теорія ймовірності і математична статистика

519.2

..

2003

519.2+600.1 / .. , 2003. . 244: . .: 114 .

, .

, , , , , , , , , , . : , , , , , , .

.

: .. , .-. , , .. , , , .. , ,

ISBN 966-02-2664-0 . ., , 2003

2

, , . , .

:

1) , , , , , , ;

2) . , ;

3) ; 4) ,

, .

. , .

, , , . , , , . , . , , , , . , . : , .

, : , , , , , , .

, , , 3

, . ,

. , .

.-. ... .., .. .. .. .., .. ..-.. .. ..-.. .., .

* * *

, , 2003 ., , .

: , 42, , 03187, , , : [email protected].

4

1. 1.1.

. , . .

1.1.1. 1. , . . ,

. 2.

(, , , .), .

1. , , , , .

2. . 3. ,

. 4. -

() . . .

. . , . , ' - .

1. , , . , (, , , ), , (2, 3, 4, 5). 5. () ,

. 1. ,

. A . A , , ,

AA .

2. A B , A , . B 1.

. ,

, .

, .

5

6. )(Ap A

AL , A , : L)(Ap

L

LA= . (1.1.1) 2. ,

, )(Ap

32

64 = .

. . 7. )(ApN A

, N

AN A , :

)(ApNN

N A= . (1.1.2) 2. ( )

) , .

N (ApN

)(Ap

8. , )

)(Ap

(ApN A N : )(Ap = . (1.1.3) )(lim ApN

N . ,

. . 3.

, 30- XX .. . . .

1.1.2. 1. .

. . .

, . 2. A ,

. A A . .

. 3. A ,

(B

BA , BA ). 4. ,

; , ; , , , .

6

, .

1. , , , . . , () .

. ' ( +), (

)( ) \ ().

' A 21 AA = , A , , , , .

1A 2A

1A 2A

A 21 AA = , A , , . 1A 2A AAB \== , A , ,

, B A .

. ' ( ) . , , ,

, Ss S sA

s . ' , , , , , .

Ss

sAsA

Ss

sAsA

2. , ,

sA 1+ sxs10 y 5,00 s ( ]5,0,0[= Ss ) (. 1.1.1).

,

Ss

sA5,10 x , 10 y ,

,

Ss

sA15,0 x , 10 y .

. 1.1.1.

', . 5. , :

1) ;

7

2) ; 3) A , A ; 4) 1A , 2A 21 AA 21 AA .

1.1.3.

1. )(A A , A , , 0)( A , 0)( = ( ), , nA ,...2,1=n =

n

nn

n

AA )()( . (1.1.4) 1. ,

, , , , . , .

2. (1.1.4) . 2. )(

A , 0)()\( == AA . 3. )( ,

n (), .

Nn ,...,2,1( =,...)2,1=n

1. .

1. ( ) , .

2. , A , - (-). 4. -

, , :

1) ; (1.1.5) 2) A AA \= ; (1.1.6) 3) nA , ,...2,1=n =1n nA . (1.1.7) 1.

. 2. (1.1.5) (1.1.7) ,

nA , ,...2,1=n =1n nA . (1.1.8) 3. ( (1.1.5), (1.1.6)).

4. (1.1.5) (1.1.8) , - , nA ( ) ,

,...2,1=n , , .

5. -. 8

6. - , .

7. - -. 5. ,

- ( ( , )). . . ,

, -.

6. () ( , )), . , ( , , ). 7. ,

3) . 1)( =p 1. , 1)(0 Ap 0)( =p . 2. 1)( =Ap , A ,

, . 0)( =Bp B 3. 1A

( ) ,

:

2A )( 21 AAp 21 AA 0)( 1 Ap1A )/( 12 AAp 2A

1A

)/()()( 12121 AApApAAp = . (1.1.9) . , .

, , ,

)/( 12 AAp 0)( 1 =Ap 4. ,

: ) , , ; 1+ nn AA 1n )(lim)(

1

nn

n

n ApAp = =

) , , . nn AA +1 1n )(lim)(1

nn

n

n ApAp = =

5. ( ),

1A 2A

)()()( 2121 ApApAAp = . , , ,

)()/( 212 ApAAp =0)( 1 Ap )()/( 121 ApAAp = 0)( 2 Ap .

. , 1A , .

2A

6. , .

MAAA ,...,, 21

)()...()()...( 2121 MM ApApApAAAp =.

. . 7.

, MAAA ,...,, 21 =MAAA ...21 .

1.1.5. 1. .

: MAAA ,...,, 21

= =Mm mAp1 1)( . (1.1.10) 1. ( ). M 2. ,

: 1)()( =+ ApAp . (1.1.11)

2 (). A ( ).

B

.)()()()( BApBpApBAp += (1.1.12) 3. , . MAAA ,...,, 21

10

) = = = ++= Mm Mnm MMnmmMm m AApAApApAp 1 1, 111 )...()1(...)()()( ; (1.1.13)

) = = = ++= Mm Mnm MMnmmMm m AApAApApAp 1 1, 111 )...()1(...)()()( . (1.1.14) 4 (). M AAA ,...,, 21 M .

).../().../()()( 111211

= = MMMm m AAApAApApAp . (1.1.15) . , ,)()(

11

== = Mm MMm m ApAp ( 6 1.1.4). 5 ( ). A

, (). M

HHH ,..., 21)(Ap A

)(Ap = . (1.1.16) )/()(

1

m

M

m

m HApHp= 6 ( ). (), . ,

,..., 21 HH

mH A :

== = )/()( )/()()( )()/( nn mmmm HApHp HApHpAp AHpAHp . (1.1.17) 1n. , . 7 ( ).

( N ).

N

A . A p , pq = 1 . , N A , n

nn

NN qpCnP =)( N n , (1.1.18) ' : nNC N n

)!(!

!

nNn

NC nN = .

. (1.1.18) . 8 ( ).

, NA .

A - ,

m mp

1 ( 1,m mq p m N= = ). , NA n , == += Nm mmNn nN zpqznP 10 )()( .

11

1.2. 1.2.1.

1. ,

X . x

X

)(=x , . .

1. , , x . X 2.

, , .

, , .

3. )(=x =x . ,

X , .

4. , , , . 1. .

( , , , ..) .

. , , , . 3. ,

, , .

. . 2.

. , () . .

1.2.2. 1. () : X1)

( ) { }F x P X x= < ; (1.2.1) 2)

f xdF x

dx( )

( )= ; (1.2.2) 3) ,

e j x , ' :

12

= dxexfjQ xj )()( , (1.2.3) P{*} ; j .

. .

1.

)(xF

)( jQ , , , .

)(xf

2. , .

X

)(xF )(xF

. : 1, 0, 1 1/3. ,

X

12 += nn XY ,...1,0=n 3.

, . , ,

X

ix ,...)2,1( =i - ( 1): )()( ii xxpxf = , (1.2.4)

. ( ) .

ip ix ip

...,2,1=i

, , , , .

)(xf

4. ' , . , . , : = x dxxfxF ,)()( (1.2.5) ':

.)(2

1)( = dejQxf xj (1.2.6)

5. , , , , = )()( xdFesM sxx (1.2.7)

13

= )()( xdFss xx . (1.2.8) 2. ,

. , , . 1.2.1, 1.2.2 1.2.3, , .

1.2.1

0)( xF 1)(0 xF )(xF

1.2.2

0)( xf

1)( = dxxf

2

1

1 2 2 1{ } ( ) ( ) (

x

x

P x X x F x F x f x dx < = = )

1.2.3

1)0()( = QjQ

( ) X )()(* jQjQ =

, , . 1.2.4 1.2.5. [25] [69].

1.2.4

1,0,

1 == IiI

pi

10,,0,)1( == IiCp iIiiIi ,...1,0),exp(

!== i

ip

i

i 10,...,1,0,)1( == ip ii

14

. 1.2.4

0,0,,0, == INnNni

C

CCp

n

N

in

IN

i

I

i

pqiqiippi ==+= 1,1,0,)1( 10,...,2,1,...,1,0,)1(1 === + miCp imi imi

. : nNC)!(!

!

nNn

NC nN = .

1.2.5

bxa

abxf = ,1)(

() 0),,(,2

)(exp

2

1)(

2

2 > = xmxxf 0,0,0,

2

))/(ln(exp

2

1)(

2

2 >>>= mxmxxxf

0,0),exp()( >>= xxxf

- ,...2,1,0,)2/(2

)2/exp()(

2/

2/)2( = = xxxxf 0,

)(

)/exp()/()(

1 = xcb bxbxxf c , ( ), c

( )

b 0>b0>c

0,0],1,0[,)()()()1(

)(11 >> += xxxxf

0,0,0),)/(exp()(1 >>= bcxbx

b

cxxf c

c

c

,0,exp)(

2)( 2

2

12

2>= xxmxmmxf m

m ( ), m 0>m ( 0> )

,0,2

exp)(202

22

2> += xxIxxxf ( 0> ), ( 0> )

15

. 1.2.5

()

++=

),(,0

),,2/)((,)/()(4

),2/)(,(,)/()(4

)( 2

2

bax

bbaxabxb

baaxabax

xf

0),,(,)(

1)(

2

0

2>+= hxxxh hxf

),()(

2

210

xfxbxbb

ax

dx

xdf ++ = 210 ,,, bbba

0,0],1,0[

,1)2/(

)2/)1(()(

2/)1(2

>> + +=

+

x mx

mm

mxf

m

),,(,21

)( = xexf mx ,m

0,,)(1 >=

+ mxx

m

mxf

. - ( 2). )(m

1.2.3. () ,

, , . , , , . . 1. )]([ XM )(X

X )(xf= dxxfxXM )()()]([ . (1.2.9) 1. : ,

(1.2.9) . 2. )]([ XM

, )(X . )(xf 2.

xm X

)(xf

XX =)( :

16

== dxxxfXMmx )(][ . (1.2.10) 3. ][XDDx =

:

X

)(xf2)()( xmXX = == dxxfmxmXMD xxx )()(])[( 22 . (1.2.11)

4. () x : X xx D= .

1. ,

xm X

)(xf

xD x .

2. xm Xx , ;

. xD

3. () () , . 5.

: x

xm . 6.

, , .

ex

X )(xf

5,0)( =xF 7. px p

, pxF =)( p )10(

( ), ( ). 9. m -

X XX =)( , - , ||)( XX = - , )()( xmXX = - , ||)( xmXX = .

1. . .

2. ' , :

;01 = ;222 xx mmD ==;23 2233 xx mmmm += .364 422344 xxx mmmmmm +=

3. , : xD

= . 0 ; )1...(5312/ xD

, : xm xD

;1 xmm = ;22 xx mDm +=;3 33 xxx mDmm += .63 4224 xxxx mdmDm ++=

4. . , , 2 . . ( , )

.

)( jQ =1+ ,!

)(

1

mj= (1.2.12)

0

)( == dj jQdm . (1.2.13)

(1.2.3) .

1. (1.2.12) .

2. , , :

)(ln jQ = .)(!1

= j (1.2.14) 18

10. (1.2.14) . :

0

)(ln == dj jQd . (1.2.15) 1. (1.2.14) ,

' : = =1 )(!exp)( jjQ . (1.2.16)

2. , ,

mm ,...,1m ,...,1 . :

;11 xmm == ;2

2

22 xx Dmm === ;23 3

3

233 =+= xx mmmm .361243 324

42

23

2

244 =+= xxx mmmmmmm 3.

: 0

)( == sxsd

sMdm

; 0)(ln == sx

ds

sMd

; (1.2.17) , ,

=)(sM x ;!0

= sm (ln =)sM x .!1= s (1.2.18)

11.

2/3

2

3

2/3

2

31 == ; (1.2.19)

2

2

4

2

2

42 3 == . (1.2.20)

1. . 01 = . . 02 = .

2. 1 2 2132 + 2/)3( 2 + .

12. . 13. () X )(xf == dxxfxfxfMH x )(log)()]([log 22 . (1.2.21) 1. ,

. X

19

2. , , . ,

, , .

X

0=xH X],[ ba

3. .

2

xX

xx eH 2log 2= . 14.

( X 0=x ) )]1)...(1([][ ][][ +== XXXMXMm . (1.2.22)

. '

:

][mX )(sx

)1()(][ xm = ; ==+ 0 ][ !)1( smsx .

15. X

])[( ][][ xmXM = . (1.2.23)

1.3. 1.3.1.

1. N - XG :

1) N - 1 1( ) { , , } { }N N NF x P X x X x P X x= < <

1.3.1

11 1( ) { ,..., };

( ) ... ( )N

N N

xx

N N

NF x P X x X x

F x f x dx

= < = xmxxf

= mxxF )( ,

dttxx = 2exp21)( 2

= 2exp)(

22 jmjQ . 1. , m . 2. 7.

. 1.5.1. . ,4=m 1;75,0;5,0;25,0= ( 1 4)

1. , ,

m . 3 .

m

%4,0

)(xf 3m . , , ,

%7,99

997,03( m , )3+m . 34

2. )0( =m , 0 -.

3. :

X

= nn

n

n

n

dX

XgdM

dD

XgMd2

2 )(2

)]([;

,)(

)]([)]([ += dXXdgDMXgmMXXgM , )(Xg X D .

4. - . XXg =)( . 1.

, : 1X 2X

)],,(exp[),( 212212 xxPCxxf = (1.5.1) , , , , , , - .

C )0( >C ),( 212 xxP1x 2x

1x 2x ),(( 212 xxP )0 1.

(1.5.1) :

21 , XX

.)1(2

)())((2)(exp

12

1),(

22

2

2

1

2

22

2

1221121

2

11

2

2

2

21

212

+=

r

mxmxmxrmx

rxxf

(1.5.2)

+++= )2(21exp),( 2222212121212211212 rjmjmjjQ ,(1.5.3) , 1m 2m 1X 2X 1 2 , r .

1. , (1.5.2) :

)],()(2

1exp[

||2

1),( 1212 mxRmx

Rxxf T

GGGG = (1.5.4) -,

, -, , ,

Txxx ),( 21=G1X 2X

Tmmm ),( 21=G1X 2X T

1R ,

2221

1211

RR

RRR = ,

. || R R

35

2.

).,()(),(

2122121212 jjQ

r

jjQ nn

n = ,

. 1.

, . 1X 2X

. , (. 9 1.2).

2. ( ) (. 1.5.2),

1X 2X

0,)1()())((

2)( 22

2

2

2

22

21

2211

2

1

2

11 >=+ CCrmxmxmxrmx (1.5.4)

T 1 2( ) ( ) ,x m R x m C C =G G G G 0.> (1.5.5)

. 1.5.2. 1X 2X

1. , , ,

),( 21 mm

1m 2m

1 , 2 , r C . 1 , 2 r .

=

=.4/

,22

1

21

212

2

2

1

21

rarctg (1.5.6)

, . D90

2. (1.5.4) (1.5.6) , , .

C

36

3. (1.5.4), (1.5.5) , , , )0( =r , .

4. (1.5.6) , =1( )2 : 4/ = . =1( )2 )0( =r , C .

3. () () , :

1X 2X

1Y 2Y

,cos)(sin)(

,sin)(cos)(

22112

22111 mXmXY mXmXY += += (1.5.7) (1.5.6).

4. - .

5. , , , ' .

. , , .

6. , , :

1X 2X

1 2 1 1 2 1 2 1 1

2

22 2 1 12 22

2 12

( / ) ( , ) / ( )

1 1exp ( ) .

2 (1 )2 (1 )

f x X x f x x f x

x m r x mrr

= = = =

(1.5.8)

. (1.5.8) , , :

12 / xxm 1X

12 / xxD

)( 111

22/ 12

mxrmm xx += , . 222/ )1(12 rD xx = 2. ,

, . 2. ,,

, : 1X NX

)],()(2

1exp[

||)2(

1),...,( 1

2/12/1mxRmx

Rxxf T

NNN

GGGG = (1.5.9)

37

TNxxx ),...,( 1=G -, ,, , 1X NX TNmmm ),...,( 1=G -, , : R

NNN

N

RR

RR

R

...

.........

...

1

111= . (1.5.10)

1.5.2. , ' , ,

. , , . , , . 1. ,

: .0,

2exp)(

2

2

21>= xxxxf (1.5.11)

1. , (1.5.11),

.0,2

exp1)(2

2

1 >= xxxF (1.5.12) 2.

2

=xm , ,

222 xx mD = =x , 2

3 33=m .

3. 1 . 1.5.3.

. 1.5.3. (1) (2 5). 1 5 1= 4,0=

1. ),( 21 XXX =G 38

( = =0) .

1xm

2xm

2 X XG , ]2,0( .

. .

.2

exp2

1),(

2

2

2

2

1

2212 += xxxxf (1.5.13)

xx

xxJ == cossin sincos),(2 .

XG ( ),x

.2

exp2

),(2

2

22 = xxxf (1.5.14) (1.5.14)

]2,0( x ),0[ XG (1.5.11)

21)(1 =f . . , ,

, -

1X 2X2

1xm

2xm ),( 21 XXX =G , (1.5.11). R

1( xm )2xm

2. ( ) ,

,0,2

exp)(202

22

21> += xxIxxxf (1.5.15)

, 2 )0( > , .

)(0 I 1. ,

(1.5.15), += x dxxIxxxF 0 202 2221 2exp)( , (1.5.16)

. 2.

39

,0,4

12

exp)(4

22

2

22

21> + += xxxxxf

, >>

.0,8

12

)(exp

2

1)(

2

2

2

1 > + = xxxxxf 3.

+ += 22

2

2

12

2

2

2

02

2

4exp

42421

2 IImx , (1.5.17)

2222 xx mD += , (1.5.18)

. (*)1I 4. >>

xm . . 1.5.3 ( 2 5).

2xD 2.

),( 21 XXX =G , .

1xm

2xm 2

X XG

22

21 xxmm += , , 2

2

01 02

22

0 02

1 cos( )( ) exp cos( )

2 2 2

exp sin ( ) , | | ,2

f F

= +

1

2arctg0x

x

m

m= , . [*]F 1 . 1.5.2.

1.5.2 1

24, 40, 47, 49

3, 16, 25, 32, 69, 74, 75, 84, 100 5, 10, 15, 20, 22, 61, 88, 89, 105 55, 56, 66, 75, 82, 91, 96, 97

40

2. 2.1.

2.1.1. 1.

, - (

)(tX

Tt T ) . . S ( ). 2. - (

) () i )(tX

x ti ( ) ,...)2,1( =i , i - t T x S . 1.

, . ,: .

2. - () , , . , .

3. . . . 2.1.1.

. 2.1.1. )(tX

. . . X t( ) x ti ( ) - - .

i

i X t( )

x ti ( ) ( , ,...)i = 1 2 . (

X t( )

t0 X t( )0 ). x ti ( )0 - t -

i

i X t( )0 . 0. . , , ' S , T , . . 2.1.1 2.1.2

41

(), .

2.1.1 ( 1)

S ( ) NRS ( ) NZS N

S (S R

) ZS (S R N ) NZS

T

() (T R ) (T ) R N

. , R Z .

2.1.2 ( 2)

1. ( 1- )

2. S

( )

3. 4.

(S , T ) (S , T )

5.

(S T )

6. ( )

42

() , . , ,

. . .

2.1.2. . N -

.

F x tN ( ; )

G G,

f x tN ( ; )GG

Q j tN ( ; )GG ,

. 2.1.3. 2.1.3

11 1( ; ) { ( ) ,..., ( ) };

( ; ) ... ( ; )N

N N

xx

N N

NF x t P X t x X t x

F x t f x t dx

= < ),...;,...( 11 MMx ,t t,xt / xm

),1),(/( MmtxtX m = , .

)

)

,...,;...( 11 MMx tt,x,t / xD

4. ) ,

.

(tDm

,...;,...( 11 MMx ,t t,xt / xm

)(tDD

),...,;...( 11 MMx tt,x,t / xD

53

. 2.2.2. :

; )(t mx

)(tX )(t mx )(tx )(tm

x ;

; )(tX

),...,...( 11 MMx ,t, t,xt / xm )(tX),...;,...( 11 MMx ,t t,xt / xm ),...,;,...,/( 11 MMx ttxxt

),...;,...( 11 MMx ,t t,xt / xm

)(tX

. ) ) :

(tDx )(tX

(tDm )(tDD

)(tDx = + . (2.2.26) )(tDm )(tDD

' ),1),(/( MmtxtX m = . 6. ),...,;...,( 11 MMx tt,x, / xttR

, :

)],,...,;.../'(),...,;.../([

),...,;...,(

11

0

11

0

11

MMMMx

MMx

tt,x,xtXtt,x,xtXM

tt,x, / xttR

==

(2.2.27)

) : ,...,;.../( 110

MM tt,x,xtX

).,...,;.../(),...,;,...,/(

),...,;.../(

1111

11

0

MMxMM

MM

tt,x,xtmttxxtX

tt,x,xtX = = (2.2.28) . 2.2.3.

. ),...,( 1...1 NttN ),...,;,...,/,...,( 111...1 MMNM ttxxttNM ++ . (2.2.13)

),...,;,...,( 11 NNN ttjjQ ),...,;,...,/,...,;,...,( 1111 MMNMNMMN ttxxttjjQ ++ . 54

, , . , .

2.2.3 )(tX

),...,;,...,/( 11 MMx ttxxtm =

=

],...,;,...,/)([ 11 MMx ttxxtXM

tt,X,t / XmDtD MMxxxm M )],...,;...([)( 11,...,1=

],...,;,...,/)([

),...,;...(

11

11

MMx

MMx

ttxxt XD

t t,x,t / xD= = tt,X,t / XDMtD MMxxxD M )],...,;...([)( 11,...,1=

)],...,;.../'(),...,;.../([

),...,;...,(

11

0

11

0

11

MMMMx

MMx

tt,x,xtXtt,x,xtXM

tt,x, / xttR

==

2.3. 1.

)(tX , () : . ))(),...,(()( 1 tXtXtX H=G

. (, , , , .).

. ( , , ). 2. ,

.

)(1 tX )(2 tX

)(1 tX

)(2 tX

.

)(1 tX )(2 tX

),;,( 21212 ttxxf N );( 11 txf N );( 22 txf N .

3. ,

))(),...,(()( 1 tXtXtX H=G

),...,;,...,( 11 NH ttxxf

55

== Hh hhNNH txfttxxf 111 );(),...,;,...,( , (2.3.1) ( ),

N

H . 1. ,

-. , . ' .

2. , ',

. ))(),...,(()( 1 tXtXtX H=G

4. )(tmx , :

))(),...,(()( 1 tXtXtX H=G)](tXMtmx

G[)( = . (2.3.2)

5. )(tD x , )(tXG :

)]([)( tXDtDxG= . (2.3.3)

6.

)',( ttRxG

)(tX HH

.),1,(

)]'()([)',(00

Hlh

tXtXMttR lhhl == (2.3.4) . )

. ' . .

',( ttRhl

7.

( , ')( , ') ( , 1, )

( , ') ( , ')

hlhl

hh ll

R t tr t t h l H

R t t R t t= = . (2.3.5)

. 2.3.1.

. () ' , , .

56

2.3.1 ()

()

, HhtDttR hhh 1, ,)(),( == -

),'(),( ttRttR lhhl = 1)

2)

1),()()(),( ttr ttttRhl lhhl

. , , , tt = .

2.4. 2.4.1.

. . - D () (. 2.4.1), , , , .

)(tx

)(ty

. 2.4.1.

. , , . 2.4.1.

2.4.1

1. , , , . 2. L ,

: 1) ; )]([)]([)]()([ 2121 txLtxLtxtxL +=+2) )]([)]([ txLtxL = .

57

1. . 3. D ,

L

)(t : )()]([)]([ ttxLtxD += .

4. , , . 5. () ,

t .

)(ty t

)(tx

6. , , .

)(ty t )(tx

t

2. , . - . - () , .

1. , .

2. .

)(tX

)(tY

2.4.2. .

. . , , .

)(tX )(tY )(tX

)(tY

, , ( , ). . 1.

)(xy = , )(yx = . ) ;(1 tyf y )(tY )(tX

dy

ydtyftyf xy

)());(();( 11

= , (2.4.1) ) . );((1 tyf x )(tX 58

2. )(xy = , , Q

Qqyx q ,1),( == . )

,(1 tyf

y )(tY

)(tX == Qq qqxy dy ydtyftyf 1 11 )());(();( . (2.4.2) (2.4.1) (2.4.2) ,

(1.3.1) (1.3.2). ,

, . .

)(tXG

)(tYG

3. )(xy GGG = , :

),...,( 111 Hyyx = ,

),...,( 1 HHH yyx = , H ),...,( 1 Hxxx =G ),...,( 1 Hyyy =G . H - );( tyf yH G )(tY

G )(tXG

)());(),...,(();( 1 yJtyyftyf HH

x

H

y

H

GGGG = , (2.4.3) ) ;( txf xH G H - , )(tX

G)( yJ HG

:

H

HH

H

H

yy

yy

yJ

=

"

""""""

G

1

1

1

1

)( . (2.4.4)

. H ,...,1 , (2.4.3) . 4. 1. -

N

),...,;,...,( 11 NNy

N ttyyf )(tY

)(tX = = Nn n nNNxNNNyN yyttyyfttyyf 11111 )(),...,);(),...,((),...,;,...,( , (2.4.5) - . ),...,;,...,( 11 NNxN ttxxf N )(tX

. .

)(tX )(tY

N Ntt ,...,1

)(tXG

)(tYG ,

59

( )Y )(tX n (tn Nntn ,1, = ). , - , - , . -

(2.4.3), ' - - . , :

N

)(tXG

)(tYG

N

)(tX )(tY N

),...,;,...,( 11 NNy

N ttyyf )(tY

)(tX N

)(tYG

N )(tXG

=== N...1 , (2.4.3) . (2.4.3) (2.4.5). 5.

),( 211 xxy = , 22 xy = , : ),( 2111 yyx = ,

22 yx = . )

: ;( 11 tyf

y)(1 tY

))(),(()( 21 tXtXtX =G2

1

211221211

),();),,(();( dy

y

yytyyyftyf xy = , (2.4.6)

) ;,( 212 txxf x )(tXG

.

. 3.

1

1

2

1

1

1

10

)(y

yyyJ == G .

(2.4.3)

1

12212212 );),,(();,(

ytyyyftyyf xy = .

) , (2.4.6).

);,( 212 tyyfy

;( 11 tyfy

. (2.4.6) , , , (. 2.4.2).

)(tY

)(1 tX )(2 tX

. , . )(tY

1. )(xy = . )(tm y - [ ] dxtxfxtYMtm xy == );()()()( 1 . (2.4.7)

60

2.4.2 '

);(1 tyfy

)(tY

);,( 212 txxfx ))(),(()( 21 tXtXtX =G

);(1 tyf y )(tY

21 xxy += );(1 tyf y = 2222 );,( dxtxxyf x 21 xxy = );(1 tyf y = 2222 );,( dxtxxyf x +

21 xxy = );(1 tyf y = 22222 );,( xdxtxxyf x 2

1

x

xy = );(1 tyf y = 22222 );,( dxxtxyxf x

. (2.4.7) , )(tY

dxtxfxtm xy = );()()( 1 . 2. )(xy = .

)(ty - . ( )[ ] [ ] dxtxftmxtmtYMt xyyy == );()()()()()( 1 .

3. , ( qQ

q

q x=1 q , Qq ,1= ). . . 2.4.3.

1. , .

)(1 tX )(2 tX

)(tY

2. QqtX q ,1),( = : )(tY

),(),(1

2 ttRttRqx

Q

q

qy= = ,

)()(1

2 tDtDqx

Q

q

qy == . 3.

,

61

.

)(1 tX )(2 tX

2.4.3 ,

)(tm y

),( ttR y ),( ttD y ,

)(tY

)(tmqx

)(tRhq xx

)(tDqx

)(tX q

q

Q

q

q xy == 1

)()(

1

tmtmqx

Q

q

qy == ),(),(),(

1

2 ttRttRttRhqqq xxh

Q

q hq

qxxqy+= =

),()()(1

2 ttRtDtDhqq xxh

hq

qx

Q

q

qy = +=

2.4.3.

. )(tY )(tX

, ( ) .

)(tY

)(tX

- , , , . . .

)(tX

)(tX )(Y . : [ ])()( tXLY t= , (2.4.8) , t . tL 1.

(), 1D 2D[ ][ ] [ ][ ]xDDxDD 1221 = .

1. . 2. M .

)(Y (. 2.4.4). )(tX

62

2.4.4

[ ])()( tmLm xty =

[ ]),(),( ttRLLR xtty = == |),()( yy RD

:

)()( txdt

dty = ,

= dttxy )()( .

3. . 2.4.5 2.4.6 , .

2.4.5

)()( tm

dt

dtm xy =

tt

ttRttR xy = ),(),( 2

ttxytt

ttRtD = = |),()( 2

2.4.6

( ) ( )y xm m

= t dt = ' ),(),( tdtdttRR xy

= tdtdttRD xy ),()(

2 . 2.4.7.

63

2.4.7 2

24, 40, 47, 49

3, 16, 25, 32, 40, 69, 73, 75, 84

5, 10, 15, 20, 22, 29, 61, 88, 89, 105 38, 55, 56, 66, 74, 82, 84, 86, 92, 94

64

3. ,

, .

' . , . () .

3.1. 1.

(), -

)(tX

)(),...,( 1 NtXtX

,)()(2

1exp

)2(

1);(

1 = xxTxx

NN mxRmx

Rtxf

GGGGGG (3.1.1) - -, - ;

T

Ntxtxx ))(),...,(( 1=G N n)( ntX

T

Nxxx tmtmm ))(),...,(( 1=G - - , -

;

N n

)( ntX )},({ mnxx ttRR = NN , ,

)( ntX

)( mtX );,1,( Nmn = , ; .

1xR xR T

, .

1. ) ) . , . , .

(tmx ',( ttRx

2. , .

. , - )

0)',( =ttRx'tt N ;( txf N GG

( ): );()...;();( 1111 NNN txftxftxf =GG . (3.1.2)

, , );( txfN GG (3.1.2). , .

0)',( =ttRx'tt

3. ,

)(tX

)),1( ),()(/( MmtxtXtX mm == . . -

MN >

65

),...,;,...,(

),...,;,...,(),...,;,...,/,...,;,...,(

11

11

1111

MMM

NNN

MMNMNMMNttxxf

ttxxfttxxttxxf =++ . (3.1.3)

, (3.1.3) (3.1.1). , . , . 4. -

() () .

)(tX

. , , ,

. . , . , .

0),( =mnx ttR mn tt )(tX

5. . .

6. .

7. , (3.1.1) : = GGGGGG xTxTN RmjtjQ 21exp);( .

8. .

3.2. 3.2.1.

1. . . . 1.

(), - -

)(tX

N

N 112 ,..., tttt N t , .

Ntt ,...,1

)(tX

, , .

1. - - , -

N

t N =),...,;,...,( 11 NNN ttxxf ),...,;,...,( 11 ++ NNN ttxxf , (3.2.1) 66

. 2.

, .

)(tX t

)();( 11 xftxf = 3.

)(tX 12 tt = , t =),;,( 21212 ttxxf );,( 212 xxf .

, . 3.2.1. 3.2.1

=)(tmx const=xm

const)( == xx DtD )()(),( 2121 xxx RttRttR ==

x

x

xD

Rr

)()(

=

1. ' - , t .

2>N2>N

2. (. 1 1). 1.

, , (3.2.1) .

)(tX

K

KN tKN

. (3.2.1) KN = , , ,

. 1 , .

KN < KN KN =

1. , - t , :

)(tX

N

N

),...,;,...,(lim 11 ++ NNN ttxxf . 1.

, 1 .

)(tX

T

Tt 1.

() , - ,

)(tX

N

N 0T=),...,;,...,( 11 NNN ttxxf ),...,;,...,( 0011 kTtkTtxxf NNN ++ , 67

,....1,0 =k

. . 2.

, -

)(tX )(tY

)( MN +N M

112 ,..., tttt N ; , , .

1

'

1

'

1 ,..., tttt M Ntt ,...,1)(tX ''1 ,..., Mtt )(tY

1. , , , =+ )',...,',,...,;,...,,,...,( 1111 MNMNMN ttttyyxxf

)',...,',,...,;,...,,,...,( 1111 ++++= + MNMNMN ttttyyxxf . 2.

, . )(tX )(tY

. 3.

, ( ), :

)(tX =)(tmx const=xmt

)(),( 2121 ttKttK xx = . 1.

. , , . (. 3.2.1).

. 3.2.1.

2. .

3. , .

, .

68

4. , , :

)(tX )(tY

t

122121 ),()]()([),( ttKtYtXMttK xyxyxy === . (3.2.2) .

, . 2.

, . .

, 3.2.2. 3.2.2

() ()

| ( )|R Dx x ; 1|)(| xr

0= : xx DR =)0( ; 1)0( =xr : )()( = xx RR ; )()( = xx rr

, )()( = yxxy RR ; )()( = yxxy rr 0= , -

. , , , ,

|| , (. 3.2.2).

)(xR

0

. 3.2.2. () () . 3.5.3. 3.2.2, .

3. . , , . , , .

. , .

69

. , , ..

3.2.2. 1.

. . Y t t X t g t( ) ( ) ( ) ( )= + , ( ), ( )t g t , ,

X t( )

)(tg ( )t . :

);()()( tgmttm xy += R t t t t R t ty x( , ) ( ) ( ) ( );1 2 1 2 2 1= D t t Dy x( ) ( ) ;= 2 r t t r t ty x( , ) ( ).1 2 2 1=

2. , . , . 1.

X t( )

]))()([(),( 22121 tXtXMttB xx = . (3.2.3) 2.

, X t( ) )()()( tXtXtX += ,

consttXMm == )]([ , (3.2.4) 2)]()([)( mtXtXMR += . (3.2.5)

1. .

2. ;0=m (3.2.6)

)()()(2)( += xxx RRRR ; (3.2.7) )]()0([2)( xxx RRB = , (3.2.8)

(3.2.3) (3.2.5). 3.

, .

3.2.3. , ,

, . 1. )(xfy = (

) 0),,( = yx , , .

1.

70

=

=0

),,(

,0),,(

yx

yx

(3.2.9)

. 2. ,

. 3. (3.2.9) ,

. 4. , ,

, yx, , , 0,0 ''

'

'

yy

xx. (3.2.10)

. fttAy 2cos)(= ( , )

f t

)(tAy = )(tAy = , t

kf =

,...).1,0(2

12 =+= kt

kf

1. ),(cos)( ttAy = , ,

)(tA

t ),( t ,t , )(tAy = )(tAy = .

. )(cos)( ttAy = , , )(tA )(t .

2. k

)(

xR ,

( ) , .

. :

1) , ( ) ( 0,1);

2) , )(xr )( x : ( ) drxk =

0

, (3.2.11)

.)( 0

dxk = (3.2.12)

.

71

3.2.4. 1. ))(),...,(()( 1 NtXtXtX =G

, , ) ( )

,( 21 ttK xG

12 tt = .

. . 3.2.3.

3.2.3

)()( lhhl RR = lhhl DDR )(

1)( hlr

3.3. ()

, . . . . 1. () ,

- , , , , - . 1.

. 2. ()

, ( 3)

=22

)(1

l.i.m.

T

T

xT

mdttXT

. (3.3.1)

2. , , )

)(tX

xm

,( 21 ttRx

=22

2

2

212120),(

1lim

T

T

T

T

xT

dtdtttRT

. (3.3.2)

. 1. (3.3.2)

0),(lim 2112

= ttRxtt , (3.3.3) 72

)

,( 21 ttRx

12 tt . (3.3.3) ,

(3.3.2). 2. (3.3.2)

: 0)(-1

1lim

0

= T xT dRTT . lim ( ) =Rx 0 . 3. 2 . ,

. , , . 3.

X t( )

Rx ( ) , Rx ( ) dttXtX

T

T

TT

)()( 1

l.i.m.02

2

0 += . 3.

Rx ( ) =++ T xxxT dRRRtT 0 002 0)]()()([-1 1lim - 0 .

3.4.

. . 1.

) :

),( 21 ttK x

(tX

)].()([),( 2*

121 tXtXMttK x = (3.4.1)

2. ) :

),( 21 ttRx

(tX

))]()())(()([(),( 2*

2

*

1121 tmtXtmtXMttR xxx = . (3.4.2) 1.

: )()(),(),( 2

*

12121 tmtmttRttK xxxx += . (3.4.3) 2. ,

: . 1 2 1 2( , ) ( , )x xK t t R t t=

73

3. ) ,

,

(tX1t 2t

0),( 21 =ttRx)()(),( 2

*

121 tmtmttK xxx = . (3.4.4) 4. )

, ,

(tX1t 2t

0),( 21 =ttK x)()(),( 2

*

121 tmtmttR xxx = . (3.4.5) 5.

) :

),( 21 ttK xy

)

)

)

(tX (tY

)].()([),( 2*

121 tYtXMttK xy = (3.4.6)

6. ) :

),( 21 ttRxy

(tX (tY

))]()())(()([(),( 2*

2

*

1121 tmtYtmtXMttR yxxy = . (3.4.7) 1.

) :

(tX (tY 1t 2t

),(),(

),(),(),(

2121

2121

21ttKttK

ttKttKttK

yyx

xyx

= , (3.4.8)

),(),(

),(),(),(

2121

2121

21ttRttR

ttRttRttR

yyx

xyx

= . (3.4.9)

7. ) , : , (3.4.9) .

)

)

(tX (tY1t 2t

),( 21 ttRxy 0),( 21 =ttRxy

8. ) , : , (3.4.8) .

(tX (tY1t 2t

),( 21 ttK xy 0),( 21 =ttK xy

9.

)()(

),(),(

21

2121

tDtD

ttRttr

xx

xx

= , (3.4.10) )()(

),(),(

21

21

21tDtD

ttRttr

yx

xy

xy

= . (3.4.11)

. .

. , . . 3.4.1.

74

3.4.1 ,

, ,

),( 21 ttRxy ),( 21 ttRx

),( 21 ttrxy ),( 21 ttrx)(tX )(tY

/ 1 ),( 21 ttRxy = ; = ),( 12* ttRyx ),( 21 ttrxy ),( 12* ttryx

)()(|),(| 21

2

21 tDtDttR yxxy ; 1|),(| 21 ttrxy( 1 1( ) ( )Y t aX t b= + ,

) a

b

2

3

Ntt ,...,1 Nzz ,...,1*

1,

),( nmnm

N

nm

x zzttR =

. 1, 3 , 2 .

3.5. 3.5.1.

1. T , :

x tT ( )

== )2exp()( tTjAtxT , (3.5.1) =

T

tT

j

T dtetxT

A

2

)(1 . (3.5.2)

(3.5.1) (3.5.2) () v

T

v =2 . A - AA = - , )arg( A= - . 1. A , ,

; ; - .

A 2. x(t) ,

: deAtx tj= )(21)( , (3.5.3)

75

dtetxA tj = )()( . (3.5.4) . ,

x(t), x(t) , x(t) [ )0()0(2

1 ++ txtx ] , x(t) - x(t) ( ).

)(tx . . 1. ,

(. 3.5.1). )(txT

)(tx

. 3.5.1. () ()

2. , )(A , )( ,

. )()( * = AA. (3.5.3) (3.5.4) f 2= ( f

) , . , .

)(tx )(A

)(+A 2.

, )(+A )(tx

=>=+=+ .0)0( ,0)(2)()()( A AAAA (3.5.5) 1.

: )(tx

)(tx =0

1( ) cos ( )

2A t d + + + ,

; : )(+A )( +)(+A = )(+A , =)( + ( )arg ( )A + ,

76

=>==

++ +

.0)(

,0)(2)()( )(

dttx

dtetx

eAA

tj

j

2. T :

)(txT

)(tx =0

2exp( )A j t

T

+ += + , +A + : +A = +A , =+ ( )arg A+ ,

=>==

++ +

.0)(1,0)(2

2

T

T

T

tT

j

T

j

dttxT

dtetxT

eAA

3.5.2.

. 30- . .. . 1. ()

, ' ':

)( fAx

)(tX )( fAx )(tX

dtetXfA ftjx2)()( = . 1.

, .

1. )

)

( fma

)(tX

xm ()( fmfm xa = , )( f -.

2.

)()(),( 12121 fffSffK xa = , (3.5.6) , ( ).

deKfS fjxx 2)()( = 3. (3.5.6) ,

. 4. ,

, .

77

5. , , , , . 2. ,

) ,

)(tX ( fS x

)(xK ' ': deKfS fjxx 2)()( = , (3.5.7) dfefSK fjxx

2)()( = . (3.5.8) (3.5.7), (3.5.8) . . )

)

( fS x

1. (3.5.7) (xK , ) ) :

( fS x

)(tX )(0 fSx

(0

tX xm

)(tX

)( fS x = + , (3.5.9) )(0 fSx

)(2 fmx .

)( fS x )(0 fSx

)

2. (3.5.8) ,

dffSRDx

xx )()0( 0== , (3.5.10)

. )(0 fSx

. (3.5.10) , .

)(0 fSx

3. ) , . .

)( fAx

( fS x

4. , ' )

)(0 fSx

(0 fAx

(0 tX

)(0 fSx

=TT

1lim )]()([ *00 fAfAM xx , (3.5.11)

78

T . )(0 tX . 1.

( ) , ) 0. (3.5.6) (

)(tX

( fS x

fff == 21 ). 0),( ffK a 2. )(xK

, , (3.5.7), , =

)(tX

)( fS x )( fS x . 3.

(3.5.7), (3.5.8) ) :

)(tX

)(0 fSx

(0

tX

dfRfS xx

2cos)(2)(0

0 = , (3.5.12) dfffSR

xx 2cos)(2)(

0

0= . (3.5.13) 4. (3.5.12), (3.5.13), (3.5.5),

, :

)( fS x+

=>=

+

,0)(2

,02cos)(4)(

0

0

fdK

fdfK

fS

x

x

x

(3.5.14)

dfffSK xx 2cos)()(0

+= . (3.5.15) . (3.5.14), (3.5.15)

, (3.5.7), (3.5.8), (3.5.12) (3.5.13) . 3.

, :

)(0

tX

)(0 fsx

)(0 fSx

)(0 fsx

=0

0 )(

x

x

D

fS, (3.5.16)

. 0x

D

1. (3.5.7), (3.5.8),

79

derfs fjxx

2)()(0

= , (3.5.17) dfefsr fj

xx

2)()( 0= , (3.5.18) )

)(0 fsx

(0

tX )(xr . 2.

: )(0 fs

x

0)(0 fsx

, , 1)(0 = dffsx )(xr . , .

3. , , . , 2, . 3. ( )

, : ef

dffSS

fx

e )(1

00

0= , (3.5.19) . )(max0 fSS x

f=

5. ef k :

k

ef 1= . (3.5.20)

3.5.3. . 1. ,

0ff e

. 3.5.2. -

2. ( , , . . 3.5.2), )

0f

(xR )(cQ 02cos f : 02cos)()( fQR cx = (. 3.2.2, ). 2.

,

)(tN

constN

fS x ==2

)( 0 . (3.5.22)

. 1. -,

)(

2)( 0 NK x = . (3.5.23)

(3.5.6). 1.

, (3.5.23) )(xR .

2. (3.5.23) 1 , , , , 0=k .

2. . (3.5.19), (3.5.22).

3. . (3.5.23).

. () , . , . 3.

, , .

)()()( tNtXtY = )(tX)(tN

1.

81

)(2

]|)([|),( 1202

121 ttN

tXMttRy = . 2.

)()()( tNtXtY += )()()()( 21 tNtXtXtY += , , .

)(tX )(1 tX

)(2 tX

3.5.4. , ,

. 1. ()

,

)(tX )(tY

)( fS xy

)(xyK : deKfS fjxyxy 2)()( = , (3.5.24) dfefSK fjxyxy

2)()( = . (3.5.25) 1.

, .

)( fS x

)(tX )( fS xy

2. ) , .

( fS x )( fS xy

)()( * fSfS yxxy = . (3.5.26) (3.5.24).

3. , )()(|)(| 2 fSfSfS yxxy . (3.5.27)

2. )(tX )(tY

)()(

|)(|)(

2

2

fSfS

fSf

yx

xy

xy

= . (3.5.28) 1. (3.5.27) ,

. , ,

)(2 fxy ]1,0[)(tX )(tY 20 ( )xyf f 0 = ,

, . 1)(2 =fxy 2.

. . , , , .

)(2 fxy)(2 xyr )(2 xyr

)(tX )(tY )(2 xyr 3.

, , .

)(tY

)(tX

)(2 fxy 82

, .

)(tY )(tX )(tY

)(1)( 2 ff xyxy = )(tY

3.6.

3.6.1. 1.

, : 111 ),()()( dttthtxty = , (3.6.1)

, , , .

)(tx )(ty ),( 1tth

1. () -: ),( 1tth

11122 ),()(),( dttthtttth = . (3.6.2)

. 2. -

(3.6.1). 3. - .

. . ),( 1tth 2. ( ),

. .

1. . , .

2. ,

),( 1tth

1, tt )(),( 1 htth = , 1tt = . (3.6.1) : dhtxdttthtxty )()()()()( 111 == . (3.6.3) 3. ,

. 4. ,

- . . dhtxdhtxty |)(||)(||)()(||)(| = , (3.6.4)

)(h 83

1. , , ,

)

)(2 K)(2 K

)

)(2 K )

(K)(th

)(1 K)(1 th )(2 th

)(K = , (3.6.11) )(1 K11211 )()()( dttththth = .

2. , , ,

(K)(1 K (2 K

)(K =)()(1

)(

21

1 KKK + . (3.6.12) 3.

)(h )(g . 5. )(g

, . . )(g

)(h : dhtg t )()(

0

= , dttdgth )()( = . (3.6.13) 4.

) , :

(K )( pH jpdehpH p +== ,)()(

0

. 1. , ,

) (

(K)( pH jp = ).

2. , ) ( pH 0 . . 5.

0 0

( ) ( )... ( ) ... ( )

M N

M NM N

d y t d x ta a y t b

dt dt+ + = + + b x t ,

, ma nb ),1,,1( NnMm == , , : , ,, . )0(y )0'y )0()1( My

3.6.1. - : ,

85

, , , . .

, . )(th )

)

(K - ,

(K

)(th

e . 3.6.1. 3.6.1

)(K

)(th

0K || 0 >||

t

t

K

)sin(

0

0K

+

+2

02

0

20

2

0 t

t

t

K

0

0

cos

2

)2

sin(

RC - j+ tet )(1 2

)( 0 + j t0cos et

t)(1

22

21 e

2 2)(1 t 2e

22t 22

22

21 e

20 )(

t

et

t

2

cos

2

)(122

0

24

86

: 1. =

1. . 3.6.3 , , , . ,

87

, , ( t). , , T, 1, 2, 3 . 3.6.3 :

)(th

0t

= Txy dhmm0

)( , = T T xy ddhhRR0 0

212112 )()())(()( , 1 1

0

( ) ( ) ( )

T

xy x 1R R h d = . , , .

Tt >0 2. 4 . 3.6.3 ,

. Tt >0

3.6.3. (3.5.23), ,

: 1

0

1

0 )()(2

)( dhhNRy += , )(2)( 20 KNS y = , )(

2)( 0 hNRxy = .

)(yR , yD)(yS

. 3.6.4. 3.6.4

)(yR ,

yD

)(yS )(

yR

yD

)(

yS

)sin(yD 2 020 NK 2 02

0NK

|| 0 >||

0cos2

sin

yD

2

2 020 NK 2

0

2

0NK

+

+22

22

00

00

0 RC- eDy

4

0N

)(2 22

2

0 +N

88

0cos

eD

y

2

0N

))((2 2

0

2

2

0 +N

2

22 eD

y 24 0N 22 204 eN

i 02 cos22eD y 22 0N 22 2)0(04 eN .

RC- ' . ' .

, 4 , , . 3 . 3.6.5.

3.6.5 3

3, 24, 40, 47, 49 16, 32, 42, 73, 75, 84, 100 5, 10, 15, 20, 23, 61, 88, 89, 105 27, 33, 35, 37, 38, 54, 55, 56, 66, 72, 74, 82, 85,

86, 92, 94, 97, 104

89

4. 4.1.

1. , )(tX Tt ,

- Nttt

, , ,

NX Nx

110 ,...,, NXXX 110 ,..., Nxxx},...,/{ 10 NN xxxP = . }/{ 1NN xxP

. 4.2.1.

2. ,

),...,/( 101 NN xxxF = ,1,),,...,/( 11 += MMNxxxF NMNN . 3. - NX

G

,..., 10 XXGG

,

),...,/( 10 NNM xxxF GGG = )/( 1NNM xxF GG . 1.

- ),...,( 1 NMNN XXX +=G . 2 3. .

. 2. N-

},...,{

00 NlNlXXP == = }{

00 lXP = = ==Nn lnln nn XXP1 1 }/{ 1 .

1. . 4. k mt l

(n m) nt},/{),( kmlnkl XXPnm ===

k , Sl . k l

( k , l LlkS ,1,; = ) , .

mt nt

),( nm LL 91

5. , : 1) '; 2) - . 3. . 4.

) (mPk k m- ),1( Lk = ),( nmkl l n- . ) (nPl l n-

)0;,...,1(),,()()(1

nmLlnmmPnP kl

L

k

kl == = . (4.2.2) (4.2.2)

),()()( nmmPnP = , P(n) - L , )(nPl Ll ,1= , n- . 5. k m- l n-

kl(m,n), k m- q r- ( nrm

6. P= P(N). Nlim

. . 7. , : 1) ; 2) ,

( ). 0t.

. 8.

, P(n) ,

nt

nt PnP =)( n. 8. ,

PP = . (4.2.6)

1. (4.2.5). 2. (4.2.6) ,

P .

3. .

. 4.2.1.

4.2.1

)(),( mnnm =

NPNP )0()( = PnP =)( n

PP = 1) )(lim NPP

N =2)

P

. - . 4.2.2.

l

. . 9. l ,

- , - l - . l ,

93

- , - . 10. l ,

, l , , ,

. 11. l , k ,

l k , k l , .

. 4.2.2. -

: 1 ; 2 ; 3 . l

12. , . 13. r

, ...,, 21 nn l . l r , , , .

1>r1=r

14. l , , . . 15. (),

. . ( ,

) ( ). L

4.3. 1. )(),...,(

00 NNtXXtXX ==

)(tX Ntt t

1. (4.3.1) , :

)(tX

= = Nn nnnnNNN txtxtxfttxxf 1 1100100 );/;();(),...,;,...,( , (4.3.2) = ) .

);/;( 11 nnnn txtx ;/;( 111 nnnn txtxf1nx 1nt nx nt

2. (4.3.1) , - , . . 4.3.1.

4.3.1

' 0);/;( txtx = 1);/;( dxtxtx )();/;(lim xxtxtx

tt=

= xdtxtxtxtxtxtx );/;();/;();/;( 0000

. .

1. )

)(tX

;/;( 00 txtx 0x 0 t x t

0 001 00000 );/;(! );();/;( x txtxtx Atxtxt = = , (4.3.3) [ ] ));/();/((1lim);( 000000

000 txtXtxttXM

ttx A

t+= .

. );/();/( 000000 txtXtxttX + , t .

);( 00 tx A - .

1. : .

)();();( 00 tXttXtX + ++= xdtxttxttxtxtxtx );/;();/;();/;( 000000 . (4.3.4)

++=+ xdtxttxttxtxttxtx );/;();/;();/;( 0000000 (4.3.5) 95

( , ). (4.3.4) (4.3.5), :

[ ] .);/;();/;();/;( );/;();/;( 000000 0000 xdtxttxttxtxttxtx ttxtxtxtx +++= =+ (4.3.6) (4.3.6) :

.);/;(

!

)'(

);/;();/;(

0

00

1

0

000

x ttxtxxx

ttxtxttxtx

+==++= (4.3.7)

(4.3.7) (4.3.6), ( ). (4.3.3).

t0t

2. (4.3.3) , 3 , () :

),;/;();(2

1);/;();(

);/;(

002

0

2

0000

0

00

00

0

txtxx

txbtxtxx

txa

txtxt

+==

(4.3.8)

) , .

;();( 00100 txAtxa =);();( 00200 txAtxb =

3. , (4.3.8), [ ] [ ]);/;();(2

1);/;();();/;( 002

2

0000 txtxtxbx

txtxtxax

txtxt += (4.3.9)

.

1. (4.3.8) (4.3.9) . 2. (4.3.9)

[ ] [ );();(2

1);();();( 12

2

11 txftxbx

txftxax

txft += ] . (4.3.10)

. - 0tt >0000011 );/;();();( dxtxtxtxftxf = .

, (4.3.9) ) , , (4.3.10).

;( 001 txf 0x

4. , (4.3.8), (4.3.9), . 5.

, ) ,

;/;( 00 txtxt 0t 0tt = : );/();/;( 000 xxtxtx = .

1. . a b

96

2. (4.3.10) , (. 3.2) [ ] Cxfxaxfxb

dx

d += )()(2)()( 11 , (4.3.11) C , . 2.

, . . .

);/;();(),;,( 112211121212 txtxtxfttxxf = . );,(),;,( 21221212 xxfttxxf = ,

, )();( 11111 xftxf = 12 tt = . , );/();/;( 121122 xxtxtx = , . 6.

, : 1) ; );(lim)( 11 txfxf

tf =

2) .

1. .

2. .

4.4. 4.4.1. ()

1. )(),(),( tntxgtxh

dt

dx += , ,

),( txh ),( txg

yxLtygtxgtyhtxh + ),(),(),(),( )0( >= constL , )(tn . ,

. (. 4.4.1) m.

. . . t . , , ,

)(tn

- ),( ttR , ,

)(tn[ ] ;0)( =tnM . (4.4.1) )(

2

1),( 0 ttNttR = . (4.4.2)

97

)(tv

)()(

tndt

tdvm = . (4.4.3)

, 0=t , , (4.4.3) , .

. 4.4.1. ,

2. , )(tv

),()(

tndt

tdv = 0)0( =v , (4.4.4) . )(tn

1. (4.4.4) , 1

0

1 )()( dttntv

t= . (4.4.5) 2. . 4.4.1. . ,

, . 3. , ,

,

)(tX T

Nttt

4.4.1

/ 1 [ ] 0)( =tVM 2 [ ]

2)()()( 021

0

21

0

2 tNddnnMt

t t

v == 3

2

1

00

1( ; ) exp( )

vf v t

N tN t= 4 dntvtv t

t

+= 32

)()()( 23

5 0);( =tva 6

2);( 0

Ntvb =

7

0 1 0

1 0 0 1 1 1

( ,..., ; , , )

( ; )... ( ; )

N N N N N

N N N N

f v v v t t t

f v t f v v t t

1 == 8 [ ] )())(),...,(/( 110 = NNN tvtvtvtVM : 1. 3 , ,

' . )(tn )(tv )(tn

2. 4 , , .

3t

2t

2t

3. 5 (4.4.5) (4.4.1), 6 (4.4.5) (4.4.2).

4. 7 , .

5. 8 4 .

4.4.2. . ,

)(tX

)()()(

tntxdt

tdx =+ , (4.4.6) , , , .

)(tn

. , , , (4.4.6) .

)(tX

)(tn

, , . )(tX , (4.4.6),

, C . (4.4.6) tCetx =)( dneetx tt =

0

)()(

(

99

(4.4.6)). (4.4.6) 0)0( xx = dneeextx ttt +=

0

0 )()( . (4.4.7)

0, 1123 =>> tttt . (4.4.7) , dneeetxtx t

t

ttt)()()(

3

2

323 )(

23 += . (4.4.8) (4.4.8) ,

, , .

)(tx 3t

1t 2t

. 4.4.2.

2. = eR 2)( .

4.4.2

textm = 0)(

[ ]tet 222 1)( = , 4 022 N=

0),min(;

),1(),(

2112

22

21 == = ttttt eettR t

( ) t = eR 2)( ( ; ) ( )a x t x t=

2);( 02

Ntxb =

. , , , . 4.4.2.

. 4.4.2. , ,

() )(tm )(2 t

)(R () 100

1.

2

1

2

0

2

11 );(

2)];([

);(

x

txfNtxxf

xt

txf += . '

= )(2 ))((exp)(2 1);( 22

1t

tmx

ttxf .

2. .

= 22

12

exp2

1)( xxf .

4.5. ' 4.5.1. '

(4.3.8) (4.3.10) . '

. 1.

) ,

;/;( 00 txtx);(1 txf x .

2. . 1. [ );();(

2

1);();();( 11 txftxb

xtxftxatxG ]= (4.5.1)

. 1. '

, t x . ( )

);(1 txf

x . - , . , , ,

t

);();( 1 txftxa

[ );();(2

11 txftxb

x ] . 2. : 1) .

; 0);();( 21 == txGtxG

1x 2x

2) . .

0);();( 2111 == txftxf1x 2x

2. 0);();(1 =+ txGxtxft . (4.5.2)

101

1. (4.3.10), .

2. , , , , .

);(1 txf

);( txG

3. (4.5.2) , consttxG =);( . (4.5.1) [ ] Gxfxaxfxb

dx

d2)()(2)()( 11 = . (4.5.3)

. ' (4.5.3) = xx zxxx dzdyyb yaxb Gdyyb yaxbCxf 1 11 )( )(2exp)(2)( )(2exp)()(1 . ,

, , .

G

C 1x )(1 xf )(1 xf

. ' . , , , , , , , . , .

4.5.2. ' '

, . '

(4.3.10) 2

0

01

2

0

0101

)0,(,

)0,(),0,(

x

xf

x

xfxf

0),(),( 11 == tdftcf . '

. . ;

dxct ,0xt tntn =

,cxixi += === xcdIIin ;,0,...;1,0 , [ ]* . . );(1 txf , (4.3.10)

:

102

).;();();(2

1

);();();();();(2

1);(

12

2

112

2

1

txftxax

txbx

txfx

txatxbx

txfx

txbt

txf

++ +=

(4.5.4)

: );(1 txf

t

ff

t

txf ninitx ni + ,1,,1 );( ,

x

ff

x

txf ninitx ni + 2);( ,1,1,1 ,

2

,1,,1

,2

1

2 2);(

x

fff

x

txf nininitx ni + + ,

. ),(1, nini txff = , (4.5.4)

: );(),;( txbtxa

0,,1,,,,1,1, ++= ++ nffff ninininininini , (4.5.5) ninini ,,, ,, , . );(),;( txbtxa (4.5.5)

. ),(1 txf '

x t '. ' , , , , '. , . ' ( ). . , 0);( txb

);(

2

txb

xt

, , ,

SxN )}

)({ ktXP

(,),( 10 NtXtX Sxx N 10 ,,

},...,;,...,/;{ 1010 NNNN ttxxtxP = . ;/;{ 11 NNNN txtxP 1.

= })(tPk = k t ),1( Lk = )',( ttkl l ' . = }t )'(tPl )'({ ltXP = l : 't

)'(tPl ==Lk 1 )(tPk )',( ttkl . (4.6.1) . 4.6.1. 1. 1 4

, 1, 2 4 .

2. , , , - t , .

4.6.1

/ 1 ' )',( ttkl 0

2 = ==Ll kl Lktt1 ,1,1)',( 3 klkl

tttt =+ ),(lim 000 *)

4

)',( 0 ttkl = , =Lq qlkq tttt1 0 )',(),( '0 ttt

2. , (4.6.2), () .

)(taql

2. '

== L

lq

qqlll tata

1

)()( . (4.6.3)

2 (4.6.3). )',( ttql ),1,( Llq = ' . ,

t t

ttt +=' ttattt llll ++ )(1),( ; (4.6.4)

ttattt qlql + )(),( ( lq ). (4.6.5) (4.6.4), (4.6.5)

, , t , , , (4.6.2). (4.6.3) (4.6.4), (4.6.5) . 3.

: 0

1

000

0

,,1,,),()(),( ttLlktttattt

L

q

qlkqkl >== = . (4.6.6) 3 2. 4. )

' (tPl l

t Lq ,1= : t

LltatPtPdt

d L

q

qlql ,1,)()()(1

== = . (4.6.7) . (4.6.2)

. (4.6.1), (4.6.7).

)( 0tPk

k

. (4.6.7) , ' )

.

L

(tPl

)(taql

3. , )',( ttkl , ' : t t

)',( ttkl = )( kl , tt = ' .

1. (4.6.4), (4.6.5) , : . constakl =

2. (4.6.2), (4.6.6) :

105

LlkaL

q

qlkqkl ,1,,)()(1

== = , Llka

L

q

qlkqkl ,1,),()(1

== = . 4. ,

=)(tP },1),({ LltPl = , , .

)(lim tPPt =

. . 5.

, : =

)(tP

)(tP P .

1. .

2. .

4.7.

' , , , , , , , . ' . , , ' ,

. - , . , , , ,

. , ' . , , .

. 4.7.1 , .

4.7.1 ,

),...,/( 101 NN xxxF = ,...2,1),/( 11 = NxxF NN

),...,/( 101 NN xxxF = ,...2,1),,...,/( 11 = NxxxF NMNN

106

. 4.7.1

),...,;,...,/;( 10101 NNNN ttxxtxF = );/;( 111 NNNN txtxF

),...,;,...,/;( 11111 NNNN ttxxtxF = );/;( 111 NNNN txtxF

0xm = , = eR 2)(

textm = 0)( ,

0),min(;),1(),( 211222

21 === ttttteettR t

()

-

4.8. ,

)(tX

T Tt ,

S

),1( LlSl = , ...

10

),1,( Lkj = . ) (twt i ),1( Li =

),()(1

tftw ik

L

k

iki == . (4.8.1) .

. . (4.8.1) ,

iT

m

i k ),1( Lk =

,1

iki T

L

k

ikT mm == (4.8.2) . dttftm ikTik =

0

)(

2. jk

)(tf jk

),1,( Lkj = . )(tij ,

t

j , 0

0 =ti , dtftt kjL

k

t

ikikiijij )()()()(1 0

+= = , ),1( Lji , (4.8.3)

,

{ tTPt ii >= )( } ti : =

t

ii dttwt )()( ,

)(twt i . .

. - . )(tij

1. (4.8.3) , (

t

j = i ), , , t

,)(

1k

j

T

L

k

k

Tj

ij

mP

mPt == (4.8.4)

),1( LkPk = k ,

jTm

k , (4.8.2).

. ' (4.8.4) t i , . 0t 4 . 4.8.1.

4.8.1 4

24, 40, 47, 49 32, 41, 42, 73, 77, 78, 99 7, 10, 15, 20, 22, 30, 61, 87, 88, 89, 105 6, 38, 57, 82, 85, 86, 92, 94, 95, 98, 104

109

5. 5.1. 1. ,

. . . 1. :

; ; .. 2. (,

) , . 3. , ,

, , , . 4. ,

, , . 2. '

, ', , '.

. . 1.

(. 5.1.1): ),...;;,(

111...1 MMMkkttttttP

M++ ,

, ],[111ttt + , , ,

, , , ; ],[

MMMttt +

Mkk ,...,

1

N - ;

),...,(1 NN

ttfN

tt ,...,1

N - ),...,( 1 NNf 112211 ;...;;0 === NNN ttttt .

. 5.1.1.

5. , ),;;,( 111...1 ++++++ MMMkk ttttttP M Mkk 1

110

[ ] ++++++ MMM tttttt ,;;, 111

M , =++++++ ),;;,( 111...1 MMMkk ttttttP M

.),;;,(...1 MMMiiikk ttttttP M ++ 6. ,

),( tttPk

+ ],[ ttt +

: )2( k ),(

1tttP +>>+ ),(

1tttP ,....3,2 ),,( =+ ktttPk

. . 7. (

), - , , , , , ,

),;...;,( 111...1 MMMkk ttttttP M ++ = . = +Mm mmmk tttP m1 ),(

. 5.1.2.

. 5.1.2.

1. :

),...,( 1 NNf = . =Nn nnf1 )(1 )( , (. 5.4).

2. .

3. 1. , ,

, , . 2. , , ,

, , .

111

3. , , , : .

. ' . ' T , . 2.

, , , (. 5.1.3):

;

),(21

ttm ),(21

ttD

],[21

tt

n ;

nm nD

1n )(t , ; )(t , , . .

),( Tttm +

),( TttD + t , ),( Tttm + = , = . )(Tm ),( TttD + )(TD 8. )(t

t ),0()( tmtm = ],0[ t :

t

tmttmt

t += )()(lim)( 0 .

. 5.1.3. ()

112

9. ( ) )(t

= +1 ),(k k tttPt

: 0tt

tttP

t kk

t += = 10 ),(lim)( .

1. , , == )(;)( tt .

2. , )(t )(t . 3. )(t , )(t .

5.2. 1.

. 1. .

, )(TPk

],0[ T (. 5.2.1), :

k

.!

)()( T

k

k ek

TTP =

. 5.2.1. K ],0[ T

2. . (. 5.2.2): 1) ; 2) 0

; 3) )(1 nnnn tt += + ,

, , 1+nt

nt ],[ nnn tt +

, )(),(),( 1011 nfff : = ef )(1 .

. 5.2.2. : - ; 0 ;

n

113

, , . 5.2.1.

5.2.1

T TTm =)( T TTD =)( , 0 , n 10 === nmmm , 0 , n 210 === nDDD

3. . . :

, .

5.3. 1. , M

),,1( Mmm = .1

== Mm m 1. ,

. , , M , M .

2. , 5 10 . 1.

. 2. )(t . ,

, : ),(

21ttP

k],[

21tt k

),(2121

21

!

),(),(

ttak

k ek

ttattP

= , . = 2

1

)(),(21

t

t

dtttta 3.

( ],[

21tt

)(t ).

],[21

tt

],[21

ttt :

cpef cp =)(1 ,

cp

)(t , . 114

. )(1 f . . 5.3.1.

5.3.1 ],[

21tt

( )

],[

21tt

),(),(2121

ttattm = ],[

21tt ),(),(

2121ttattD =

cp

m 1=

21

cp

D =

5.4. . ,

1,2,...)=( nn ,

),...,( 1 NNf = . =Nn nnf1 )(1 )(. ,

. . . . .

5.4.1. 1.

, , . )()()( 1)(1)2(1 fff N ===

1. :

N ,,

1

, . 2. (.

1 3 5.3). 1. 5.4

, . , .

2. N . , , L

115

, . . .

11,..., NLL

3. - N == Nn nNN fff 2 11)1(11 )()(),...,( , ) (

( 1)1(

1 f0=t , ,

), 1

t

)(1 f (. 5.4.1).

4. )( 1)1(1 f )(1 f , , .

. 5.4.1. : 1

, , ; 0t= t1

- ; 0 ; n .

1. )( 1)1(1 f )(1 f ' : = 1

0

11

)1(

1 ])(1[)(

dff ,

m1== . 2. 0

, (. . 5.4.1).

)( 0)1(

1 f 1. n

, , ]

1+nt,[ nnn tt + (. . 5.4.1),

. ) , )(~1 nf ( 1)1(1 f)(1 f .

116

. . 3 ().

, . . , ( ). , .

. , , , .

5.4.2. 1. - ,

, k (. 5.4.2).

k

. 5.4.2. ( 3- ): ; n

1. ,

k

, ,

k : == kn n 1 ; nef n

=)(1 , 0n . 1. 1, 1 . 2. . 3. :

, . k

k 1=k . , k . 1. - k

0,

)!1(

)()(

1 = ekeek eekf , . 2. -

k )(

,TP

kl

l ],0[ T T

kl

lkn

n

kl en

TTP +== 1)1(, !)()( .

. 5.4.1.

117

5.4.1

km ke =, == kn nkm 1, 1

2, kD ke = == kn nkD 1 2, 1 k

= 11

1

= = kn n

. .

k

, . ,

;

2

D

mk =

1

m = , . m D

5.4.3. 1. ,

k

,

k

knn 1,=, , nnef

nn

=)(1

.

1. ,

k

k constn

== . 1.

k

==

=

=

k

nk

inin

ni

k

nnk

nef

1

1,

1 )(

)(

.

. 5.4.1.

118

5.4.4. . ,

: = + (. 5.4.3).

. 5.4.3.

1. ' , ' . , . .

2. : 1) :

. 5.4.4.

)(),( tPtttP kk =+ ),( 21 ttPk

===

N

n

n

n

NN

f

f

1

)(

1

1

)(

),...,(

N

N

kk

kk

PP

P

...1

1...= =

)(

)(

1

)(

1 ff nn= =

= i

ke =

1-

k -

-

M

N

N

kk

kk

PP

P

...1

1...= =

5.5. ' ,

. ,

. , , ' , INTERNET, , , , , , , . , , ,

, : .

120

. . . .

. '

, , , . ,

. 5 . 5.5.1.

5.5.1 5

24, 40, 47, 49 16, 32 20, 23, 88, 89, 105, 109, 113 38, 79, 83

121

6. 1. -

(). 2. ,

, .

6.1.

6.1.1. 1.

(). 1. , .

( ) .

. ,

( ), .

X

nx )(tX

nx t ,...2,1=n 2. ,

, , . , . 2. . 3. ,

, .

Nxx ,...,1

N

4. . 5. ,

. N

6. , , , .

Nxx ,...,1 )(1 xF

)(1

xF

7. , , -

N

N

G = = ,

.........

...

...)2()2(

1

)1()1(

1

N

N

xx

xx

),,(1 N

XX

( ). 1. ,

G , xG , G .

122

2. (

),,(1 N

XX n

X Nn ,1= ) , . )(

1 nxF

3. - )N (xFN G G

:

)(1 n

xFn

x

)(xFNG

= . =Nn nxF1 1 )( 8.

xLG

G . . ,

xLG

1( )nf x nX

G:

xL G = . (6.1.1) =Nn nxf1 1 )(

, .

)(1 n

xf

nX

. . XG . 9.

),...,(1 N

XXYY = XG . . .

, , .

),...,(1 N

XXY

xL G

)(1

xf

10. (), .

1. , .

2. , , , ,

)(1 xF

m D . , , , ,

)(*

1 xF*m

*D . 3. ,

, . , .

123

6.1.2. . .

1. () , (

nx Nn ,1= ).

2. , :

j

jN jx N

N

Nj

j= .

1. , , (

j

G ), .

2. , .

jp X

jx

3. : j

1) =1;

jj

*2) N .

j

jp

3. :

jx

j Jjx

jj,1),,( = .

4. , ' ( , ),

jx *

j j = J,1 .

1. (6; 3; 5; 2; 3; 2; 5; 2; 5; 5). , , : (2; 2; 2; 3; 3; 5; 5; 5; 5; 6). (2;

10

3), (3;

5

1), (5;

10

4), (

10

1;6 ).

. 6.1.1.

. 6.1.1.

. , J . , , , , .

jx

124

5. X, .

1. X , . , 10 20.

2. , . , ,

: .

3. , , , . 6. , -

, , - , : =

*

ip i

iN

i N *i

pN

Ni

.

7. , . 8.

i -

*

if

*

ip

ix : =*

if

i

i

x

p * . 1. ,

, , , .

2. .

*

if

9. , . ,

- . i *i

p

. . .

2. 100 . (. 6.1.1), (. 6.1.2) (. 6.1.3).

6.1.1

[-4,5:-3,5]

(-3,5:-

2,5]

(-2,5:-

1,5]

(-1,5: -

0,5] (-0,5: 0,5]

(0,5:

1,5]

(1,5:

2,5]

(2,5:

3,5] *

jp 0,012 0,050 0,144 0,266 0,240 0,176 0,092 0,020

125

. 6.1.2.

. 6.1.3.

10. , )(* xF x X xX .

1. , .

)(* xF

)(xF

2.

)(* xF

x .

3. , .

)(* xF

11. ,

)(* xF

xX .

. 6.1.4.

126

. 6.1.4. ( 2)

6.1.3. 1.

),...,( 1 NXXX =G (),

N*m

*m = =Nn nXN 11 . (6.1.2) 2.

),...,( 1 NXXX =G ( )

*

xm = =Nn nXN 11 . (6.1.3)

3. ),...,(

1 NXXX =G N

* = = Nn xn mXN 1 * )(1 . (6.1.4) 4. ),...,( 1 NXXX =G

*

xD = = Nn xn mXN 1 2* )(1 . (6.1.5)

6.2. .

. . , .

N

. .

6.2.1.

127

, . . X <

. (6.2.1) . X

. )(xF = )()( 2 xdFmxD xx . ,

xx m xmx xx xdFmxD )()( 2 . . 22)( >

xmx

)()( 22 >= > xmxx mxPxdFD x , (6.2.1).

6.2.2. .

(

NXX ,...,

1

xx mm ,...,1

xx ,...,DD 1 NnCD nx ,1, = NyN mY . (6.2.2) . .

=ym yD NY =Nn nXN 11 : == Nn xy n mNm 11 ; == Nn xy nN DND 121 . CD

nx ,

N

CD y .

, N

Y

128

( ) 2 2NN yN y D C Y m N > . N (6.2.2). . X

. .

X, . (6.2.2) :

NXX ,...,

1

01

lim1

= >= mXN Nn xnN . .

.

6.2.3. . ,

' ' . . N

ap

N

Na

: a

p

0lim = > pNN aaN . (6.2.3) . ,

. : 1 0 =(1 ).

NXX ,...,

1

ap

aq

ap

= ,

nX

nxm

ap

nxD =(1 )

ap 2

ap +(0 )

ap 2

aq =

ap

aq < .

N

XX ,...,1

N

Na

A . , , (6.2.3).

6 . 6.2.1. 6.2.1

6

3, 11, 24, 47, 49, 70, 93, 108 13, 16, 32, 42, 80, 91 5, 21, 22, 45, 46, 50, 59, 61, 90, 101, 105 1, 28, 31, 52, 55, 56, 62, 65, 66, 71, 84,

97, 106, 107

129

7.

7.1.

, . : ( ) , , () . .

XG

N . =NXX ,...,1 () X

),...,/( 1 KxF ),...,;( 1 KxF K ,...,1 . ,

, Nxx ,...,1 X X N1 ,..., . K ,...,1 1. ),...,/( 1 KxF ,

, K ,...,1 , . ),...,;( 1 KxF 2. , ,

. .

( ), G

),,( 1 K G = *G G . ,

K,...,1

),...,/( 1 KxF ),...,;( 1 KxF : X X N1 ,...,K ,...,1

).,...,(

................................

);,...,(

1

111

NKK

N

XX

XX

==

1. , ;

XG

xG K,...,1

, . K ,...,1 2. . , , .

3. , 7.5.

7.2. 7.2.1.

1. G G , / - , :

GG

130

GGG = ]/[M . (7.2.1) .

( ) GGGG = ]/[0 M . . G , , . XGG 1.

]/[2

2 GGG = M . :

2

0

22 += , (7.2.2) [ ]( )[ ] 2**2 = MM .

2 , , , .

22

0 G 2. . , .

X

3. , , , , . , , , . .

7.2.1. 7.2.1

, == Nn nx XNm 1* 1m m x x= = Nn xnx mXND 1 2** )(11 = = Nn xnx mXND 1 2* )(1xD

x ( ) X

= = NnN xmnXNkx 1 2)*(11* =+ = Nn xN mnXNkx 11 2)(1* xyR

= = Nn ynxn*xy mYmXNR 1 ** ))((11 = = Nn ynxn*xy mYmXNR 1 ))((1

131

. . Nk

=

2

2

1

2

1

N

N-N

k N , - ( 2).

. 7.2.2.

)( Nk

7.2.2 Nk

N N N Nk Nk Nk Nk NkN N

3 1,1284 6 1,0506 12 1,0230 25 1,0104 40 1,0064

4 1,0853 7 1,0423 15 1,0181 30 1,0087 45 1,0056

5 1,0640 10 1,0280 20 1,0134 35 1,0072 50 1,0051

GG 2. , G /G N , : GGG = ]/[lim MN . (7.2.3)

7.2.2. GG. , :

0}/{lim => GGGPN , (7.2.4) N 0> ' , . 1. ,

( ). (7.2.4) 2*lim / 0

NM = G GG .

2. , . N

7.2.3. . -. . 1.

*

e , *e N , - : *i

]/)[( 2 eM < , =1, 2,. (7.2.5) ]/)[( 2 iM i 132

1. ])([ 2 M D [ * ]. .

2.

]/[ eD < , i =1, 2,. (7.2.6) ]/[ iD 3. , ,

. , , , .

X

2. ,

l

*

e * :

]/)[(

]/)[(2

2 = MMl e . (7.2.7) . [0,1].

, , =1. l 3. ,

*

N . 4. )('l

1 2 :

]/)[(

]/)[()('

2

2

2

1 = MMl . (7.2.8) )('l. , . [ ]/*D , .

*XG

xG 1. N

. )/(xf N G , xG .

/*[ ]( )[ ] // 2** MM

+

/)/(ln

1

2

2

0

XfM N

G[ ]/*D = , (7.2.9) 133

D ( ), ; M , , ; XG* 0 .

. [ ] xdxfxM N GGG )/()(/ ** = . (7.2.10)

0

* )/()( += xdxfx N GGG . (7.2.11) (7.2.11) :

+= 0* 1)/()( xdxfx N GGG . (7.2.12)

= )/(ln)/()/( xfxfxf NNN GGG . (7.2.13)

(7.2.12) :

+= 0* 1)/()/(ln)( xdxfxfx NN GGGG

+= 0* 1/)/(ln XfM NG

. (7.2.14)

1)/( = xdxf N GG . (7.2.15)

(7.2.15) : 0

)/( = xdxf N GG . (7.2.16) (7.2.13), (7.2.16)

0)/()/(ln = xdxfxf NN GGG ,

0/

)/(ln = XfM NG

. (7.2.17)

[ ]/*M : (7.2.17) [ ] 0/)/(ln /* = XfMM N G . (7.2.18) 134

(7.2.14) (7.2.18): [ ]( ) /)/(ln /** XfMM N G + 01 = . (7.2.19) ,

(7.2.9). 5.

= /)/(ln2

XfMJ NN

G . (7.2.20)

1. : = /)/(ln 2

2 XfMJ NN

G. (7.2.21)

. , (7.2.13):

2

2

2

2 )/(ln)/(

)/(ln)/()/( += xfxfxfxfxf NNNNN GGGGG . (7.2.13):

2

22

2

2 )/(ln)/(

)/(ln)/(

)/( + = xfxfxfxfxf NNNNNGGGGG

.

xG . (7.2.16),

xdxfxf

xdxfxf

NN

NN GGGGGG )/()/(ln)/()/(ln

2

22 = ,

=

222

)/(ln)/(ln xfMxfM NN GG . (7.2.20), (7.2.21). 2. (7.2.20)

:

[ ]NJ

D

2

0

*

1

/

+ . (7.2.22) 3. [ ]

NJD

1/* . (7.2.23)

4. ( )[ ] /2* M [ ] = /*M , :

[ ]NJ

D

2

0

*

1

/

+ ( )[ ] /2*M . (7.2.24) 135

, * . * 5. (7.2.24) , 0 (-1). , . 6. :

1

2

01

JN

+ ( )[ ] /2*M [ ] /*D , (7.2.25) : 1J

= 2121 )/(ln XfMJ . (7.2.25) (7.2.21), (7.2.24). 7. (. 1) , . 6. ,

e( )[ ] /2* M

NJ

2

01 += . (7.2.26) 2. 6

, :

e += )/(ln1* xfJ NNeG

. (7.2.27)

2, , 6. 6. ,

e[ ]( )[ ] [ ] 1// ,/ *2* * == = Nee e JDM M (7.2.28)

1. 1 6 . 2. 6

(7.2.24), 1 (7.2.26).

3. 6 , (7.2.24) , .

4. 6 , , , , , , .

5. , , .

136

,

20 , (.

(7.2.2)) 2 .

6. 6 , )/(ln xf N G (7.2.27), )/( xf N G : [ ] )()()( 21 xhkkexp e G +)/( xf N G = ,

)(xhG

xG )(1 k )(2 k , . 7. , , ,

. , , , , . . . .

*G GK . I 7.

),1,(

)/(ln)/(lnKji

XfXfMI

j

N

i

N

ij = = GGGGG

, (7.2.29)

G - K -

N

)/( GGXf N - .

N

XG 1. ,

.),1,()/(ln

)/(ln)/(ln

2

KjiXf

M

XfXfM

ji

N

j

N

i

N

= ==

GGG

GGGGG

(7.2.29) : ),1,(

)/(ln2Kji

XfMI

ji

N

ij = = GGG

. (7.2.30)

2. -

XG

N

)(GGS , R

T1( ) ( ) ( , 1, )ij

i j

S SI R i j

= K=G GG

. (7.2.31)

]/))([(* GGGGGG TMR = (7.2.32)

, 1.

137

3. xG .

XG

),...,( 1 K =G D KK

),1,()(1 0 Kjid ij

ij =+= G , ),1()(0 Kii = G )(0 GG .

1T TU R U U D I D U G 1 , (7.2.33) U - . K

1. (7.2.33) .

2. (7.2.33) 1T TU R U U I U G . (7.2.34)

3. (7.2.34) , *G11 = URU T G 1, .

=UIU T*

eG 8. , **1 ,, Kee , ,

eR G1

1T TU R U U I U =G . (7.2.35)

1. , (7.2.35) , .

**

1 ,, Kee N 2. , (7.2.35),

. ),,( **1* Keee G = :

TbI 1* += GG , (7.2.36) ),1(

)/(lnKi

xf

i

N = GG

b . 3.

. .

4. , , (7.2.35), .

*

eG 8. , **1 ,, Kee ,

N*

iG**1 ,, Kii URUURU

ie

TT 11 > GG ),2,1( =i . (7.2.37) .

xG 4.

N , . ( ) )(1 f 1+N

138

),(1 xf N G+ , 2

1

2 );( + xf N G + );(1 xf N G xG , 0);()(lim 1

* = + xdxf N GG . [ ] 12* )( NJM , (7.2.38) , : NJ

= +2

1 );(ln XfMJ NN

G =

+ 212 );(ln Xf

M NG

, (7.2.39) =

M (7.2.38) , (7.2.39) XG* .

1. (7.2.38) , .

* 2. (7.2.38), (7.2.39)

KG .

UIUURU TT 1*G , (7.2.40)

KK : *GR [ ]TM )()( ** GGGG*GR = , (7.2.41) I

= ++ jKNiKNij XfXfMI );(ln);(lnGGGG

, (7.2.42)

*G (7.2.41) G , (7.2.42) XG G .

3. (. 8, 1 8 8).

4. G 8 );( xf K G

G . 5. ,

.

7.2.4.

139

. G G , - )/,...,( *1 GNN xxfN X G .

1. , , )/( *GGxf N , GG , .

mN

xxn

N

* = =1 1. n , m

Nxx

n

N

* = =1 1 11N n . 2. , ' .

. 3. . 4. ,

. , ' , . G.

),,( 1 K G = )G /,,( 1 NN xxf : , (7.2.43) )()/ 111 NNN xxhfxxf ,...,(=)/,...,(

xG ) , /1 (f ,

, h x xN( 1,..., ) . G. : )xf K GG /(G , xG , )(xGG .

. 7.2.3. 7.2.3

X

- (

)

,

,

-

-

-

== Nn nx XNm 1* 1 xm

+ + + +

= = Nn xnx mXND 1 2** )(1 + ( ) 1l + xD = = Nn xnx mXND 1 2** )(11 + + + )( 12 ll <

140

N

Np A=* + + + +

. . )( 12 ll <

7.3. 7.3.1.

. );(1 xf X , . , . , ,

X - : = dxxfxm );( . :

)( =m . (7.3.1) ' (7.3.1) ,

)(1 m= , (7.3.2) , 1 .

(7.3.2). : mv

*= = Nn vnv XN 11* 1 . 1.

. 2.

*

, .

X

K K,...,1Kmm ,...,1 ,...,1

(7.3.3) =

=).,...,(

.................................

);,...,(

1

111

KK

m

m

' , , , , *mm ,...,*1mm ,...,1

=

=).,...,(

...............................

);,...,(

**

1

*

**

11

*

1

KK

mm

mm

3. , , . , ,

141

.

,1

7.3.2. . N - )(GG

XL

( ) ,

X X N1 ,...,G , . *G G G , .

*G. G , .

X

. , )(GG

XL

: == 1 1 )/()( n nX xfL GGG . (7.3.4) (7.3.4). == Nn nX xfL 1 1 )/(ln)(ln GGG . (7.3.5) .

(7.3.4) (7.3.5) . *G

==

=

=N

n K

n

N

n

n

xf

xf

1

1

1 1

1

.0)/(ln

.................................

;0)/(ln

G

G

(7.3.6)

1. (7.3.6) '. ' , . ' ', - ( ).

2. . (7.3.6)