,,. N (, ), N(, ),,, ( X, X,, Xm ),

270 7 67 ( Y, Y,, Y ),,, H : = ( = ). N(, ), N(, ), 6.7 8, S * x S * y F( m -, - ). F = S * x H : = S * y = S * x, S * y. =, F = S * x S * y = S * x S * y F( m -, - ). H : =,, F F( m -, - ),, F, F( m -, - ),. F :, ( m -, - ), F F ( m -, - ) F - ( m -, - ), P{ F < F ( m -, - ) } = P{ F > F - ( m -, - ) } =, F = S * x F > F - ( m -, - ) H, H. F S * y, F < F ( m -, - ), F, F., () k = m - k = -, p = - F - ( m -, - ) ; () F ( m -, - ), :, ( -, m - ), F - ( -, m - ),, F ( m -, - ) = F -?. ( -, m - ). 6.6, F : F F( m -, - ), F F( -, m - ). P F > F - ( -, m - ) P{ F < F =, P F < ( m -, - )} =, F - ( -, m - ) =.

271 68 F ( m -, - ) = F - ( -, m - ). 5 N (, ) N(, ), m =, S x S y = 5.9; = 9, =..H : = ( =.5). m =, S x = 5.9, S * x = m m - S x = 5.9 = 39., - = 9, S y =., S * y = - S y = 9. = 6., 9 - F = S* x S * y = =.5. =.5, ( m -, - ) = (9,8 ), F, F - ( m -, - ) = 4.36, F ( m -, - ) = F - ( -, m - ) = 4. =.44,.44 < F =.5 < 4.36, H : =. H 7..6,., H, H., W, W, H, H.,,, : N(, ),, X = 96( ), S *? = ( ), :, H : =, H :,,,,,,,?, H :, H : <.,.,.,. N (, ), >, ( X, X,, X ), H : ( H : < ). N(, ), 6.7 4, X - t( - ). S * T = X - S * = X - + S * -. S *

272 H :, S *, T, t( - ),,, T, t( - ),. - H :, <, T t( - ) S *,, T, t( - ),. :, -, t t - ( - ), P{ T < - t - ( - )} =, T = X -, T < - t - ( - ) H, H. 6 N(, ), =, X = 96( ), S * = ( ), :?( =.5) H : ( H : < ). =, X = 96, S * =, S * T = X - = S * 96 - = =.5, - = 9, t, t - ( - ) =.79, T = > -.79 = -.. t - ( - ), H :, N (, ), N (, ),,, ( X, X,, X m ), ( Y, Y,, Y ),,, H : ( H : > ). N(, ), N(, ), 6.7 8, S * x S * y F( m -, - ). F = S * x H : S * y = S * x, S * y., F F( m -, - ),, F, F( m -, - ),.

273 7 H :, >, F F( m -, - ),, F, F( m -, - ),. :, ( m -, - ), F F F - ( m -, - ), P{ F > F > F - ( m -, - ) H, H. F - ( m -, - )} =, F = S * x, 7, 5, S x 6, S y S * y = 5.68, = 3., N(, ) N(, ), :? (=.5) H : >, H :,,., H H, H : ( H : > ). m = 5, S x = 5.68, S * x = 6, S y = 3., S * y F = = = S * x S * = y m m - S x = - S y = =.5, ( m -, - ) = 4.38, =.944, = = ( 4,5), F, F - ( m -, - ) = 5.9, F = < 5.9, H :, H : >, :.,,,.,.,. (),, ; (), p = - p =,, p = - p =, ; (3), H : * * ( H : * > * ),, >, H, H ; (4), H : * * ( H : * < * ),, <, H, H.

274 7 7,., 7,,., :,,,,.,.,, 6.7,. 7 H = = = U = X - N(, ) T = X - S * t ( - ) = ) = S + ( X - ( ) = S ( - ) = =, U =, T = = X - Y m + X - Y S w m + N(, ) t( m + - ), F = S x + ( X - ) F ( m, ) S y + ( Y - ), F = S * x S * y F ( m -, - ) ,,.,,.,,. [ T, T ], ( 95% 99% ),,

275 7,.,., - ( - = ), T T, P{ T T } = -, [ T, T ], -, T, T.,,. 7.3.,, N(, ), >, ( X, X,, X ), -. P N(, ), 6.7 4, X - t( - ). S *,, -, t t - ( - ), X - S * > t - ( - ) =, P X - S * t - ( - ) = -, P X - t - ( - ) S * X + t - ( - ) S* = -. T S * = X - t -, T S * = X + t -, P{ T T } = -,, [ T, T ] -. : N (, ), 95%. =, X = 35.47, S * = =.95,=.5, =.5, - =.975, - =, t t - ( - ) =.. t - S * = = 4.593, T = X - t - S * = = 3.84, T = X + t - S * = = % [3.84, 4.] ,, N(, ),, ( X, X,, X ),

276 N(, ), 6.7, S ( - ). -, -, ( - ) ( - ), P S T = S - -., P{ T < ( - ) = S P > - P ( - ) S - ( - ) = -, P S - ( - ) S ( - ) = -. ( - ) =,, T = S, P{ T T } = -,, [ T, T ] T } = - P{ T T } = -,,, - [ T, T ]. N (, ), =, S * = 7.95, 95%. =, S * = 7.95, S = ( - ) S * = ( - ) 7.95 = =.95,=.5, =.5, - =.975, - =,, ( - ) = 3.86, - ( - ) =.9. T = S - = S = 6.3, T = = = % [ 6.3,5.66]. T = 6.3 = 5., T = 5.66 =.7,, 95% [5.,.7 ] ,, N (, ), N(, ),,, =, ( X, X,, X m ), ( Y, Y,, Y ),,, - -. N(, ), N (, ), =, 6.7 6, ( X - Y ) - ( - ) S w m + t( m + - ),, S w = ms x + S y m + -., m + -, t t - ( m + - ),

277 74 P P{ T. S x P P ( X - Y ) - ( - ) S w m + ( X - Y ) - ( - ) S w m + > t - ( m + - ) =, t - ( m + - ) = -, X - Y - t - S w m + - X - Y + t - S w m + T T = X - Y - t - ( m + - ) S w m +, = X - Y + t - ( m + - ) S w m + = -. - T } = -,, [ T, T ] , X =.45, =.89; 5, Y =.39, S y =.9. N(, ) N (, ), =, - 95%. S w = ms x + S y m + - = =.9, =.5, m + - = 7, t, t - ( m + - ) =.3646, t - ( m + - ) Sw m + = =.463, T T = X - Y - t - ( m + - ) S w m + = X - Y + t - ( m + - ) S w m % [ -.3,.79 ]. = = -.3 = = ,, N (, ), N(, ),,, ( X, X,, X m ), ( Y, Y,, Y ),,, -. N(, ), N(, ), 6.7 8,

278 7 75 S * x S * y F( m -, - ). -, ( m -, - ), F F F ( m -, - ) F - ( m -, - ), P S * x S * y < F ( m -, - ) = P S * x S * y > F - ( m -, - ) =, P F ( m -, - ) S * x S * y F - ( m -, - ) = -, P F - S * x S * y ( m -, - ) S * y S * x F ( m -, - ) = -. T = S* x S * y, T F = S* x S * y, P{ T - F T } = -,, [ T, T ] -. 4, ( : ) : 55, 54, 6, 5 635, 58, 595, 66, 64, N(, ) N (, ), 95%. m = 4, S * x = 4, = 5, S * y = 7.5, S * x S * y = = =.95,=.5, =.5, - =.975, ( m -, - ) = (3, 4), F, F ( m -, - ) = F - ( m -, - ) = 9.98, F - ( -, m - ) = 5. =.66. T = S * x S * y =.64 F =.67, T = S * x S * y =.64 F.66 = % [.67, 9.9 ] , : () U; () -, U,,, W W, P{ U W } = - ;

279 76 (3) U W T T, P{ T T } = -, [ T, T ] -.,,.,,. 7 [ T, T ] = T, = X u - N(, ) S * T, = X t - t ( - ) = T = S + ( X - ) - ) T = S + ( X - ( ) T = S -, T = S ( - ), T, = X - Y u - m + N(, ) -, T, = X - Y t - S w = m + t( m + - ), T = T = S x + ( X - ) F - S y + ( Y - ) S x + ( X - ) S y F + ( Y - ) F ( m, ), T = S * x S * y, T F = S * x - S * y F ( m -, - ) F, 7,,.,,,.,,. 6.7,.

280 ,,,,,.,,,,,.,,. 7.4.,.: a = a < a < a < < a r = b, [ a, b] r., ( a k -, a k ] k, k, k. ( a k -, a k ], k a k - a k -,.,. p k = P{ a k - < a k } ( a k -, a k ], ( ak -, ak ] ( ak -, ak ],,k pk ( k =,,, r)., F ( x)., k,, F ( x) p k = P{ a k - < a k }. F ( x), k p k, k p k ; F ( x), k p k.,. k ( a k -, a k ] ( k =,,, r), H : F ( x),, F ( x). F ( x) p k = P{ a k - < a k }, ( k =,,, r)., H,, = r k = ( k - p k ) pk ( r - ) ; H,,, ( r - ),. :, r -, - ( r - ), P{ > - ( r - ) } =,, > - ( r - ) H, H.

281 78 k. (), k = r - p = - - ( r - ) ; (), 5, k 5 ( k =,,, r), < 5, ; (3), = r k = k p k -,? r = k = ( k - p k ) p k r k - k p k + ( p k ) = k = p k = r k = k p k r - k k = r + k = p k = r k k = p k -.,, 3,4,,,,3, 4 36,7, 5, :?( =.5), H : P{= k} = 4, ( k =,, 3,4)..5 <.5 <.5 < 3.5 < 4.5, 4 ( k -.5, k +.5], ( k =,, 3,4 ). H, p k = P{ k -.5 < k +.5} = P{= k} = 4, ( k =,, 3, 4 ). (.5,.5] (.5,.5] (.5, 3.5 ] ( 3.5,4.5 ] k p k = r k = k - = p k = =.5, r - = 3,, - ( r - ) = 7.85, = 9.36 > 7.85, H : P{= k} = 4,, : k ( a k -, a k ] ( k =,,, r),

282 7 79 H : F ( x),, F ( x), m,,, m.,,, m,,, m, F ( x), ( a k -, a k ] p k = P { a k - < a k }, ( k =,,, r)., H,, = r k = k p k - ( r - m - ) ; H,,, ( r - m - ),. :, r - m -,, - ( r - m - ), P{ > - ( r - m - )} =,, > - ( r - m - ) H, H., : x k 3 k :?( =.5 ) H : P(),..6.4,,, ^= X. ^= x = r k x k = ( ) =.45. k = -.5 <.5 <.5 <.5 < +, 4. H,: ^p = ^P{ -.5 <.5} = ^P{= } = ^! e - ^ =.45! e -.45 =.63763, ^p = ^P{.5 <.5} = ^P{= } = ^! e - ^ =.45! e -.45 ^p 3 = ^P{.5 <.5} = ^P{= } = ^! e - ^ =.45! e -.45 =.8693, =.6456, ^p 4 = ^P{.5 < < + } = - ^P{.5} = - ^p - ^p - ^p 3 = =.88.

283 8 = r k k = = ^p k = =.5, r - m - =,, - ( r - m - ) = 5.99, = < 5.99, H : P(),. 3 5 : X = 77.5, S = 4.7.,?( =.5) H : N(, ),,,.,.6.4,,,,X S, = X = 77.5, = S = < 6 < 7 < 8 < 9 < +, 5. ( a k -, a k ] p k : p k ak - ak - - = P { a k - < a k } = P { a k } - { a P k - } = -., : ( a k -, a k ] ( -, 6 ] ( 6, 7] (7, 8] (8, 9] (9, + ) k p k = r k = k p k - = =.738. =.5, r - m - =,, - ( r - m - ) = 5.99, =.738 < 5.99, H : N(, ),.

284 ,., ;,,,.,?????,.,,.. r : A, A,, A r, s : B, B,, B s.,, ( A i, B j ) ij ( i =,,, r, j =,,, s). B B j B s A j s A i i ij is i A r r rj rs r j s, s r i = ij ( i =,,, r), j = j = i = r = i =, ( ). H :. H,, s i= j = j. ij ( j =,,, s), P{ Ai, Bj} = P{ Ai } P{ Bj }, P{ Ai, Bj} ij, P{ Ai } i, P{ Bj } j ij j i, ij i j ;, H,, ij i j.

285 8, H,, = r s i = j = ij - i j i j ( ( r - ) ( s - ) ) ; H,,, ( ( r - ) ( s - ) ),. :, ( r - ) ( s - ), - ( ( r - ) ( s - ) ), P{ > - ( ( r - ) ( s - ) ) } =,, > - ( ( r - ) ( s - ) ) H, H. (), k = ( r - ) ( s - ) p = - - ( ( r - ) ( s - ) ) ; (),. r s ij i = j = = i j -.? = r s i = j = ij - i j i j r = s i = j = i j ij - ij i j + i j r s ij i = j = = = i j r s ij i = j = i j r - s ij + i = j = - + r i = s i j j = r s ij i = j = = i j -., 339, : B B B 3 A A :?( =.5 )

286 7 83,, H :. r s ij i = j = = i j - = = =.5, ( r - ) ( s - ) =,, - ( ( r - ) ( s - ) ) = 5.99, = > 5.99, H :, :. 7. N(, ),,=.( cm)., =.5(cm).75,.54( cm),:?( =.5) 7., 5, ( : % ) : 3.5, 3.7, 3.4, 3.6, 3.4., :3.5?(=.5) 7.3, ( : ms) N(, ),, =. 5,,X = 6.4, S * =.5) = 44.77, :,,?( 7.4, 5, ( : cm) :.5,.6,.,.4,.5,.3,.3,.,.9,.6,.8,.5,.7,.,.7. N(, ).: () =.5?( =.5) () =.5?( =.5) 7.5,,, :.9,.8,.,., -., 4.4, 5.5,.6, 4.6, 3.4 ;, :.7, -.6, -., -., -., 3.4, 3.7,.8,.,..,.:?( =.5) 7.6, (, : % ),,, : 78., 7.4, 76., 74.3, 77.4, 78.4, 76., 75.5, 76.7, 77.3 ;, :

287 84 79., 8., 77.3, 79., 8., 78., 79., 77.3, 79., 8., 8., 8..,.: ()?( =.5) ()?( =.5) 7.7,( : % ) : 54, 533, 55, 5, 544, 53, 536, 59, , 577, 58, 575, 556, 54, 56, 53, 57, 56,?( =.5) 7.8 N(, ), N(, ),,, =, ( X, X,, X m ), ( Y, Y,, Y ),,.6.7 5,, H : = ( H : ). 7.9 ( :). 5, : 69, 7, 56, 65, 54. 5?( =.5) 7. N(, ),, ( X, X,, X ),6.7,, H : ( H : > ). 7. ( :),,.5. 9, S =.7,,,?( =.5) 7.,, 46, 56 ; 55, 48.,.:,?( =.5 ) 7.3 ( : mm), 8 9, : 5., 4.5, 5., 5.5, 4.8, 5., 5., , 5., 4.8, 5., 5., 5., 4.8, 5., 4.8 :?( =.5) 7.4 ( : % ) N(, ),, =.8.5, : 95%. 4.8, 4.4, 4.4, 4.35, ( :gcm 3 ) 6, X =.75,S =.9. N(, ),:

288 7 85 () 95% ; () 95%. 7.6 N(, ), 9, S * = (m s), 95 %. 7.7 N(, ), N(, ),,, ( X, X,, X m ), ( Y, Y,, Y ),,,6.7 5,, A B ( : ) N(, ), N(, ),,, =., : A 46, 55, 64, 6, 6, 66, 74, 8 B 58, 64, 75, 64, , S x =.549 S y =.65,, 95 %. 7. 3,: ?( =.5) 7. Rutherford Geige, 68 (7.5 ),, : :,?( =.5) 7.,, : :?( =.5)

289 86. () = {,,,,}, A = {8,8,,}; () = {3,4,5,,8}, A = {7,8,,7}, B = {3,4,,8} ; (3) = {(,,3), (,,4), (,,5 ), (,3,4), (,3,5), (,4,5), (,3,4), (,3,5), (,4,5), (3,4,5)}, A = {(,,3), (,,4), (,,5), (,3,4), (,3,5), (,4,5) }; (4) = {,,,}, A = {,,,,5}; (5) = {( x, y, z) x + y + z =, x, y, z > }.. () i i ( i =,,, 8), = {,,, 8 }, () AB = {, 4 } 4 ;.3 () A; A + B = {,, 3, 4, 6, 8 } 4 ; B = {, 3, 5, 7 } ; A - B = {, 3 } 4 ; BC = {,, 3, 4, 5, 7, 8 } 3 ; B + C = {, 5, 7 } 3. () A( BC + BC + BC) A( B + C) ; (3) ABC + ABC + ABC; (4) ABC + ABC + ABC + ABC AB + AC + BC; (5) ABC A + B + C; (6) ABC + ABC + ABC + ABC BC + AC + AB ; (7) ABC A + B + C..4 () AB = {,3,5,7} ; () A( BC) = {,3,4,6,7,8,9,}..6 P = () P = 5 9 () P = P( A ) = 7 5, P( A ) = 4 5, P( A 3 ) = ( a) P = ; ( b) P = ( - ).. ( ) P = 7 ; () P 6 = 6 7 ; (3) P 6 3 = -. A i = { i}, i =,,3. P( A ) = 3 8 ; P( A ) = 9 6 ; P( A 3 ) = P = 4..3 = {( p, q) p, q }, S = 4.

290 87 () P( A) = 3 4. ( ) P( B) = P( ABC) =..5 P( A) = K - ( K - ) N..6 P( B) = - p..7 ( ) - c; () - a - b + c ; (3) b - c; ( 4) - a + c..8.3 ; P = ( + ).. () = ; () = A = { }, B = { }, B = { - },, P( B A) =.75 ; P( B A) =.5. P( B A) > P( B P = - ( - a) ( - b) ( - c)..9 ( a) p =.3; ( b) =.5..3 P = p = 3..3 ( ).6386; () 49. A),,..33 =.94,= C ,= C P{= x i } () P{= x i }..3.6 () P{ 4} =.4;.3 P{= }.4346; P{= } = ,9 43, 4 43, P{= y j }.6.3. P{> 3} =.

291 88.5 P = P{ } = ().68, ( ) () P{ 3} = () P{= k} = () 7 e- 3 ; ( ) P{ 3 } = 3 3 k - 3 ( k =,,3, ) ; - 7 e e P{= x i } P{= k} = C r- k - p r q k - r ( k = r, r +, r +,, q = - p), r = ( ) ; ().36 ; (3 ) F( x) =.5.69( )..6 ( ) ; ( ). F( x) =.7 ( ) A = ; () ( x) =.8 ( ) - ex x < e - x x + x x x > x < x < ; (3) P{ -.3 < <.7} =.49., ; ( ).5 ; (3) ( x) = ( x ( - < x < + ). + ).9 P{ x < < x } = x - 4. ( ) e - ; () e -.. T E() ; ( ) P{ T > 6 T > 8} = e ( ).757; ().895; (3).88; (4).4..3 p = P{6 84}.686;.5 P( A).64 ; () P( B A) () a = 5 ;. () 4 9 P{= y j } ( y) = e y - ey y ( -, + )

292 89.9 ( y) = 4 a 3 m y y m e - ma y > y.3 () ( y) = y - ( y - ) e- 4 y > y () ( y) = e - y y > y.3 ( y) = - y < y < 3. ().5 ; ().4; (3) () P{= i} a ab a ( a + b) ( a + b) a + b ab b b ( a + b) ( a + b) a + b P{= j} a a + b b a + b () P{= i} a( a - ) ( a + b) ( a + b - ) ab ( a + b) ( a + b - ) a a + b ab ( a + b) ( a + b - ) b( b - ) ( a + b) ( a + b - ) b a + b P{= j} a a + b b a + b 3.3 P{= j} 3 P{= i}

293 P{= j} 3 P{= i} P{= i,= j} = ( - p) i+ j p ( i, j =,,,). 3.7 () A =, B = C = ; () ( x, y) = 6 ( x + 4) ( y + 9) ; (3) F ( x) = + arcta x, F ( y) = + arcta y 3, ( x) = ( x + 4 ), 3.8 () ( x) = 3 ( y) = ( y + 9 ). () P = - < x < () A = 6; ( ) F( x, y) = (3) ( x) = e - x x > x (4) P{ (,) D} e - k. 3. ( y) = < y < ( - e x ) ( - e - 3y ) x >, y > ( y) = 3e - 3 y y > y 3 P ( y j ) P ( x i ) 3 3.

294 P{= x i } ( x y) = ( y + ) xe - x( y+ ) ( x > ), ( y x) = 3.4 P{> } = ( ) ; () e ( ) ( x, y) = 3.7 p = () () ; (3 ) e - y x, y > xe - xy ( y > ),. ; () P{ } =.45. P{ = x i } P{ = y j } P{= z k } ( z) = z z - z + 3 z - 3 z - 3 z =, ( z) = 3. F ( z) =, ( z) = 3.3 ( ) ( z) = < z < z 3 ze - z z > z - ( e- z - e - z - ze - z z > z e- z z > z - e - z ) z > z ( z) = () ( z) = ze - z z > z ( z + ).

295 9 3.4 T E( 3). 3.5 ( z) = 6 e - 3 z ( - e - z ) ( - e - z ) z > z 4. E= ,. a, 3 m 4 a. 4 ( a + b) ( a + b ) () a 6( ) ; () a = 3( ). 4.8 () E= 3 ; ( ) E=. 4.9 () E= +, D= - ; () E=, D= ( - ). 4. E= -., D=.76, E( + ) = E=, D=. 4. ) a =, b = ; ) E=, D=. 4.3 P{ - E< D} = D= P{ < < 6} E= 6 5, D= E(+ e - ) = 4 3, D( 3- ) = E=.5, E=.5, E si (+ ) =.5, E(max(,) ) =., D(max(, ) ) = E= 7 6, E= 7 6, E= E= v 3 = ( a + b) ( a + b ) 4., v 4 = a4 + a 3 b + a b + ab 3 + b ( ) = ; () D(+ ) = E(+ + ) =, D(+ + ) = ( ) E= 3, E=,Cov(, ) = ; ().

296 ( ) E= E= 3, Cov(, ) =, = ; ( ). 4.3 =. 4.3 E= 4, D=. 4.3 E =, D = P{ } = P{ > 35} = () P{79 9}.977; () () P{ 5} = C k 5 (. ) k (.99) 5 - k ; k = () P{ 5} =.66 ; P{ 5} = ()..( ) ; X = 58, S = 5.84, S = 634.5, S * = 6.93, S * = () Y = X - a ; ( ) S y = S x b b. 6.3 ^a = X - 3 S, ^b = X + 3 S. 6.4 () ^= X ; () ^= X. 6.5 () ^= 6.6 ^p = X. 6.7 ^= X i. i = X - X ; ( ) ^= - i = l X i. 6.8 ^a = 3 i = X i. 6.9 EX =, DX =, E( S ) = - 6. ^..

297 94 7. U =.8868 >.96 = u.975, H = T =.343 <.7764 = t.975 (4), H = ( 4 ) =.4 < = 4.86 <.975 (4) = , H =. 7.4 () T = -.9 <.448 = t.975 (4), H =.5; () = > 6.9 =.975 ( 4 ), H = T =.868 <.9 = t.975 (8), H =. 7.6 () F.5 (9,) =.56 < F =.666 < F.975 (9,) = 3.59, H = ; () T = >.86 = t.975 (), H =. 7.7 F =.7 <.9 = F.5 (8,9 ), H =. 7.9 T = 3.86 >.38 = t.95 (4), H 5, H > = 7.64 > 5.57 =.95 (8), H.5, H > T = < = - t.95 (8), H, H <. 7.3 F = 3.66 > 3.5 = F.95 (7,8), H, H >. 7.4 [4.7,4.46]. 7.5 () [.689,.7] ; () [.,.464]. 7.6 [55.,444] ;[7.43,. ] [ T, T ], T = X - Y - u - m +, T [ ,95.]. 7.9 [.,3.6]. = X - Y + u - m = 8.96 <.7 =.95 (5), H,. 7. ^= X = =.97 < 8.37 =.95 (), H,. 7. = 7.39 > 3.84 =.95 (), H,.

298 95 b(, p) P{= k} = p k ( - p) - k k =, p p( - p) b(, p) P{= k} = Cp k k ( - p) - k k =,,, p p( - p) P () P{= k} = k k! e - k =,,, g( p) P{= k} = ( - p) k - p k =,, p - p p H(, M, N) P{= k} = C k M C - k N- M C N k =,,, M N M N - M N N - N - U ( a, b) ( x) = b - a a x b a + b ( b - a ) E () ( x) = e - x x > x N(, ) ( x) = ( x - ) e- ( ) ( x) = x - e - x x > x t t( ) ( x) = + + x - + ( > ) - ( > ) F F ( m, ) ( x) = m + m m m x m - ( mx + ) m+ x > - ( > ) ( m + - ) m( - ) ( - 4) ( > 4) x

299 96 P{= k} = k e- k! k k

300 97 k

301 98 3 N(,) ( x) = P{ x} = x e - t d t - x N(,) ( x) x

302 99 4 N(,) N(,) p = ( x) = P{ x} = x e - t d t - p = ( x) N(,) x p

303 3 p

304 3 5 t T t( k) P{ T t p ( k) } = p p k t t p ( k) k p

305 3 k p 6 ( k) P{ p ( k)} = p p k p ( k)

306 33 7 F F F( k, k ) P{ F F p ( k, k ) } = p p ( k, k ) F F p ( k, k ) p k k

307 34 p k k

308 35 p k k

309 36 p k k

310 37,.. :,,.. :, 3.. :, ( ). :, :, :, :, :, :,988.. :,983 SHELDON ROSS,.. :,98 G.P.WADSWORTH,.. :, :, :, :, 6.. :,994