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2003
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p(A)
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3) p( ) 1 .
A1
p(A) 1 , B
, A1
(
A2 A1
p( A) 1 A .
0
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A2
) p ( A2 / A1 )
p( )
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p ( A1
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p ( A1 A2 ) p ( A2 / A1 )
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An 1 , n 1 ,
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lim p( An ) .
n
n 1
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An , n 1 ,
1
p(
n
n 1
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p ( A1 )
A2 p ( A1 A2 ) p ( A1 ) p ( A2 ) . p ( A1 / A2 ) p ( A1 ) , p ( A2 )
( p ( A2 / A1 )
A1
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,
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AM
A1 , A2 ,..., AM :
.
M
p ( Am ) 1 .
(1.1.10)
m 1
(M
1. 2. : 2(
).
,
).
p( A) p( A ) 1 . A B
(1.1.11) (
). 3.
p ( A B) A1 , A2 ,..., AM
p( A) ,
p( B )
p ( A B) .
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(1.1.12) .
) M
M
p(
M
Am )
p ( Am ) m 1
m 1
p ( Am
An ) ... ( 1) M 1 p( A1
...
AM ) ; (1.1.13)
p ( Am
An ) ... ( 1) M 1 p ( A1
...
AM ) . (1.1.14)
m ,n 1
) M
M
p(
M
Am )
p ( Am ) m 1
m 1
4(
m, n 1
M
A1 , A2 ,..., AM
).
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p(
Am )
p( A1 ) p( A2 / A1 )... p( AM / A1
AM 1 ) .
...
(1.1.15)
m 1
,
. M
M
p(
Am )
p( AM ), m 1
m 1
(
6
1.1.4).
5 (
). (
).
A H 1 , H 2 ,...H M , p(A)
A
M
p(A) =
p( H m ) p ( A / H m ) .
(1.1.16)
m 1
6 (
H 1 , H 2 ,...
). (
),
.
Hm , A
: p ( H m A) p ( A)
p( H m / A)
p ( H m ) p( A / H m )
.
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p ( H n ) p( A / H n ) n 1
.
, .
7 (
N
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).
A. ,
p, A PN (n)
N
C Nn
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A q 1 p.
n , C Nn p n q N n , N! n : C Nn . n!( N n)! (1.1.18)
(1.1.18)
. N A.
). ,
A qm A
pm ,
m-
1 pm (m 1, N ). n ,
,
N
N
N
PN (n) z n n 0
(q m
pm z) .
m 1
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1.2. 1.2.1.
X
1. , .
X
x
x
( ),
. . ,
1. ,
X.
x
2. ,
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x
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1.2.2. 1.
(
)
X
: 1)
F ( x)
P{X
x} ;
f ( x)
dF ( x ) ; dx
(1.2.1)
2)
3)
(1.2.2)
,
e
j x
,
:
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'
f ( x )e j x dx ,
Q( j ) ; j
P{*}
(1.2.3) .
. . 1. Q( j ) f (x) . 2. F (x) ,
F (x) ,
, ,
X F (x) .
X
.
: 1, 0, 1
1/3.
Yn
X 2n 1 ,
n
0, 1,...
3. ,
.
X xi (i 1, 2,...)
,
, f ( x)
pi ( x xi .
pi ( i 1, 2, ... )
(
xi ) ,
1): (1.2.4) pi
.
,
f (x)
, ,
,
. 4. '
,
.
, .
, : x
F ( x)
f ( x)dx,
(1.2.5)
' : 1 2
f ( x) 5.
Q ( j )e
j x
d .
(1.2.6)
, ,
, , M x ( s)
e sx dF ( x)
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(1.2.7)
x
s x dF ( x) .
( s)
(1.2.8) ,
2. ,
. 1.2.1, 1.2.2
. 1.2.3,
,
, . 1.2.1
F (x) 0 0 F ( x) 1
F (x) 1.2.2
f (x)
0
f ( x)dx 1 x2
P{x1
X
x2 } F ( x2 ) F ( x1 )
f ( x )dx x1
1.2.3
(
X)
Q( j )
Q ( 0) 1
Q( j )
Q* ( j )
, . 1.2.4 [25]
,
1.2.5. [69]. 1.2.4
1 , i 0, I 1 I )I i , i 0, I , 0
pi C Ii
pi
i
(1
1
i
pi
pi
(1
i!
exp(
), i
)i , i
0,1,..., 0
0,1,...
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1
. 1.2.4
.
pi
C Ii C Nn C Nn
i I
pi
ip (1 i )q, i
pi
C mi
, i
m i 1
C Nn
0, n, N
0, N
I
0
0,1, q 1 p
)i , i
(1
n
0,1,..., m 1,2,..., 0
1
N! . n!( N n)!
n
: CN
1.2.5
f ( x)
(
1 2
f ( x)
)
exp
1
f ( x)
exp
2 x
f ( x) x(
f ( x)
-
2) / 2
2
x
f ( x) 2 ( m)
f ( x) (
1
(ln( x / m)) 2 ,x 2 2
/2
x),
0, x
exp( x / 2) , x ( / 2)
0,
(1 x) 1 ( ( ) ( )
m
m
x 2 m 1 exp
2
m 0 )
2
(m f ( x)
)
x 2
0,
0
0 1, 2,...
, x [0,1],
cx c 1 exp( ( x / b) c ), x c b m
0, m
0
( x / b) c 1 exp( x / b) ,x 0 , b (c ) ( b 0 ), c
b 0 )
f ( x)
, a x b b a ( x m) 2 , x ( , ), 2 2
exp(
f ( x) (c
1
exp
x2 2
0, c
x2 , x 0 ),
0
0, b 0,
2 2
«
0,
(
I0 »( 0 )
2
x ,
x
0 ),
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0,
0
. 1.2.5
(
4( x a ) /(b a) 2 , x (a, (a b) / 2), 4(b x ) /(b a ) 2 , x ((a b) / 2, b), 0, x (a, b)
f ( x)
)
h , x ( , ), h h ( x x0 ) 2 df ( x) x a f ( x ), dx b0 b1 x b2 x 2
1
f ( x)
2
0
a, b0 , b1 , b2 ((m 1) / 2)
f ( x)
( m / 2) m x [0,1], 0, 0
f ( x)
m,
(m)
-
(
1.2.3.
( m 1) / 2
,
x m
x (
,
, ),
m m x
f ( x) .
1 e 2
1
x2 m
(
1
, x
m,
0
2).
) ,
, , . ,
,
.
,
.
M [ ( X )] f (x)
1.
X M [ ( X )]
(X )
( x) f ( x )dx .
(1.2.9) :
1. (1.2.9)
,
.
M [ ( X )]
2.
(X )
,
f (x) . mx
2.
f (x) (X )
X:
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X
mx
M[X ] Dx
3.
xf ( x)dx .
(1.2.10)
X
D[ X ]
f (x) (X )
2
(X
mx ) : Dx
M [( X
mx ) 2 ]
( x m x ) 2 f ( x)dx .
(1.2.11)
(
4.
X
:
)
x
Dx .
x
X f (x) ,
mx
1. Dx
x
.
X
mx
2. ,
x
Dx
; . (
3. (
)
,
) .
5. :
mx
.
x
xe
6.
X,
f (x) F (x)
7.
F ( x)
(
p,
0,5 . xp (0
,
p
p p 1) . 1. 2. x 0.25 , x 0.5 , x 0.75 x0.1 , x 0.2 ,
, x 0.9 , . 3. ( . , . 8. xm f (x) . 1. . 2. ), ( ) 3. , .
0,5: x e
x 0.5 . , x 0.02 ,
, x 0.09
x 0.01 , )
F ( x) X , (
), .
,
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p
( ( 9.
),
). m
-
X
(X ) -
-
(X ) | X | ,
(X ) -
(X
X ,
mx ) , (X ) | X
mx | ,
. .
1. . '
2.
,
: 1
0;
3
m3
3m x m2
3. ,
2mx2 ;
Dx
m2
4
m4
4m3 m x
/2
1) D x
;
Dx ,
m2
mx ;
3m x Dx
3mx4 .
.
mx
m3 4.
6m2 m x2 Dx :
1 3 5...( 0
m1
mx2 ;
2
3 x
m ; m4
3D
2 x
: 2 x
Dx
m ; 2 x
6m d x
m x4 . .
, .
,
2.
(
,
)
.
Q( j ) =1+
m
(j ) m , ! 1
d Q( j ) j d
0
(1.2.12)
.
(1.2.13)
(1.2.3) . (1.2.12)
1. . 2.
, ,
:
ln Q( j ) = 1
!
(j ) .
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(1.2.14)
10. (1.2.14) .
: d ln Q ( j ) j d (1.2.14)
1.
0
.
(1.2.15) , :
' Q( j )
exp
(j )
!
1
.
(1.2.16)
2. m
,
1 1
m1
mx ;
2
m2
m x2
3
m3
3m2 m x
4
m4
3m22
,...,
m1 ,..., m :
.
,
Dx ;
2
2m 3x
3
;
4m3 mx 12m2 m x2
6m x4
4
3
3 2
.
3. : d M x ( s) ds
m ,
d ln M x (s ) ds
;
s 0
;
(1.2.17)
s .
(1.2.18)
s 0
, M x (s) =
m s ; ! 0
ln M x ( s ) = 1
!
11. 3 3/ 2 2
1
2
3 3/ 2 2
;
4 2 2
3
(1.2.19) 4 2 2
.
(1.2.20)
1. .
1
«
» 2
2
3
(
2
.
0.
2. 3) / 2 . 12.
1
2 1
2
. (
13.
X
0.
)
f (x) Hx
M [log 2 f ( x)]
(1.2.21) ,
1.
X
f ( x ) log 2 f ( x)dx .
.
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Hx 0. [ a, b] ,
X
,
2. X
,
, ,
. 2 x
3.
X Hx
. 14.
log 2 2 e
x
X ( x 0) [ ] m[ ] M [ X ] M [ X ( X 1)...( X X
1)] .
m[
.
.
(1.2.22)
]
'
x
(s)
: m[ x
( ) x
]
( s 1)
(1) ;
m[
s ]
!
0
.
15.
X M [( X
[ ]
mx )[ ] ] .
(1.2.23)
1.3. 1.3.1.
N-
1.
X :
1) N -
FN ( x )
P{ X 1
x1,
, XN
xN } P{ X
2) N -
x} ; ,
(1.3.1)
N
FN ( x ) ; x1 x2 ... x N
f N ( x)
(1.3.2)
3) N QN ( j )
... f N ( x )e j x dx .
'
(
.
(1.3.3) . 1.3.1).
. . 1.3.2 1.3.4. Z
2.
X
Z Z
N-
Z
2N -
X
jY
( X ,Y ) , jY
( X ,Y ) .
. . .
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1.3.1
FN ( x )
P{ X 1 x1
FN ( x )
x1,..., X N
xN };
xN
...
f N ( x )dx N
FN ( x ) ; x1 ... x N
fN (x) f N (x)
1 (2 )
N
... Q N ( j )e
j x
d
... f N ( x )e j x dx
QN ( j )
1.3.2
FN (x )
0
0
FN ( x ) 1
FN ( x )
xn
FN (x )
-
FM ( x1 ,..., x M )
x FN ( x1 ,..., x M , ,..., )
M
N
1.3.3
f N (x )
0
... f N ( x )dx f N (x )
1 -
xn f M ( x1 ,..., x M ) = ... f N ( x1 ,..., x N )dx M 1 ...dx N
M
N
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1.3.4
QN ( j )
Q( j )
1
j
QM ( j
n
1 ,..., j
M
) = QM ( j
1
,..., j
M
M
,0,...,0)
N
1.3.2. .
N-
Z
( X ,Y ) ,
M-
( N M )-
X
Y.
( N M )-
1. ( N M )-
Y
FN
M-
,
M
( y / x) ,
X
x.
( N M )-
2. ( N M )-
Y fN
M
( y / x) , x.
,
X
( N M )-
3. QN
M
(j
1
,..., j
N M
( N M )-
Y / x) ,
,
X
x.
FN
. QN
M
(j
1
,..., j
N
M
( y / x) ,
f N M ( y / x) ' M / x)
: y1
FN
yN
M ( y / x ) = ...
M
fN
M
(u / x )du1 ...du N
M
;
(1.3.4)
N M
fN
QN
M
(j
1
FN M ( y / x ) ; y1 ... y N M
M ( y / x) =
,..., j
N M
/ x ) = ... f N
1.
Z
M
(1.3.5)
( y / x ) e j y dy .
(1.3.6)
f N ( x , y)
(X ,Y )
'
fN
Y
M
( y / x)
f M (x)
X f N ( x, y) = f N
M
( y / x) f M (x) , .
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(1.3.7)
X Y f 2 ( x, y )
4.
,
f (x) f ( y ) f 2 ( x, y ) = f (x) f ( y) .
X
Y: (1.3.8)
.
X
Y F2 ( x, y ) = F (x) F ( y) ; Q2 ( j x , j y ) = Q ( j x ) Q ( j
y
(1.3.9) (1.3.10)
).
X 1 ,..., X N
5. ( f N ( x1 ,..., x N ) f ( x1 ) ,
, f ( x N ) :
),
f N ( x1 ,..., x N ) = f ( x1 )
f ( x N ) .
(1.3.11)
. X 1 ,..., X N FN ( x1 ,..., x N ) = F ( x1 )
F ( x N ) ; Q N ( j 1 ,..., j N ) = Q ( j 1 )
Q ( j N ) . 6. X 1 ,..., X N
(1.3.12) (1.3.13) X i , X j , ( i, j
,
1, N )
. X 1 ,..., X N . X1, X 2 , X 3
1. . .
,
f 2 ( x1 , x 2 ) = f ( x1 ) f ( x 2 ) , f 2 ( x1 , x3 ) = f ( x1 ) f ( x3 ) , f 2 ( x 2 , x3 ) = f ( x2 ) f ( x3 ) .
f 3 ( x1 , x 2 , x3 ) = f ( x1 ) f ( x 2 ) f ( x3 ) . , f 3 ( x1 , x 2 , x3 ) = f 2 ( x1 , x 2 ) f ( x3 / x1 , x 2 ) = = f ( x1 ) f ( x2 ) f ( x3 / x1 , x 2 ) . , , X3 ,
( f ( x3 / x1 , x 2 ) = f ( x3 ) ),
X1, X 2 , X 3
, .
2. . . 2. ( X 1 ,..., X N ) f N ( x1 ,..., x N )
f N ( x1 ,..., x N ) N f1 ( x N / x1 ,..., x N 1 )... f 1 ( x 2 / x1 ) f 1 ( x1 ) ,
f1 ( x n / x1 ,..., x n 1 ) (n f 1 ( x1 ) .
2, N )
(1.3.7).
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(1.3.14)
( X 1 ,..., X N )
. (1.3.14)
,
(1.3.11).
1.3.3.
(
) 1.3.3
1. ,
,
. . , 1.
(X )
N-
M-
f N ( x1 ,..., x N ) M [ ( X )]
2 1.3.1. MX ( X 1 ,..., X N ) ,
( x1 ,..., x N ) f N ( x1 ,..., x N )d x1 ...dx N ,
...
(1.3.15)
.
M [ ( X )]
1.
(X )
X
(
) f N (x )
, .
X
N (X )
2.
M .
mx
2.
X
N(X ) X :
mx
N-
,
M[X ].
(1.3.16) Z
. mz
mx
Dx
D[ X ]
3.
X (X )
Z
X
N-
,
m xn ) 2 , n 1, N ) .
(1.3.17)
Dz N -
. jY
N-
(( X n
m xn ) 2
,
j (Yn
jY
jm y .
N-
(( X n
X
m yn ) 2 , n 1, N ) .
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(
4.
X
)
N-
x
N-
, Dx . mx
1.
X, Dx
x
. mx
2.
X
x
X,
, Dx
X. ,
.
X,
,
.
/
, mx
,
x
2
, ( / )2
Dx
,
/
)2 .
(
3.
N-
z
Z
X
jY
N-
, Dz ,
Dz .
m 1 ...
5.
1
N
...
N
X
N-
, m 1 ... n
M [ X 1 1 ... X NN ] ,
N
(1.3.18)
(n 1, N ) . m 1 ... N
1.
QN ( j QN ( j
1
1
,..., j
,..., j
N
N
)
: m 1 ...
)= 1 ,...,
N
1!...
0
(j N!
N
1
) 1 ...( j
N
)
N
. (1.3.19)
, 1
m 1 ... N = ( j )
1
...
...
N
N
QN ( j 1 1 ...
1
,..., j N
N
) 1
...
N
0
.(1.3.20)
N
-
2. 1
,...,
N
,
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, .
6.
1 ... N
...
1
N-
N
X : ln Q N ( j
1
,..., j
N
1 ... N
1
= ( j) N
1 ...
1
...
(j 1 !... N !
)= 1 ,...,
N
...
N
N
0
ln Q N ( j 1 1 ...
1
1
) 1 ...( j
,..., j
N
N
)
N
.
) 1
N
...
N
0
(1.3.21)
.
(1.3.22)
N
7.
1 ... N
...
1
N
X
N-
, 0 1 ...
0
1
N
M [ X 1 ... X N ] ,
N
(1.3.23)
0
X
X
mx
. 8.
11
X1
,
X2 m11 ,
x1
x2
,
, 11
r x1
.
(1.3.24)
x2
1. X1
,
X2
.
.
2.
X1
11
'
X2
m x1
m11
mx2 11
m11
mx1 mx2 ,
(1.3.25) .
3.
r ,
11
' .
'
11
r
'
r
0.
1. ,
4. , . Z
9. X jY :
m11
m11
M [ XY ] ,
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11
:
M [( X
11
mx )(Y
m y )] . K
10. Z1
X1
jY1 , Z 2
X2
jY2
K11 K 21
K
K 12 , K 22
M [ Z n Z m* ] ( n, m 1,2 ),
K nm R
R11 R21
R Rnm
M [( Z n
R12 , R22
m z n )( Z m*
m *z )] ( n, m 1,2 ). m
R11 , R22
.
R 2
R11 | D[Z 1 ] | M [( X 1
m x1 )
R22 | D[ Z 2 ] | M [( X 2
m x2 ) 2 X1
11. (
(Y1
Z1 Z 2 : m y1 ) 2 ] ;
(Y2
m y2 ) 2 ] .
X2
),
0,
11
m11
m x1 m x2 .
(1.3.26)
. . , .
' ,
,
( 1.4). X1
12. , 1.
X2
0.
m11
.
m x1 , m x2
, ,
.
2.
N
, 13. (
14. (
'
. X 1 ,
, X N
), M [ X n X m* ]
M [ X n ]M [ X m* ], n
m, n, m 1, N , . X 1 ,
, X N
(1.3.27)
), M [ X n X m* ]
0, n
m, n, m 1, N .
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(1.3.28)
(
15.
Y x,
) X
,
Hy/x
M [log 2 f ( y / x)].
(1.3.29)
Z
1.
( X ,Y )
Hz M [log 2 f ( x, y )] Hz Hx Hy/x .
.
Z
2.
( X ,Y )
:
Hz ,
Hx X
Hy , Y
2
3.
X (X ,
(1.3.30) . ,
X, X ) X
X, X .
X
4.
J ( x, y)
Hx
Hy
Hz .
(1.3.31)
X
5.
Hx
H x1
H x2 / x1
( X 1 ,
, X N )
... H xN / x1 ,..., x N 1 ,
(1.3.14).
Y
(1.3.32)
X
,
,
Hx
H x1
16. f N ( x1 ,..., x N )
C,
H x2
... H x N .
C
(1.3.33) .
. . . , ,
. '
1. ,
, .
2. , my / x
M [Y / x ] ;
Dy / x
D[Y / x ] .
:
1.4. 1.4.1. 1.
,
y
' (
)
'
x
B,
B. 28
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B
1.
,
x. : y
1.
lg x, y
x5 . B
,
2. ,
y
y
x.
: 1) y = x(t )dt ; 2) y
2.
max x(t ) .
B
3.
,
. : y( )
3.
x (t )dt , y(t )
lg x(t ) .
4.
.
5.
. , . .
2
. .
. . 1.4.1.
1.4.1
( )
y |x|
4.
y
x2 ,
,
y ,
sin( x x . .
,
),
,
,
. .
. ,
, . .
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1.4.2. 1.
y x Y
f 1x ( x )
X (x) , f 1y ( y)
( y) . f1 y ( y )
f1x ( ( y ))
d ( y) . dy
(1.4.1)
X
.
X
x
X
, dx y Y
x
Y
Y
'
f1 y ( y ) dy
y dy ,
f 1x ( x) dx .
(1.4.1). f 1x ( x )
X
2.
y
Q
(x) , x
q
( y ), q 1, Q .
f 1 y ( y) f 1x ( x )
Y X
Q
f1 y ( y )
f 1x (
q
( y ))
d
q 1
(1.4.2)
q
( y)
:
.
dy
,
(1.4.2)
(1.4.1).
X
1. x
y
f1 ( x ) my
my
(x) . ( x ) f1 x ( x)dx ,
MY
(1.4.3) .
(1.4.3)
1.
,
Y ( x) f1 x ( x)dx .
my
(1.4.4)
2.
y
(y
(x) Y
(1.4.4) ,
,
ax b ),
my X
mx
'
my
amx
b. X
2. x
y
f1 ( x ) y
1.
(1.4.5)
M Y
y
(x) .
my
( x) m y (1.4.6)
f 1x ( x)dx .
,
Y
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(1.4.6)
Dy
2
my
( x) m y
(1.4.7) Dy
2. ax b ),
(y
M Y
2
f1x ( x )dx .
(1.4.7)
y '
,
Y
(x)
X
Dx
a 2 Dx .
Dy
(1.4.8)
.
.
X
. (a, b] ( f1x ( x ) 1 (b a)) .
y
F1 y ( y ) .
y x
(x) , Y
(x)
( y) .
(1.4.1)
.
Y dx . b a dy x c (b a ) F1 y ( y) , c y y ( F1 ( ) 0, F1 ( ) 1 ) c x a F1 y ( y ) b a 1
f1 y ( y )
,
y F1
y
y
( x)
F1
y
. a.
x a , b a
(1.4.9)
F1 y ( y ) .
, '
1.
F1 y ( y ) . ,
(0,1] ,
,
(1.4.9). ,
2.
y
a (b a ) F1x x X Y
(1.4.10) F1x x
( a, b ] . 1.4.3.
X
,
Y. 1. f Hx (x )
H-
X
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y
H-
(x ) ,
Y. ,
x1
1
xH
( y1 ,..., y H ) ; ( y1 ,..., y H ) .
H
f Hy ( y )
Hf Hy ( y )
f Hx ( 1 ( y ),...,
J H ( y)
H
Y
( y )) J H ( y ) ,
(1.4.11)
: 1
1
y1
yH
J H ( y)
. H
H
y1
yH
.
1
,...,
(1.4.12)
,
H
(1.4.11)
,
(1.4.2).
X
2. x 2
f ( x1 , x 2 ) ( x1 , x 2 ) , y 2 x2 ,
y1
( X1, X 2 )
, : ( y1 , y 2 ) , y2 .
x1 x2
1
f 1 y ( y1 ) f1 y ( y1 )
:
Y1
f 2x ( ( y1 , y 2 ), y 2 )
1
( y1 , y 2 ) dy 2 . y1
(1.4.13)
1.
.
J ( y)
1
1
y1 0
y2 1
1
y1
.
(1.4.11) f 2y ( y1 , y 2 )
f 2x ( ( y1 , y 2 ), y 2 )
1
y1
. f 2y ( y1 , y 2 )
f1 y ( y1 ) , .
(1.4.13). (1.4.13)
Y, ,
, X1
X2 (
f1 y ( y )
'
Y, 32 Create PDF files without this message by purchasing novaPDF printer (http://www.novapdf.com)
. 1.4.2). 1.4.2
X
( X1, X 2 ) ,
x 2
f ( x1 , x 2 ) f1 y ( y ) y
x1
x2
f1 y ( y ) =
f 2x ( y
x 2 , x 2 )dx 2
y
x1
x2
f 1y ( y) =
f 2x ( y
x 2 , x 2 )dx2
y
x1 x 2
f 1y ( y) =
x1 x2
y
f1 y ( y ) =
f 2x (
dx y , x2 ) 2 x2 x2
f 2x ( yx 2 , x 2 ) x 2 dx 2
. ,
. Q
.
y
q
xq ,
q
.
q 1
. . 1.4.3. 1.4.3
Y
my
Dy
m xq
D xq
Q
Xq
y
q
xq
q 1
Q
my
q
m xq
q 1 Q 2 q
Dy
D xq
q
q 1
h
R x q xh
q h
. R xq x h
.
X1,
1.
X2
Y ,
X1, X 2
.
2. Xq
Y
(q 1, Q)
:
Q 2 q
Dy
D xq .
q 1
3. .
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1.5. 1.5.1. , . 1.
,
. .
, ,
. 1.5.1.
. 1.5.1. 1.5.1
f ( x)
1 2
( x m) 2 , 2 2
exp
x m
F ( x) x
1
( x)
(
, ),
2
Q( j )
exp jm
m
. 1. 2.
, 7.
. 1.5.1.
m
0
,
t2 dt 2
exp
2
x
2
2
.
.
4,
0,25; 0,5; 0,75; 1 (
1 4)
1. , .
m, m
3
0,4% f (x)
m 3
,
.
99,7% . ,
(m 3 , m 3 ) .
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, 0,997
(m ,
2. -
0) 0
.
X
3. : d M [ g ( X )] dD n n
M [ Xg ( X )]
d 2n g ( X ) ; dX 2 n
2 nM
mM [ g ( X )] DM
g (X ) .
dg ( X ) , dX X, D
4. -
g( X )
. X1
1.
X2
, f 2 ( x1 , x 2 ) C
, -
x1
x2
: (1.5.1) , ,
C exp[ P2 ( x1 , x 2 )], , (C 0) , P2 ( x1 , x 2 ) x1 x 2 , ( P2 ( x1 , x 2 )
0) .
1.
X 1, X 2
(1.5.1)
Q2 ( j
1
,j
2 2 2
2
: 1
f 2 ( x1 , x 2 ) exp
( x1
)
1
1 r2
2
m1 ) 2
exp jm1
2r
1
1
jm2
2
( x1
2
2 1
1 ( 2
2
2 1
m1 )( x 2 m2 ) 2 2 2 (1 r ) 2 1
2 1
2r
1
2
m2
m1
X .
1
, r 1.
2 2
2
X1
(1.5.2)
m2 ) 2
( x2
.
2 2
X2 ,
) ,(1.5.3) 1
2
. ,
(1.5.2)
: f 2 ( x1 , x 2 ) x X1
1 1 exp[ ( x m) T R 1 ( x m)], 2 2 |R|
( x1 , x 2 ) T , T X 2 , m (m1 , m2 ) X1 X 2 , T R
|R|
(1.5.4)
, , R R11 R21
1
R12 , R22
R.
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,
2. n
Q2 ( j r
1 n
,j
2
)
(
1
2
1
2
) n Q2 ( j
1
,j
2
). ,
. X1
1. ,
X2
. . ,
9
(
1.2). X1 X2 . 1.5.2),
2. ( ( x1
m1 ) 2
2r
2 1
( x1
.
m1 )( x 2 1
m2 )
( x2
)(
m2 ) 2 2 2
2
(1 r 2 )C 2 , C
( x m )T R 1 ( x m ) C 2 , C
0
0.
(1.5.4)
(1.5.5)
X1
. 1.5.2.
X2 ( m1 , m 2 ) ,
1. 1
,
2
m1 , m 2 , r
,
C. 1
,
2
r. 2r 1 arctg 2 2 1 /4
1
2 2 2
1
1
2
2
,
(1.5.6)
.
,
90 . 2.
(1.5.4) (1.5.6)
, ,
C
.
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(1.5.4), (1.5.5)
3.
, , ,
(
(r 0) , 4. 2)
1
. (1.5.6) :
, /4. (r
(
1
2
)
0) ,
C.
(
3. X2
X1
) (
Y1 ,
)
Y2
: Y1
(X1
Y2
m1 ) cos
(X1
(X 2
m2 ) sin ,
m1 ) sin (X 2 (1.5.6).
(1.5.7)
m 2 ) cos ,
-
4. .
,
5. , ,
'
. ,
. ,
.
6.
X1
X2
, ,
f1 ( x2 / X 1
x1 )
f 2 ( x1 , x2 ) / f1 ( x1 )
1 2 (1 r 2 )
2
1 x2 2 2 2 (1 r 2 )
exp
(1.5.8)
.
: 2
m2
r
2
( x1 m1 )
.
(1.5.8)
1
,
m x2 / x1
X1,
Dx2 / x1
: m x2 / x1
m2
r
2
( x1
m1 ) ,
D x2 / x1
2 2
(1 r ) 2 .
1
,
2. , .
X 1 ,
, X N
2. , f N ( x1 ,..., x N )
: 1 (2 ) N / 2 | R |1 / 2
1 exp[ ( x m) T R 1 ( x m)], 2
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(1.5.9)
( x1 ,..., x N ) T
x
-
, T
X 1 ,
, X N , m
(m1 ,..., m N ) , R R11 ... R1N R ... ... ... . R N 1 ... RNN
1.5.2.
,
, : (1.5.10)
'
,
, .
,
,
.
, ,
. ,
1. f 1 ( x)
x 2
exp
: x2 , x 2 2
0.
(1.5.11)
0.
(1.5.12)
,
1. (1.5.11),
F1 ( x) 1 exp
x2 , x 2 2
2.
Dx x
2
mx 2
2
,
m x2 ,
,
m3
3
3
2
.
3. . 1.5.3.
1
. 1.5.3.
(2 5).
(1)
0,4
1
1.
X
( X1, X 2 )
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15
( m x1 = m x2 =0) 2
X
.
X
,
(0,2 ] .
.
. 1
f 2 ( x1 , x 2 )
2
2
x sin x cos
cos sin
J 2 ( x, )
x12 x 22 . 2 2
exp
(1.5.13)
x.
X ( x, ) x
f 2 ( x, )
2
2
x2 . 2 2
exp
(1.5.14) (1.5.14)
(0, 2 ]
[0, )
x
X 1 2
f1 ( )
. ,
. X1
(1.5.11)
,
X2 2
m x1
mx2 ,
X
-
R
(mx1 , m x2 )
( X1, X 2 ) (1.5.11).
(
2.
)
, x
f1 ( x )
2
exp
«
x2 2 2
»,
2
I0
2
x
, x
2
(1.5.15)
0,
(
0) , I 0 ( )
. ,
1. (1.5.15), x
F1 ( x )
x 2
0
exp
x2 2
2
I0
2
x 2
dx ,
. 2.
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(1.5.16)
x
f1 ( x )
2
x2 2
exp
, 1 exp 2
f 1 ( x)
(x 2
2
2
1
2
)2
x2
, x
4
4
2
1
2
x
8 x
0,
, x
0.
3. 2
mx
1
2
2
2
2
I0
4
2
2
2
2
2
I1
4
2
2
exp
4
,
2
(1.5.17)
Dx
2
2
2
mx2 ,
(1.5.18)
I 1 (*)
. 4.
Dx . 1.5.3 (
mx
2
. 2 5).
2.
X X m x21
( X1, X 2 ) m x1 , mx2
2
.
X 2
m x22 , f1 ( )
,
1 exp 2
2
2
cos(
2
0
2
)
F
cos(
0
)
2
exp
0
arctg
m x2 m x1
2
2
sin 2 (
0
) , |
0
, 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
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2. 2.1. 2.1.1.
X (t )
1. t
, T
(
T
)
. S
.
(
). 2. i )
(
X (t ) ( xi ( t ) (i 1,2,...) , t T
)
ix
S. 1. ,
. ,:
. -
2. (
)
,
,
.
,
. 3. .
. . 2.1.1.
X (t )
. 2.1.1.
.
.
X (t ) .
i-
x i (t )
i-
X (t ) . xi ( t ) (i
1,2,...)
X (t ) . (
t0
). t0
xi ( t 0 )
i-
X (t 0 )
i-
X (t 0 ) .
. . ,
,
' ,
.
S, T . 2.1.1 2.1.2
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(
),
. 2.1.1 (
( S RN ) (S R S Z )
S N S T .
1)
(
) R)
(T , Z
R
(S ZN) (S R N S ZN)
RN )
(T .
2.1.2 (
1. (
2)
2. 1-
3. (S
S
)
(
)
4. ,T
)
5. (S T
(S
,T
)
6.
(
)
)
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(
)
,
. ,
, .
.
.
2.1.2. .
N.
FN ( x ; t ) , QN ( j ; t )
f N ( x; t )
,
. 2.1.3. 2.1.3
FN ( x ; t )
P{ X (t1 )
xN };
xN
x1
FN ( x ; t )
x1 ,..., X (tN )
...
f N ( x ; t )dx N
F ( x; t ) ; x1 ... x N
f ( x; t ) 1 (2 ) N
f N ( x; t )
QN ( j ; t )
... QN ( j ; t )e
j x
d
... f N ( x; t )e j x dx
.
N
.
. , . . 2.1.4 2.1.6. 2.1.4
FN ( x; t ) 0 0 FN ( x; t ) 1
FN ( x ; t ) xn
FN ( x; t ) ( xn , tn ) FM ( x1 ,..., x M ; t1 ,..., t M ) M N
FN ( x1 ,..., x M , ,..., ; t1 ,..., t N )
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2.1.5
f N ( x; t )
0
... f N ( x; t )dx 1
f N ( x; t ) ( xn , t n ) f M ( x1 ,..., x M ; t 1 ,..., t M ) =
-
M
= ... f N ( x1 ,..., x N ; t1 ,..., t N )dx M 1 ...dx N
N
2.1.6
QN ( j ; t )
1,
QN ( j ; t ) ( j n , tn ) Q M ( j 1 ,..., j M ; t 1 ,..., t M ) = = Q N ( j 1 ,... j M ,0,...0; t1 ,..., t N )
QN (0; t ) 1 -
M
N
2.1.3.
X (t )
. X ( t1 ),..., X ( t N ) ( , ( ,
. 2.1.2). M
. (N M)
,
( N M)
,
M X (t )
).
f N ( x; t ) , f M ( x1 ,... x M ; t1 ,..., t M ) fN
M
:
( x M 1 ,..., x N ; t M 1 ,... t N / x1 ,... x M ; t1 ,... t M ) f N ( x1 ,..., x N ; t1 ,..., t N ) = f M ( x1 ,... x M ; t1 ,..., t M ) f N M ( xM 1,..., x N ; tM 1 ,...tN / x1 ,... xM ; t1 ,..., tM ) .
t1 , t 2 ,..., t N
1. ,
,
,
. t1 , t 2 ,..., t N
2. (
),
,
, N
f N ( x; t )
f1 ( x n ; t n ) , FN ( x ; t ) n 1
,
N
N
F1 ( x n ; t n ) n 1
QN ( j ; t )
Q1 ( j n 1
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n
; tn ) .
X (t )
. 2.1.2.
, , . ,
1. . fN
2. ( x ,..., x N ; t M 1 ,... t N / x1 ,... x M ; t1 ,... t M ) = f N M M 1
f N ( x1 ,..., x N ; t1 ,..., t N ) = f M ( x1 ,... x M ; t1 ,..., t M ) f N . . 2.1.7 2.1.9.
M
, ( x ,..., x N ; t M 1 ,..., t N ) M M 1 ( x M 1 ,..., x N ; t M 1 ,..., t N ) .
2.1.7
FN
0 FN
M
M
FN
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) M
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) 1
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) xn
FN
*)
M
0
(n
M
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) -
1, N )
(xn , t n )
*) FN M K ( x M K 1 ,..., x N ; t M K 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) = FN M ( ,..., , x M K 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) K M N ( xn , tn )
. ,
.
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,
2.1.8
fN
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M )
M
...
fN
M
0
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) dx M 1 ...dx N =1
fN
M
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) -
( xn , t n ) fN
*) M K
(xM
...
fN
dx M 1 ...dx M *)
K 1
M
,..., x N ; t M
,..., t N / x1 ,..., x M ; t1 ,..., t M ) =
( x M 1 ,..., x N ; t M 1 ,...t N / x1 ,..., x M ; t1 ,..., t M )
K
K
K 1
M
N
( xn , t n )
. ,
,
.
2.1.9
| QN QN
M
(j
( j M 1 ,..., j N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) | | Q N M (0,...,0; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) | 1 M
M 1
,..., j
N
; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) ( j n , tn ) ( xn , t n )
QN M K ( j M K 1 ,..., j N ; t M Q N M (0,...,0, j M K 1 ,..., j K M N
,..., t N / x1 ,..., x M ; t1 ,..., t M ) = N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M )
K 1
. 1. ,
, «
»
.
. . ,
2. , «
»
, « ,
»
: f N M ( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x K ; t1 ,..., t K ) =
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= ... f N fM
K
M
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M
(2.1.1)
( x K 1 ,..., x M ; t K 1 ,..., t M / x1 ,..., x K ; t1 ,..., t K ) dx K 1 ...dx M (0 K M N ) .
. , ). x (t1 ),..., X (t M )
3. :
( X (t / X (t1 )
x ( t m ), m 1, M ) , X (t / x1 ,..., x M ; t1 ,..., t M )
X (t / X (t m )
1.
X (t / x(t m ), m
X (t / X (t m )
x (t M )) , 1, M ) .
x (t m ), m 1, M )
,
X (t )
t1 ,..., t M . f N ( x1 ,..., x N ; t1 ,..., t N )
X (t )
, f N M ( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) X (t / X (t m )
.
1, M )
x (t m ), m
2. .
2.2. 2.2.1.
NX (t )
1. ...
1
N
m 1... N (t1 ,..., t N ) = ...
x1 1 ... x NN f N ( x1 ,..., x N ; t1 ,..., t N ) dx1 ... dx N = = M [ X 1 (t1 )...X
(t N )] ,
(2.2.1)
( n 1, N ).
n
N
X (t )
2.
x f 1 ( x; t ) dx .
m x (t ) = m1 ( t ) = M [ X (t )] =
(2.2.2)
0
X (t )
3. 0
X ( t ) = X (t ) mx (t ) . NX (t ) 1 ... N
4. 1
...
N
(2.2.3)
N-
0
X (t ) : 0 1 ...
(t1 ,..., t N ) = M [ X N
1
0
(t 1 )... X
N
(t N )].
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(2.2.4)
. , 0
M [| X (t1 ) | 1 ... | X (t N ) | N ]
0
M [| X (t1 ) | 1 ... | X (t N ) | N ] . ,
. 1. t1 ,..., t N .
2. . -
3.
(
,
N
)
N .
,
Q1 ( j ; t )
. (j ) m (t ) . ! 1
Q1 ( j ; t ) =1+
(2.2.5)
. Q1 ( j , t ) =
e j x f 1 ( x; t ) dx .
(2.2.6)
: z
e z =1+
!
1
(2.2.7) Q1 ( j ; t ) =
.
(2.2.7)
(2.2.6), ( j x) [1 ] f 1 ( x; t ) dx . ! 1
(2.2.5). , : QN ( j
1
,..., j
N
m 1 ... N (t1 ,..., t N )
; t1 ,..., t N ) = 1 ,...,
0
N
1
!...
N
4.
! N
(j
1
) 1 ...( j
)
N
.
(2.2.8) -
N.
m (t )
N
,
.
m (t )
Q1 ( j ; t ) | j
0
.
(2.2.9)
.
Q1 ( j ; t )
j
,
: Q1 ( j ; t )
(j )
Q1 ( j ; t ) 1 1
(2.2.5) (2.2.10)
!j
. 0
(2.2.9).
.
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(2.2.10)
1
m 1 ... N (t1 ,..., t N ) = ( j )
1
...
...
QN ( j
N
N
,..., j N ; t1 ,..., t N ) ... NN
1 1
1
...
1
N
, (2.2.11)
0
(1.3.20). . 5. 1 ... N
(t1 ,..., t N ) ln QN ( j
: 1
,..., j
N
1 ... N
; t1 ,..., t N ) = 1 ,..., N
1 1 ...
(t1 ,..., t N ) = ( j ) N
1
...
...
N
N
(t1 ,..., tN )
1 !...
0
ln Q N ( j
N!
,..., j ... NN 1
1
1
N
( j 1 ) 1 ...( j
N
)N,
...
N
(2.2.12)
; t1 ,..., t N ) 1
0
. (2.2.13)
1. . R1 (t i ) ,
n
0 ( n 1, N , n
i)
n
0 ( n 1, N , n
i, n
j)
n
0 (n
i, n
j, n
1, N , n
1
i i
R2 (t i , t j ) ,
i
k ) R 3 (t i , t j , t k )
1
j j
k
1
. .
. ,
R2 (t i , t j ) . 2. ,
,
.
: m1 (t ) R1 (t ) ; m11 (t1 , t 2 ) R 2 (t1 , t 2 ) R1 (t1 ) R1 (t 2 ) ; m111 (t1 , t 2 , t 3 ) R3 (t1 , t 2 , t 3 ) [ R1 (t1 ) R2 (t 2 , t 3 )
R1 (t 2 ) R2 (t1 , t 3 ) R1 (t 3 ) R2 (t1 , t 2 )]
R1 (t1 ) R1 (t 2 ) R1 (t 3 ); 3. N
, ,
.
N
.
(
) -
N
-
N
N-
. 2.2.2. m x (t ) ,
D x (t )
x
(t ) (
. 1.
X (t )
D x (t ) :
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).
D x (t )
[ x m x (t )]2 f 1 ( x; t )dx .
(2.2.14)
(
2.
X (t )
:
x
(t )
)
x
(t )
D x (t ) . ,
. .
. . 2.2.1. . X (t ) . .
X (t ) m x (t )
, x
(t )
. . 2.2.1
M[ X (t ) (t )] m x (t ) (t ) M[ A(t ) X (t )] A( t ) m x (t ) D x (t )
m 2 (t ) m x2 (t )
D[ X (t )
(t )]
D[ A(t ) X (t )]
D x (t )
[ X (t )
(t )]
[ A(t ) X (t )]
(t )
A( t )
;
A (t ) D x (t )
(t )
x
.
D x (t ) 2
D
x
A(t ) ;
(t ) x
(t )
.
(
3.
X (t) ( 2.2.1)
,
.
) R x (t , t ) 1
5 :
R x (t , t )
M [( X (t ) m x (t ))( X (t ) m x (t ))]
[ x m x (t )][ x
(2.2.15)
m x (t )] f 2 ( x, x ; t , t )dxdx .
X (t)
K x (t , t )
4.
: K x (t , t )
M [ X (t ) X (t )]
xx f 2 ( x, x ; t , t )dxdx . '
. R x (t , t )
K x (t , t ) m x (t )m x (t ) .
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(2.2.16) : (2.2.17)
(
5.
)
R x (t , t ) . x (t ) x (t )
rx (t , t )
(2.2.18) ,
1. , (
)
.
2. ,
,
.
.
. 2.2.2. 2.2.2
/
t 1
t
R x (t , t )
1) 2)
;
rx (t , t ) 1 Rx (t , t )
2
rx (t , t ) | R x (t , t ) |
3
(t )
ry (t , t )
rx (t , t ) x
(t )
x
(t ),
rx (t , t )
Y (t ) A(t ) X (t ), R y (t , t ) A(t ) A(t ) R x (t , t ),
X (t ) A( t ) 5
Rx (t , t ),
rx (t , t ' ) 1 Y (t ) X (t ) (t ), R y (t , t ) R x (t , t ),
X (t ) 4
D x (t ),
1)
ry (t , t )
rx (t , t )
; 2) 13
. ,
,
2 (
): | f1 ( x) | 2 dx
f 1 ( x ) f 2 ( x)dx | X
X
f 1 ( x), f 2 ( x)
| f 2 ( x) | 2 dx , X
X.
, 4
Y (t )
2
3
|
1
X (t )
5
(t )
Y (t )
A(t ) X (t ) .
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(2.2.20)
t ,t '
6.
R x (t , t )
, .
1. ( ,
.
. 2.2.1).
'
, .
'
,
. (
2. '
)
. : X (t ) R x (t , t ) x (t ) x (t ) .
3. , ,
aX ( t ) b ,
a b
.
. 2.2.1.
,
,
t, t'
7.
K x (t , t )
, . (2.2.17)
1.
1
, 2.
,
6 ,
, . m x (t ), m x (t )
, ,
.
2.2.3. . 1.
(N
M)
...
M 1
m
M 1 ... N
M[X
X (t )
N
(t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) M 1
(t M 1 )... X
N
,
(t N ) / x1 ,..., x M ; t1 ,..., t M ], t1 ,..., t M
xm , (m 1, M ) .
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(2.2.21)
X (t)
2. (
)
X (t )
m x (t / x1 ,..., x M ; t1 ,..., t M ) = M x [ X (t ) / x1 ,..., x M ; t1 ,..., t M ] =
(2.2.22)
= xf1 ( x; t / x1 ,..., x M ; t1 ,...t M )dx . (
.
m x (t )
)
'
M x1 ,..., xM [m x (t / X 1 ,..., X M ; t1 ,..., t M )] ,
M x1 ,..., xM
,
X 1 ,..., X M .
,
3.
Dm (t ) Dx1 ,..., xM
m x (t / x1 ,...,x M ; t1 ,...,t M ) : Dx1 ,..., xM [m x (t / X 1 ,...,X M ; t1 ,..., t M )] , ,
(2.2.23)
X 1 ,..., X M .
D x (t / x1 ,...,x M ; t1 ,..., t M ) X (t ) : D x (t / x1 ,...,x M , t1 ,..., t M ) D x [ X (t ) / x1 ,..., x M ; t1 ,..., t M ] . 5. D D (t ) D x (t / x1 ,...,x M ; t1 ,..., t M ) 4.
,
D D (t )
M x1 ,..., xM [ Dx (t / X 1 ,...,X M ; t1 ,..., t M )] ,
(2.2.25)
X 1 ,..., X M . . 2.2.2. m x (t / x1 ,...,x M ; t1 ,...,t M )
1.
X (t )
,
m x (t ) . .
t1 ,..., t M
2. m x (t / x1 ,...,x M ; t1 ,...,t M ) , 3. m x (t / x1 ,...,x M ; t1 ,...,t M ) X (t / x(t m ), m 1, M )
(2.2.24) , :
x (t m ), (m 1,M ) , D x (t / x1 ,...,x M ; t1 ,..., t M ) t tM
.
D x (t / x1 ,...,x M ; t1 ,..., t M )
,
. 4.
D x (t / x1 ,...,x M ; t1 ,..., t M )
Dm (t ) m x (t / x1 ,...,x M ; t1 ,...,t M ) D D (t ) .
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,
. 2.2.2.
m x (t )
:
X (t ) ; m x (t ) x (t ) X (t ) ; mx (t ) X (t ) ; mx (t / x1 ,...,xM , t1 ,...,tM ) m x (t / x1 ,...,x M ; t1 ,...,t M ) x (t / x1 ,..., x M ; t 1 ,..., t M ) m x (t / x1 ,...,x M ; t1 ,...,t M ) X (t )
X (t ) D D (t ) : (2.2.26)
D x (t ) Dm (t ) D x (t ) = Dm (t ) + D D (t ) . '
.
X (t / x (t m ), m 1, M ) 6.
. R x (t , t / x1 ,...,x M ; t1 ,..., t M ) ,
: R x (t , t / x1 ,...,x M ; t1 ,..., t M ) 0
(2.2.27)
0
M x [ X (t / x1 ,...,x M ; t1 ,..., t M ) X (t ' / x1 ,...,x M ; t1 ,..., t M )], 0
X (t / x1 ,...,x M ; t1 ,..., t M )
:
0
X (t / x1 ,...,x M ; t1 ,..., t M )
(2.2.28)
X (t / x1 ,..., x M ; t1 ,..., t M ) m x (t / x1 ,...,x M ; t1 ,..., t M ). . 2.2.3. ( t . 1 ,..., t N ) 1 ... N M 1 ... N
(t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) .
(2.2.13) QN ( j
1 ,..., j
QN
N ; t1 ,..., t N ) M
(j
M 1
,..., j
N
; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) .
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,
,
.
, . 2.2.3
X (t ) m x (t / x1 ,..., x M ; t1 ,..., t M ) = = M x [ X (t ) / x1 ,..., x M ; t1 ,..., t M ]
D m (t )
Dx1 ,..., xM [m x (t / X 1 ,...,X M ; t1 ,..., t M )] D x (t / x1 ,...,x M ; t1 ,..., t M ) Dx [ X (t ) / x1 ,..., x M ; t1 ,..., t M ]
D D (t )
M x1 ,..., xM [ Dx (t / X 1 ,...,X M ; t1 ,..., t M )] R x (t , t / x1 ,...,x M ; t1 ,..., t M )
0
0
M x [ X (t / x1 ,...,x M ; t1 ,..., t M ) X (t ' / x1 ,...,x M ; t1 ,..., t M )]
2.3. 1. X (t ) , : X (t ) .
(
)
( X 1 (t ),..., X H (t )) . (
,
,
,
,
.).
. ( ,
,
). X 1 (t ) X 1 (t )
2.
,
X 2 (t )
X 2 (t ) .
. X 1 (t )
X 2 (t )
f 2 N ( x1 , x 2 ; t1 , t 2 ) f N ( x1 ; t1 )
X (t )
f N (x2 ; t2 ) . 3. ( X 1 (t ),..., X H (t ))
, f NH ( x1 ,..., x ; t1 ,..., t )
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H
f NH ( x1 ,..., x ; t1 ,..., t )
f N ( xh ; t h ) ,
(2.3.1)
h 1
N
( ), H
. ,
1. , '
X (t )
.
. .
2. ( X 1 (t ),..., X H (t ))
,
'
,
. m x (t )
4.
X (t )
( X 1 (t ),..., X H (t ))
,
: M [ X (t )] .
m x (t ) 5.
(2.3.2)
D x (t ) ,
X (t ) : D x (t )
D[ X (t )] .
(2.3.3) R x (t , t ' )
6.
H H
X (t ) 0
Rhl (t , t ' ) ( h, l
0
M [ X h (t ) X l (t ' )]
(2.3.4)
1, H ).
.
Rhl (t , t ' ) . '
. .
7. rhl (t , t ')
Rhl (t, t ') Rhh (t , t ') Rll (t, t ')
( h, l
1, H ) .
(2.3.5)
. 2.3.1. .
(
)
' ,
,
.
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2.3.1 ( (
)
)
, 1)
Rhh (t , t )
D h (t ), h
Rhl (t , t )
Rlh (t ' , t )
Rhl (t , t )
2)
rhl (t , t ) ,
.
t
h
(t )
l
1, H
(t )
1
,
,
t .
2.4. 2.4.1. .
.
D
-
(
)(
,
. 2.4.1),
x(t ) ,
y (t ) ,
.
. 2.4.1.
. ,
, . 2.4.1. 2.4.1
,
1. ,
,
. L : 1) L[ x1 (t ) x 2 (t )] L[ x1 (t )] L[ x 2 (t )] ; L[ x(t )] . 2) L[ x(t )] 2.
,
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1. .
D
3.
, L
(t ) : D[ x(t )] ,
4.
L[ x(t )]
(t ) . ,
. (
5.
)
y (t )
, t
t.
x(t )
,
6. y (t ) t,
x(t )
t
. ,
2. . .
(
)
, . ,
1. . 2.
X (t ) Y (t ) . 2.4.2. .
X (t )
Y (t )
X (t )
Y (t ) . . , , . ,
, (
,
). . 1.
y x y
f 1 ( y; t )
(x) ,
( y) . Y (t )
f 1 y ( y; t )
X (t )
f1 x ( ( y ); t )
f1x ( ( y ); t )
d ( y) , dy
(2.4.1)
X (t ) .
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2.
y Q x
q
(x) ,
,
( y ), q 1, Q . f 1 y ( y, t )
Y (t )
X (t ) Q
f1 y ( y; t )
f1 x (
q
( y ); t )
d
( y)
dy
q 1
(2.4.1)
q
.
(2.4.2)
(2.4.2)
,
(1.3.1) (1.3.2). ,
X (t ) ,
Y (t ) .
.
3.
y
(x ) ,
: x1
H
x
1
( y1 ,..., y H ) ,
xH H ( y1 ,..., y H ) , ( x1 ,..., x H ) y ( y1 ,..., y H ) .
y H
Y (t )
f ( y; t ) f Hy ( y; t )
f Hx ( 1 ( y ),...,
f Hx ( x ; t ) H J H ( y)
H
HX (t )
( y ); t ) J H ( y ) ,
(2.4.3)
X (t ) , : 1
1
y1
yH
J H ( y)
. H
H
y1
yH
.
1
,...,
(2.4.4)
,
H
(2.4.3)
. 1.
4.
f Ny ( y1 ,..., y N ; t1 ,..., t N )
NY (t )
X (t ) N
f Ny ( y1 ,..., y N ; t1 ,..., t N )
f Nx ( ( y1 ),..., ( y N ); t1 ,..., t N ) n 1
f Nx ( x1 ,..., x N ; t1 ,..., t N ) N . N t1 ,..., t N . X (t )
X (t )
( yn ) , yn
(2.4.5)
X (t ) . Y (t )
Y (t ) ,
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( X n (t ) Yn (t ) , NX (t ) , Y (t ) NX (t ) , Y (t ) .
t n , n 1, N ).
N-
y N
f ( y1 ,..., y N ; t1 ,..., t N ) (2.4.3), '
X (t ) Y (t )
Y (t ) N-
,
: (2.4.3) (2.4.3)
. 5. ( x1 , x 2 ) , y 2
y1
1
...
(2.4.5).
x2 ,
: x1 x2
( y1 , y 2 ) , y2 . 1
f1 y ( y1 ; t )
X (t )
X (t ) . , N
N-
( X 1 (t ), X 2 (t ))
Y1 (t )
: f1 y ( y1 ; t )
f 2x ( ( y1 , y 2 ), y 2 ; t )
f 2x ( x1 , x 2 ; t ) .
1
( y1 , y 2 ) dy 2 , y1
(2.4.6)
X (t ) . 3.
J ( y)
1
1
y1 0
y2 1
1
y1
.
(2.4.3) f 2y ( y1 , y 2 ; t )
f 2x ( ( y1 , y 2 ), y 2 ; t )
1
y1
. f 2y ( y1 , y 2 ; t )
f1 y ( y1 ; t ) , .
(2.4.6). (2.4.6)
Y (t ) , ,
X 1 (t )
X 2 (t ) (
, .
2.4.2). . ,
Y (t ) . y
1. y
m (t ) m y (t )
-
( x ) f1 x ( x; t )dx .
M Y (t )
(x) .
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(2.4.7)
2.4.2 y
f 1 ( y; t )
'
Y (t ) f 2x ( x1 , x 2 ; t )
X (t )
( X 1 (t ), X 2 (t )) f1 y ( y; t )
Y (t )
y
x1
x2
f 1 y ( y; t ) =
f 2x ( y
x 2 , x 2 ; t )dx 2
y
x1
x2
f 1 y ( y; t ) =
f 2x ( y
x 2 , x 2 ; t )dx 2
y
x1 x 2
f 1 y ( y; t ) =
x1 x2
y
f 1 y ( y; t ) = .
(2.4.7)
f 2x (
dx y , x2 ; t ) 2 x2 x2
f 2x ( yx 2 , x 2 ; t ) x 2 dx 2 ,
Y (t ) m y (t )
( x ) f 1x ( x; t )dx .
y
2. y
y
(t )
(t )
-
(x) . .
( x) m y (t ) f 1x ( x; t )dx .
M Y (t ) m y (t )
Q
3.
q
xq , (
q
q 1
,
q 1, Q ). .
. 2.4.3. X 1 (t ) , X 2 (t )
1.
Y (t ) . X q (t ), q 1, Q
2.
Y (t ) Q
R y (t , t )
2 q
R xq ( t , t ) ,
2 q
D x q (t ) .
q 1 Q
D y (t ) q 1
3. ,
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:
X 1 (t ) .
X 2 (t )
2.4.3 m y (t ) , R y (t , t )
Y (t )
D y (t , t )
m xq (t ) ,
Rxq xh (t )
X q (t )
D xq (t ) Q
y
q
xq
q 1
Q
m y (t )
q
m x q (t )
q 1 Q 2 q
R y (t , t )
R xq xq (t , t )
q 1
q
h
R xq x h ( t , t )
q h
Q 2 q
D y (t )
D xq (t )
q 1
q
h
R x q xh ( t , t )
q h
2.4.3.
Y (t )
X (t ) . Y (t )
X (t ) , . -
(
)
,
, X (t ) ,
. . .
X (t )
Y( ). Y( )
Lt
: Lt X (t ) , t.
, 1. ),
(
(2.4.8)
D1
D2 D 2 D1 x .
D1 D 2 x
. M
1. 2.
Y( )
X (t )
. (
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. 2.4.4).
2.4.4
my ( )
Lt m x ( t )
Ry ( , ) Dy ( )
Lt Lt R x (t , t ) Ry ( , ) |
: d x (t ) , dt
y (t ) y( ) 3.
x (t )dt . . 2.4.5 2.4.6 ,
. 2.4.5
m y (t )
d m x (t ) dt 2
R y (t , t ) D y (t )
R x (t , t ) t t 2 R x (t , t ) |t t t t
2.4.6
my ( )
mx ( t )dt '
Ry ( , ) Dy ( )
2
R x (t , t )dtdt R x (t , t ) dtdt
. 2.4.7.
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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
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3. , , . '
.
,
. (
)
.
3.1. X (t )
1. ( ), X (t1 ),..., X (t N )
-
1
f N ( x; t )
1 (x 2
exp
N
(2 ) Rx
( x(t1 ),..., x(t N )) T N -
x
1
m x ) T Rx ( x m x ) ,
-
, n-
(m x (t1 ),..., mx (t N )) T , n-
X (t n ) ; m x N-
-
X (t n ) ; R x
(3.1.1)
{R x (t n , t m )}
N,
N
X (t n ) , X (t m ) (n, m 1, N ); Rx
1
.
Rx ; T
,
, . 1. mx (t )
R x (t , t ' ) .
,
.
,
. 2. ,
t
( f N ( x; t ) ,
): (3.1.2) f N ( x; t ) R x (t , t ' ) 0 ,
f1 ( x1 ; t1 )... f1 ( xN ; t N ) . ,
(3.1.2). t
t' .
X (t )
3. .
0
f N ( x; t )
N-
t'
. R x (t , t ' )
,
.
X (t / X (t m ) -
, .
x(t m ), ( m 1, M )) N
M
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fN
M
f N ( x1 ,..., x N ; t1 ,..., t N ) . (3.1.3) f M ( x1 ,..., x M ; t1 ,..., t M )
( x M 1 ,..., x N ; t M 1 ,..., t N / x1 ,..., x M ; t1 ,..., t M ) , (3.1.3)
(3.1.1). ,
.
,
. X (t )
)
4. (
( tn
0
.
,
. Rx (t n , tm )
)
,
tm , .
X (t ) . , . , . 5. . . 6. . ,
7. (3.1.1)
: QN ( j ; t )
exp j
T
mx
1 2
T
Rx
.
8. .
3.2. 3.2.1. 1. . .
.
X (t )
1. ( -
),
N-
t2
N
t,
t1 ,..., t N
t1
t1 ,..., t N
X (t ) . ,
,
. 1. t,
Nf N ( x1 ,..., x N ; t1 ,..., t N )
N-
f N ( x1 ,..., x N ; t1
,..., t N
),
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(3.2.1)
. 2.
X (t ) f 1 ( x; t ) f 1 ( x ) . 3. X (t )
t,
t,
t 2 t1 f 2 ( x1 , x 2 ; ) .
f 2 ( x1 , x 2 ; t1 , t 2 )
,
. 3.2.1. 3.2.1
m x (t )
mx
const
D x (t )
Dx
const
R x ( t1 , t 2 )
R x (t1
t2 )
Rx ( )
Rx ( ) Dx
rx ( ) '
1.
N
2
, N
t.
2 2.
( 1 .
.
1 1 ).
X (t ) K,
N N
t,
K K.
(3.2.1)
(3.2.1)
.
K,
N
,
, N
K.
1 N
,
N
K.
X (t )
1 . ,
N-
N
t,
: lim f N ( x1 ,..., x N ; t1
,..., t N
).
X (t )
1 .
T, t
1
T.
X (t )
1 . (
)
, N
f N ( x1 ,..., x N ; t1 ,..., t N )
f N ( x1 ,..., x N ; t1
kT0 ,..., t N
N-
T0 , kT0 ) ,
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K
k
0, 1,.... . .
X (t )
2.
Y (t ) (N
, t2
X (t ) ,
t1 ,..., t N ' 1
t ,..., t
' M
t1 ; t
' 1
t1 ,..., t
' M
M) -
M
N t1 ,
t1 ,..., t N
Y (t ) .
,
1.
,
,
f N M ( x1 ,..., x N , y1 ,..., y M ; t1 ,..., t N , t1 ' ,..., t M ' ) f N M ( x1 ,..., x N , y1 ,..., y M ; t1 ,..., t N , t1 ' ,..., t M ' ) . 2. X (t ) Y (t ) , . . X (t ) 3. , ( mx (t ) m x const ), t: K x ( t1 , t 2 ) K x ( t1 t 2 ) . 1. . , , . ( . 3.2.1).
. 3.2.1.
2. . 3. , . ,
.
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X (t )
4. ,
Y (t ) ,
t:
K xy (t1 , t 2 )
M xy [ X (t1 )Y (t 2 )]
K xy ( ),
t2
t1 .
(3.2.2)
. ,
.
2. ,
.
. ,
3.2.2. 3.2.2 ( (
)
) | Rx ( )| Dx ; | rx ( ) | 1 0 : R x (0)
: Rx ( )
Rx (
,
) ; rx ( )
R xy ( )
R yx (
D x ; rx (0 ) 1
rx (
) ; rxy ( )
0,
ryx (
)
, , . 3.2.2).
. ,
| |,
)
(
,
Rx ( )
0
. 3.2.2. . 3.5.3.
( )
( )
3.2.2,
3.
.
,
,
,
.
,
.
. ,
.
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.
,
.
, . .
3.2.2. 1. . Y (t )
. (t ), g( t )
(t ) X ( t )
, X (t) g (t )
g (t ) , ,
(t ) . :
m y (t ) (t )mx g (t ); Ry ( t1, t2 ) ( t1 ) ( t2 ) Rx ( t2
t1 );
2
Dy (t ) ry ( t1, t 2 )
( t ) Dx ; rx ( t2 t1 ).
2. ,
.
, .
X (t )
1.
B x (t1 , t 2 )
M x [( X (t1 )
X (t 2 )) 2 ] .
(3.2.3)
2.
X (t) ,
X (t )
X (t
X (t )
)
, m
M [ X (t )] const ,
R ( )
(3.2.4) 2
M [ X (t ) X (t
)] m .
(3.2.5)
1. . 2. m 0; R ( ) 2Rx ( ) Rx ( ) Rx ( B x ( ) 2[ R x (0) R x ( )] , (3.2.3) (3.2.5). 3. ,
(3.2.6) (3.2.7) (3.2.8)
);
. 3.2.3. , 1.
, f (x)
y
( x, y , )
)
, . (
0
, ,
. 1.
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( x, y , )
0,
( x, y, )
(3.2.9)
0
. ,
2. . (3.2.9)
3.
,
. ,
4.
x, y
,
,
,
,
'
x
x
y
y y
2k 1 (k 2t
f
A(t )
y
0.
(3.2.10)
y
. )
''
0,
'
f
A(t ) cos 2 ft (
,t
k t
f
A(t ) ,
0, 1,...). 1.
y
A(t ) cos (t , ) ,
A(t ) t, ,
(t , )
t,
y
A(t )
y
A(t ) . y
.
A(t )
(t )
, 2.
A(t ) cos (t ) ,
.
k
Rx ( ) , (
)
,
. . : 1)
, (
( 2)
0,1);
x
) , rx ( )
( ):
rx
k
d ,
(3.2.11)
( )d .
(3.2.12)
0
k
x 0
.
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3.2.4. 1.
X (t )
( X (t1 ),..., X (t N ))
,
,
K x (t1 , t 2 ) ( t1 .
t2
)
. . 3.2.3. 3.2.3
Rhl (
Rlh ( )
)
Rhl ( )
Dh Dl
rhl ( )
1
3.3. (
)
,
. . .
.
(
1.
,
)
-
, ,
, ,
.
1. . (
2.
) (
, l.i.m. T
1 T
3)
T 2
X (t )dt
mx .
(3.3.1)
T 2
X (t )
2.
mx
, R x ( t1 , t 2 )
,
lim
T
1 T2
T T 2 2
R x (t1 , t 2 )dt1 dt 2
0.
(3.3.2)
T T 2 2
. (3.3.2)
1. lim Rx (t1 , t 2 )
t 2 t1
0,
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(3.3.3)
R x (t1 , t 2 ) t2
t1 .
(3.3.3)
,
(3.3.2). (3.3.2)
2. : T
1 lim T T
10
Rx ( )d
T
lim Rx ( )
0.
0.
2
3.
.
, .
,
,
.
X (t )
3.
R ( ),
l.i.m.
Rx ( )
T
1 T
x T 2 0
0
X (t
) X (t ) dt .
T 2
3.
Rx ( ) lim
T
1 T
T
10
2
t
[ Rx ( ) R x (
-
0
0
)Rx (
0
)]d
0
.
3.4. .
. K x (t1 , t 2 )
1. X (t ) :
K x (t1 , t 2 )
M [ X (t1 ) X * (t 2 )].
(3.4.1)
Rx (t1 , t 2 )
2. X (t ) : R x (t1 , t 2 ) 1.
*
M [( X (t1 ) m x (t1 ))( X * (t 2 ) m x (t 2 ))] .
(3.4.2)
: K x (t1 , t 2 )
*
Rx (t1 , t 2 ) m x (t1 )m x (t 2 ) .
(3.4.3) ,
2. : K x (t1 , t2 )
Rx (t1 , t2 ) .
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X (t )
3. t1 R x (t1 , t 2 )
,
t2
0, K x (t1 , t 2 )
m x (t1 )m x * (t 2 ) .
(3.4.4) X (t )
4. t1 K x (t1 , t 2 )
,
t2 0,
R x ( t1 , t 2 )
m x (t1 )m x * (t 2 ) .
(3.4.5)
5.
K xy (t1 , t 2 ) X (t )
Y (t ) : M [ X (t1 )Y * (t 2 )].
K xy (t1 , t 2 )
(3.4.6) R xy (t1 , t 2 )
6. X (t )
Y (t ) : *
M [( X (t1 ) m x (t1 ))(Y * (t 2 ) m y (t 2 ))] . X (t ) Y (t ) t1 : K x (t1 , t 2 ) K xy (t1 , t 2 ) K (t1 , t 2 ) , K yx (t1 , t 2 ) K y (t1 , t 2 )
R xy (t1 , t 2 ) 1.
R(t1 , t 2 )
t2
(3.4.8)
Rx (t1 , t 2 ) Rxy (t1 , t 2 ) . R yx (t1 , t 2 ) R y (t1 , t 2 )
(3.4.9) X (t )
Y (t ) t1 : Rxy (t1 , t 2 ) 0 ,
7. , Rxy (t1 , t 2 ) (3.4.9)
(3.4.7)
t2
. X (t )
Y (t ) t1 : K xy (t1 , t 2 ) 0 ,
8. , K xy (t1 , t 2 ) (3.4.8)
t2
. 9. rx (t1 , t 2 ) rxy (t1 , t 2 )
R x ( t1 , t 2 ) D x (t1 ) D x (t 2 ) Rxy (t1 , t 2 ) D x (t1 ) D y (t 2 )
,
(3.4.10) .
(3.4.11)
.
.
. ,
.
. 3.4.1. 74 Create PDF files without this message by purchasing novaPDF printer (http://www.novapdf.com)
3.4.1
Rx (t1 , t 2 )
R xy (t1 , t 2 ) ,
rxy (t1 , t 2 ) ,
rx (t1 , t 2 )
X (t ) , Y (t ) / 1
R xy (t1 , t 2 ) = R *yx (t 2 , t1 ) ;
rxy (t1 , t 2 ) = ryx* (t 2 , t1 )
| Rxy (t1 , t 2 ) | 2 D x (t1 ) D y (t 2 ) ; 2
Y (t1 ) aX (t1 ) b , a ) z1 ,..., z N
( b t1 ,..., t N
3
| rxy (t1 , t 2 ) | 1
N
R x (t m , t n ) z m z n*
m ,n 1
1, 3
. ,
2
.
3.5. 3.5.1.
xT ( t )
T
1. ,
: 2 A exp( j t) , T
xT (t )
1 xT (t )e TT
A (3.5.1)
j
2 t T
(3.5.1)
dt .
(3.5.2)
(3.5.2) (
A . -
2 v T A
) -
arg(A )
,
v A
-
. A ,
1.
, ;
A
; -
. x(t) :
2. x(t )
1 2
A( )e j t d ,
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, (3.5.3)
A( )
j t
x(t )e
dt .
(3.5.4) ,
. x(t),
x(t) x(t)
, 1 x ( t 0) x ( t 0) 2 x (t )
x(t)
, x(t)
-
(
). . 1. x(t )
.
(
. 3.5.1).
. 3.5.1.
( )
( )
,
2.
A( ) A( ) A* ( ) . . )
,
xT (t )
( )
, (3.5.3)
2 f (f
(3.5.4)
,
,
. ,
.
x(t )
A( ) A ( ) A ( )
2. ,
x(t )
A ( )
A( )
A(
)
2 A( )
0,
A(0)
(3.5.5)
0.
x(t )
1. :
x(t ) = A ( );
1 2
A ( ) cos
t
( ) d ,
0
( )
: A ( )= A ( ) ,
( ) = arg A ( ) ,
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j t
2 x (t ) e A ( )
A ( )e
j
dt
0,
( )
x (t ) dt
0. xT (t )
2.
T
:
x(t ) =
2 t T
A exp( j 0
A
),
= arg A ,
A =A ,
:
2
A
A e
j t 2 xT (t ) e T dt TT 1 xT (t ) dt TT
j
0, 0.
3.5.2. .
30. (
1.
. . Ax ( f ) Ax ( f ) ,
)
X (t )
. '
X (t )
' : Ax ( f )
j 2 ft
X (t ) e
dt .
1. , . ma ( f )
1.
X (t )
ma ( f )
mx
(f)
mx ( f ) ,
-
. 2. K a ( f1 , f 2 ) Sx ( f )
K x ( )e
j2 f
d
S x ( f1 ) ( f 2
f1 ) ,
(3.5.6)
,
(
). (3.5.6)
3.
, . ,
4. ,
.
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5. ,
,
,
,
. ,
2.
X (t )
Sx ( f ) , Kx( )
'
' : j2 f
Sx ( f )
K x ( )e
d ,
Kx ( )
S x ( f )e j 2 f df .
(3.5.7) (3.5.8) Sx ( f ) .
(3.5.7), (3.5.8)
. Kx ( )
(3.5.7)
1.
Sx ( f )
,
X (t )
S0 ( f ) x 0
X (t ) :
X (t )
mx
S x ( f ) = S 0 ( f ) + m x2 ( f ) ,
(3.5.9)
x
Sx ( f )
S0 ( f ) x
. (3.5.8)
2. Dx
,
R x ( 0)
S 0 ( f )df ,
(3.5.10)
x
S0 ( f ) . x
.
(3.5.10)
S0 ( f )
,
x
. Ax ( f ) ,
3. Sx ( f ) .
. S0 ( f )
,
4.
'
x 0
A0 ( f )
X (t )
x
S 0 ( f ) = lim x
T
1 M [ A0 ( f ) A*0 ( f )] , x x T
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(3.5.11)
0
T
X (t ) . . X (t ) S x ( f ) 0. K a ( f , f ) 0 ).
1. (
) (3.5.6) ( 2. X (t )
f1
f2
, f
Kx( ) ,
,
(3.5.7), Sx ( f ) =Sx ( f ) .
,
X (t )
3. (3.5.7), (3.5.8)
S0 ( f ) x 0
X (t )
: S0 ( f )
2 R x ( ) cos 2 f d ,
x
(3.5.12)
0
2 S 0 ( f ) cos 2 f df .
Rx ( )
(3.5.13)
x
0
(3.5.12), (3.5.13),
4.
(3.5.5), , Sx ( f )
: 4 K x ( ) cos 2 f d
f
0,
0
Sx ( f )
(3.5.14)
2 Kx( ) d
f
0,
0
Kx ( )
S x ( f ) cos 2 f df .
(3.5.15)
0
.
(3.5.14), (3.5.15) ,
(3.5.7), (3.5.8),
(3.5.12) (3.5.13) 3.
. 0
X (t ) s0 ( f ) ,
x
S0 ( f )
:
x
S0 ( f ) s0 ( f ) = x
x
D0
,
x
D0
.
x
(3.5.7), (3.5.8),
1.
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(3.5.16)
s0 ( f ) rx ( )
j2 f
rx ( )e
x
d ,
(3.5.17)
s 0 ( f )e j 2 f df ,
(3.5.18)
x
s0 ( f )
x 0
X (t )
rx ( ) . s0 ( f )
2.
x
: s0 ( f )
s 0 ( f )df
0,
x
1,
x
rx ( )
.
,
. ,
3. , . 2,
,
. (
3.
)
,
: fe
S0
fe
1 S 0 ( f )df , S0 0 x
(3.5.19)
max S x ( f )
.
f
fe
5. k
fe
: 1
.
(3.5.20)
k
3.5.3.
. ,
1. fe f0
f0 ,
(3.5.21)
,
(
).
(3.5.21)
,
. 1. f0 (
0
)(
. 3.5.2).
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. 3.5.2.
-
2. f0
(
, ,
.
Rx ( )
. 3.5.2),
Qc ( ) R x ( ) Qc ( ) cos 2 f 0 (
cos 2 f 0 :
. 3.2.2, ). 2.
N (t )
, N0 2
Sx ( f )
const .
(3.5.22)
. -
1. N0 ( ). 2
Kx( )
, (3.5.23)
(3.5.6). 1. ,
(3.5.23) Rx ( ) . (3.5.23)
2.
1
,
, ,
,
0.
k
2. .
(3.5.19), (3.5.22). .
3. (3.5.23). (
.
)
,
. , .
3.
Y (t ) X (t ) N (t ) , , N (t )
X (t )
.
1.
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R y (t1 , t 2 )
Y (t )
2. X (t )
N (t )
N0 (t 2 2
M [| X (t1 ) | 2 ] Y (t )
t1 ) .
X (t ) , X 1 (t )
X 2 (t ) N (t ) , .
X 1 (t )
X 2 (t )
3.5.4. ,
,
. (
1.
X (t )
)
Y (t )
S xy ( f ) ,
K xy ( )
: j2 f
S xy ( f )
K xy ( )e
d ,
K xy ( )
S xy ( f )e j 2 f df .
(3.5.24) (3.5.25) Sx ( f )
1.
X (t ) ,
S xy ( f ) .
Sx ( f ) ,
2.
S xy ( f )
. S *yx ( f ) .
S xy ( f )
(3.5.26)
(3.5.24). , | S xy ( f ) | 2 S x ( f )S y ( f ) .
3.
(3.5.27)
2.
X (t )
2 xy
2 xy
1. (f)
X (t ) , Y (t )
,
(f)
Y (t ) | S xy ( f ) | 2
S x ( f )S y ( f ) (3.5.27) f
, 2 xy
0
.
(3.5.28) , [0, 1] . 2 xy
(f)
0,
( f ) 1. 2 xy
2. 2 xy
rxy2 ( )
r ( ).
X (t )
Y (t ) .
,
(f) rxy2 ( ) ,
,
.
Y (t )
3.
X (t ) ,
, 2 xy
(f)
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.
Y (t )
X (t ) 1 xy ( f )
,
2 xy
(f)
Y (t ) Y (t )
.
3.6. 3.6.1. 1. ,
: y (t )
x (t1 )h(t , t1 )dt1 ,
x(t ) , y (t )
(3.6.1) , h (t , t1 )
, .
h (t , t1 )
1. ( h(t , t 2 )
)
-
(t 2
t1 )h(t , t1 )dt1 .
: (3.6.2)
. -
2.
(3.6.1). -
3.
. . h(t , t1 ) . (
2.
),
. . 1.
. ,
. 2. t , t1 ,
(3.6.1)
h ( t , t1 )
h (t , t1 ) t t1 .
h( ) ,
: y (t )
x (t1 )h(t t1 )dt1
x(t
) h( ) d .
(3.6.3)
,
3. .
,
4. -
. . | y (t ) | |
x (t
) h( ) d |
| x (t
) || h( ) | d ,
h( )
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(3.6.4)
: | h( ) | d
.
,
5.
, h(t ) , .
, .
,
,
, ,
1.
t
,
y(t ) x (t 0 ) ,
0
, t
t0
x(t ) ( t
t 0 ).
,
2.
,
(3.6.3)
t
y (t )
x (t1 )h(t t1 )dt1
x (t
)h( )d
.
(3.6.5)
0
t,
x(t ) t
y (t )
t
x (t1 )h(t t1 )dt1
x(t
0
)h( )d .
(3.6.6)
0
(
6. K( )= K( ) e
j ( )
)
(
)
h(t ) : K( )
j
h( )e
d .
(3.6.7)
K( )
7.
K( )
-
( )
,
-
. 1.
'
' : 1 2
h( )
K ( )e j d .
.
.
Ax ( )
2.
(3.6.8) '
Ay ( )
.
.
Ay ( ) = K ( ) Ax ( ) .
(3.6.9)
Sy ( )
3.
Sx ( )
K( )
' Sy( )
2
K( ) Sx ( ) .
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(3.6.10)
K( )
1.
h(t ) K1 ( ) , K 2 ( ) h1 (t ) , h2 (t ) ,
K ( ) = K1 ( ) K 2 ( ) , h(t )
(3.6.11)
h1 (t1 )h2 (t t1 )dt1 . K( )
2. K1 ( ) , K 2 ( ) ,
K2 ( )
, K( )=
K1 ( ) . 1 K1 ( ) K 2 ( )
(3.6.12)
3.
h( )
g( ) . g( )
5. ,
.
g( )
.
h( ) : t
g (t )
h(t )
h( ) d , 0
dg (t ) . dt
(3.6.13)
4. K( )
H ( p) ,
: H ( p)
h( )e
p
d ,
p
j .
0
,
1.
,
K( ) H ( p) (
p
j ). 2.
H ( p)
, .
0.
5. d M y (t ) ... a0 y (t ) dt M (m 1, M , n 1, N ) aM
a m , bn
bN
d N x (t ) ... b0 x (t ) , dt N ,
: y (0) , y'0) ,
, y 3.6.1. -
(M 1)
,
( 0) . :
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,
, ,
, . . ,
h(t )
K( ). K( ),
-
h(t ) . 3.6.1.
e
3.6.1
K( )
h(t ) K0 0
|
K0
|
|
|
t)
sin( t
K0
K0 0
2 0
2 0
2
0
RC -
t
sin( 2
2 t
) cos
2
t
1(t) e
j
j(
0
2
1(t ) e t cos 0 t
)
t2
2
1 2
2
e
1(t )
2 0)
( 1 2
1(t )
2
2
e
2
0t
2
2 cos
2
2
e
t2
e
2 2
2
2
0t
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4 2
: 1. 1(t )
1
t
0;
0
t
0.
2.
, RC-
,
,
, .
3.6.2.
Y (t ) X (t )
'
.
, ,
' . 3.6.2 3.6.3 ( t ). 3.6.2
, , t
m y (t )
m x (t
) h( ) d
t1 t 2
K y (t1 , t 2 )
K x ( t1
1
, t2
2
) h( 1 ) h( 2 ) d 1 d
2
0 0 t2
K xy (t1 , t2 )
K x (t1 , t2
) h( ) d
0 t1 t 2
R y (t1 , t 2 )
R x ( t1
1
, t2
2
) h( 1 ) h( 2 ) d 1 d
2
0 0 t2
Rxy (t1 , t 2 )
R x ( t1 , t 2
) h( ) d
0
3.6.3 , / 1
t
m y (t )
m x h( ) d 0
2
t1 t 2
R y (t1 , t 2 )
R x (t 2
2
(t1
1
))h( 1 )h( 2 )d 1d
2
0 0 t2
3 R xy (t1 , t 2 )
Rx (t 2
t1 ) h ( ) d
0
4
K 2 ( ) S x ( ) (t 0
Sy( )
. 3.6.3
1. ,
T) , ,
.
,
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,
,
h(t )
( t).
,
t0 ,
T,
. 3.6.3
1, 2, 3
: T
my
m x h( ) d , 0
T T
Ry ( )
Rx (
(
2
1
))h( 1 )h( 2 )d 1 d
2
,
0 0 T
Rxy ( )
Rx (
1
) h( 1 ) d 1 .
.
0
t0
, ,
T
4
2.
t0
. 3.6.3
T,
. 3.6.3. (3.5.23),
, :
N0 h( 2 0
Ry ( )
)h( )d
1
Rxy ( )
Ry ( ) ,
1
Sy ( )
,
N0 2 K ( ), 2
N0 h( ) . 2 Dy
Sy ( ) . 3.6.4. 3.6.4
Ry ( ) ,
Dy
Sy ( ) Ry ( ) Dy
sin(
Dy K 02 N 0 2
)
Sy( ) K 02 N 0 2 0
| |
|
|
2 0
K N0 2 sin
2
Dy
cos
K 02 N 0 2
0
2
0
2
0
2
0
0
0 RC-
N0 4
Dy e
N0 2(
2
2
2
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2
)
2
Dye
cos 2
Dy e i
2
Dy e
N0 2
0
N0 e 4
4 2 N0 cos
N0 e 4
2 2
0
2
2
)2 )
0 2
N0
2
2
2
2
2(
N0 (
(
2 2
0) 2 2
2
. 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
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4. 4.1. 1. -
t1
X (t ) , t T ... t N ,
t2
,
T,
F1 ( x N ; t N / x1 ,..., x N 1 ; t1 ,..., t N 1 ) = F1 ( x N ; t N / x N 1 ; t N 1 ) . 1. (4.1.1) , tN . 2. (4.1.1) , X (t ) N :
X (t )
(4.1.1) tN 1 ,
N 1
F N ( x1 ,..., x N ; t 1 ,..., t N ) = F1 ( x1 ; t1 )
F1 ( x n 1 ; t n 1 / x n ; t n ) ,
(4.1.2)
n 1
N-
, t
tn
t1 , ( n
t1 2, N ) .
S. 4 ;
: ;
; . . 4.1.1.
4.1.1 S
4.2. t0 { l ,l
1, L}
t1 S (
X (t ) , ... ,
T
. 4.2.1).
t 0 , t1 ,...
(
). XN .
X (t N ),
N
0,1,...,
, F1 ( x N / x0 ,..., x N 1 ) = F1 ( x N / x N 1 ), N . (4.2.1) ,
1, 2,... .
1. (4.2.1) P{x N / x0 ,..., x N 1 }
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, XN X 0 , X 1 ,..., X N 1
xN , x0 , x1 ,...x N 1 , P{x N / x0 ,..., x N 1 } = P{x N / x N 1 } .
. 4.2.1.
2. , F1 ( x N / x0 ,..., x N 1 ) = F1 ( x N / x N
M
,..., x N 1 ), N
M,M
1,
.
-
3.
XN X 0 , X 1 ,... ,
FM ( x N / x0 ,..., x N 1 ) = FM ( x N / x N 1 ) . 1. -
XN
(X N
M 1
,..., X N ) .
2 3. . . N-
2. N
P{ X 0
l 0 ,..., X N
l N } = P{ X 0
P{ X n
l0 }
ln
/ Xn
ln
1
1
}.
n 1
1. . 4. tn ,
l
(m, n)
P{ X n
l
/ Xm
k
},
S. k
(
k
,
l
l
(n m) kl
k
tm
k
S ; k , l 1, L ) (m, n)
tm , tn
L L.
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l
,
5. 1) 2)
'
:
;
-
. .
3. 4. Pk (m)
m-
k kl
(k
(m, n) Pl (n)
1, L )
nn-
l l
.
L
Pl (n)
Pk (m)
kl
(m, n), (l
1,..., L; 0
m
n) .
(4.2.2)
k 1
P(n)
(4.2.2) P ( n) L
-
P(m) (m, n) , Pl (n) , l nk m-
5.
1, L , .
l k
kl(m,n),
m-
q
r-
(0
m
r
kq ( m, r ) ,
l
n-
n-
ql
n) q
r-
( r , n) .
L kl ( m, n )
kq
(m, r )
ql
(r , n),
(k , l
1,..., L; 0
m
n),
q 1
(m, n) (4.2.3) . , , (4.2.3)). 6.
(m, r ) (r , n) .
(4.2.3) .
(4.2.3)
, ( ,
P(0))
(
(n 1, n)
n 1, N .
P(N ) , tN ,
: N
P( N )
P ( 0)
(n 1, n) . n 1
,
5.
(m, n)
tm , tn ,
,
(nm),
(m, n)
( n m) .
,
(4.2.4) (4.2.4),
. 6. P( N )
P ( 0)
N
, .
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(4.2.5)
6. P= lim P(N). N
.
. 7.
,
1) 2)
:
; , (
t 0 ).
. . 8. , P(n)
tn
P ( n)
tn ,
P
n. ,
8.
P
P .
(4.2.6)
1. 2.
(4.2.5). ,
(4.2.6) P
. .
3. . 4.2.1.
4.2.1
(m, n) P(n)
( n m)
P
P( N )
P
lim P( N )
N
P
2)
l-
. . 4.2.2. .
.
9.
, ,
l
-
l l
N
n
P 1) P
P ( 0)
-
. ,
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-
,
-
. 10.
,
l
, l,
,
, .
11.
l
,
l
,
k
,
k
k
l
,
.
l-
. 4.2.2. :1
;2
;3
.
,
12. .
r
13. n1 , n 2 , ... ,
l
.
r,
l
r 1,
r 1.
, 14.
,
l
, . . (
15.
),
. (
.
, )
(L
).
4.3. X0
1.
X (t 0 ),..., X N
X (t )
X (t N ) t0
,
t1
tN
-
f1 ( x N ; t N / x 0 ,..., x N 1 ; t 0 ,..., t N 1 )
X (t )
... t N .
f1 ( x N ; t N / x N 1 ; t N 1 ) .
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(4.3.1)
(4.3.1)
1.
, X (t )
: N
f N ( x 0 ,..., x N ; t 0 ,..., t N )
f 1 ( x0 ; t 0 )
( xn ; t n / xn 1 ; t n 1 ) ,
(4.3.2)
n 1
( xn ; t n / xn 1 ; t n 1 ) = f 1 ( x n ; t n / x n 1 ; t n 1 ) xn
tn
1
xn
1
tn . (4.3.1)
2.
,
-
,
. . 4.3.1. 4.3.1
( x; t / x ; t )
'
0
( x; t / x ; t )dx 1 lim ( x; t / x ; t ) t
t
( x; t / x0 ; t 0 )
(x
x)
( x; t / x ; t ) ( x ; t / x 0 ; t 0 )dx
.
.
X (t ) t0
1. ( x; t / x 0 ; t 0 )
x0
x
t ( x; t / x0 ; t 0 )
t0 A ( x0 ; t 0 )
1
lim t
0
1 M ( X (t 0 t
A ( x0 ; t 0 ) !
t / x0 ; t 0 )
.
(4.3.3)
X (t 0 / x0 ; t 0 )) . X (t 0
,
( x; t / x 0 ; t 0 ) , x0
t / x0 ; t0 ) t.
X (t 0 / x 0 ; t 0 ) A ( x0 ; t 0 ) -
. : X (t 0 );
1.
( x; t / x 0 ; t 0 )
( x; t / x0 ; t 0
t)
X (t 0
t );
X (t ) .
( x; t / x ; t 0
t ) (x ; t0
t / x 0 ; t 0 ) dx .
(4.3.4)
( x; t / x 0 ; t 0
t ) ( x ; t0
t / x 0 ; t 0 ) dx
(4.3.5)
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(
, ). (4.3.4) ( x; t / x0 ; t 0 )
(4.3.5), ( x; t / x0 ; t 0
( x; t / x ; t 0
t)
: t)
( x; t / x0 ; t 0
( x; t / x ; t 0
t)
t)
( x ; t0 (4.3.6) t)
( x; t / x0 ; t0
( x ' x0 ) ( x; t / x0 ; t0 ! x0 1 (4.3.7) (4.3.6), ( t 0 ). 2. (4.3.3)
t)
(4.3.6)
t / x 0 ; t 0 ) dx . :
(4.3.7)
. t (4.3.3).
, (
3,
)
: t0
( x; t / x 0 ; t 0 )
a( x0 ; t 0 ) a( x0 ; t 0 ) A1 ( x0 ; t 0 ) b( x0 ; t 0 ) A2 ( x 0 ; t 0 ) 3.
( x; t / x0 ; t 0 ), ,
. (4.3.8),
,
( x; t / x 0 ; t 0 )
t
( x; t / x0 ; t 0 )
x0
(4.3.8)
2 1 b( x0 ; t 0 ) 2 2 x0
x
a( x; t ) ( x; t / x 0 ; t 0 )
1 2 b( x; t ) ( x; t / x 0 ; t 0 ) (4.3.9) 2 x2
. (4.3.8) (4.3.9) (4.3.9)
1. 2.
t
f1 ( x; t ) t
-
.
x
f 1 ( x; t ) ,
(4.3.9)
a ( x; t ) f1 ( x; t )
1 2 b( x; t ) f1 ( x; t ) . 2 x2
(4.3.10)
t0 f1 ( x0 ; t 0 ) ( x; t / x 0 ; t 0 )dx 0 . f 1 ( x0 ; t 0 )
x0 , ,
,
4.
.
(4.3.10). (4.3.8), (4.3.9),
. 5. ( x; t / x0 ; t 0 ) t t 0 : ( x; t / x 0 ; t 0 )
, t
t0 , a
1. b
( x / x0 ; ) .
.
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(4.3.10) 3.2)
2. (
.
d b( x ) f 1 ( x) dx C
,
2a ( x ) f 1 ( x ) C ,
,
(4.3.11)
.
2. ,
.
.
. f1 ( x1 ; t1 ) ( x 2 ; t 2 / x1 ; t1 ) . f 2 ( x1 , x 2 ; t1 , t 2 ) , ( x 2 ; t 2 / x1 ; t1 )
f 2 ( x1 , x 2 ; t1 , t 2 ) f1 ( x1 ) ,
f1 ( x1 ; t1 )
t2 .
t1 .
f 2 ( x1 , x 2 ; ) , ( x 2 / x1 ; ) ,
6. :
,
f 1 f ( x)
1)
lim f1 ( x; t ) ; t
2)
. 1. . .
2.
4.4. 4.4.1.
(
)
1. dx dt
h( x, t )
h( x, t )
g ( x , t ) n( t ) ,
g ( x, t )
,
h( x, t ) h( y, t )
g ( x, t ) g ( y, t )
n(t )
Lx
(L
y
const
0) ,
. , . (
. 4.4.1)
m.
. . t
.
n(t ) ,
. ,
, -
R(t, t ) ,
n(t )
, M n (t )
R (t , t )
0; . 1 N 0 (t 2
(4.4.1) t) .
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(4.4.2)
v(t ) m
dv(t ) dt
,
n (t ) . t
,
0
, ,
. 4.4.1.
(4.4.3)
(4.4.3) .
,
2.
v(t ) , dv (t ) dt
n(t )
v(0)
n(t ),
. (4.4.4)
1.
0,
(4.4.4)
, t
v (t )
n(t1 ) dt1 .
(4.4.5)
0
. 4.4.1.
2.
,
. ,
.
X (t ) ,
3.
T, t0
,
t1
... t N
T
4. M X (t )
X (t 0 ) , X (t1 ) X (t 0 ) ,
, X (t N ) X (t N 1 ) . X (t ) , , t T t0 t1 ... t N T M X (t N / x (t 0 ),..., x(t N 1 ))
T,
x (t N 1 ) .
3.
.
f 1 (v; t ) t
1 N0 4
2
f1 (v; t ) . v2 3
4.4.1.
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4.4.1
/ 1
M V (t )
0
t t 2 v
2
(t )
M n( 1 ) n ( 2 ) d 1 d
N 0t 2
2
0 0
3
f1 ( v; t )
4
v (t 3 )
1 v2 exp( ) N 0t N 0t t3
v (t 2 )
n( )d t2
a(v; t )
0 N0 b(v; t ) 2 f N ( v0 ,..., vN vN 1; t0 , , tN
5 6 7
f1 ( v0 ; t0 )... f1 ( vN
v N 1; t N
M V (t N / v (t 0 ),..., v (t N 1 ))
8 : 1.
3
n(t )
,
'
tN 1 ) tN 1 ) v(t N 1 )
v(t )
,
n(t )
.
2.
4
t3
,
t2 t2 .
, 3.
5 (4.4.5)
4.
(4.4.5) (4.4.2). ,
7
(4.4.1),
6
. 5.
8
4
.
4.4.2.
X (t ) ,
.
,
dx(t ) dt n(t )
,
x (t )
n(t ) ,
(4.4.6) ,
. ,
,
X (t )
, x (t )
X (t ) (4.4.6) ,
,
. n(t )
Ce
t
,
C
. .
, .
(4.4.6), (4.4.6)
t
x (t )
e
t
e n( ) d 0
(
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(4.4.6)). x ( 0) x 0
(4.4.6) t
x (t )
x0 e
t
e
t
e n( ) d .
(4.4.7)
0
t3
t2
t1 , t1
0.
(4.4.7)
, t3
x (t 3 )
( t3 t2 )
x (t 2 ) e
e
t3
e n( )d .
(4.4.8)
t2
(4.4.8) t1 ,
x(t )
,
t3 t2 ,
. . 4.4.2. 2. R( )
2
e
. 4.4.2
m(t )
x0 e
t 2
2
2
(t )
1 e 2
R(t1 , t 2 ) t2 (t
t1 ; t R( )
)
a ( x; t ) b( x; t ) .
,
2 t
2
,
e
(1 e
min(t1 , t 2 ) 2
( )
4 t ), 0
e
x(t ) 2 N0 2
, ,
. 4.4.2.
2
N0
. 4.4.2.
m(t ) ,
,
R( ) ( )
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2
(t )
1. 2
f1 ( x; t ) t
x
[ xf1 ( x; t )]
2
N0 2
f1 ( x; t ) . x2
' 1
f1 ( x; t )
(t )
2
exp
( x m(t )) 2 . 2 2 (t )
2. . 1 2
f 1 ( x)
4.5.
exp
x2 . 2 2
'
4.5.1.
' (4.3.8) (4.3.10) .
' . 1. ( x; t / x0 ; t 0 ) , x.
f 1 ( x; t )
.
2. 1. G ( x; t )
1 b( x; t ) f1 ( x; t ) 2 x
a( x; t ) f1 ( x; t )
(4.5.1)
. t
'
1. , f 1 ( x; t ) ) -
x
. ( t.
x
, .
a ( x; t ) f 1 ( x; t ) ,
, 1 b( x; t ) f1 ( x; t ) 2 x 2. 1) G ( x1 ; t ) G ( x 2 ; t ) 0 . x1 x2 ; 2) f 1 ( x1 ; t ) f 1 ( x 2 ; t ) 0 . x1 x2 . 2. t
f1 ( x; t )
, . :
x
G ( x; t )
0.
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(4.5.2)
1. (4.3.10), 2.
. , f 1 ( x; t )
G( x; t ) ,
,
,
.
3. (4.5.2)
G( x; t )
, d
const .
b( x ) f 1 ( x )
(4.5.1)
2a ( x ) f1 ( x )
2G .
(4.5.3)
dx . '
(4.5.3) x
f 1 ( x)
x
C a( y) exp 2 dy b( x ) b( y ) x1
z
2G a( y) exp 2 dy dz . b( x ) x1 b( y ) x1 , f1 ( x ) ,
G x1
C
f1 ( x ) ,
. .
' ,
. , ,
,
,
, ,
. ,
4.5.2.
.
' ' ,
. ' f 1 ( x 0 ,0) , x0
f1 ( x 0 ,0),
(4.3.10) f 1 (c , t )
f 1 (d , t )
2
f 1 ( x 0 , 0) x0
2
0.
'
t
xi
0, c
x d. t x.
xi c, n
0,1,...; i
tn 0, I ; I
d
c x
,
tn ;
*
. f 1 ( x; t )
, :
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. (4.3.10)
2 1 b( x; t ) 2 f1 ( x; t ) 2 x
f1 ( x; t ) t
x
b( x; t ) a ( x; t )
x
(4.5.4)
2
1 2
x2
b( x; t )
x
a ( x; t ) f 1 ( x; t ). :
f 1 ( x; t )
f i ,n 1 f i , n f1 ( x; t ) , xi ,t n t t f i 1,n f i 1,n f1 ( x; t ) , xi ,t n x 2 x 2 f i 1, n 2 f i , n f i f 1 ( x; t ) xi , t n 2 x x2
f i,n
f 1 ( x; t )
1, n
,
f 1 ( xi , t n ) . a( x; t ), b( x; t )
,
(4.5.4)
:
f i ,n i ,n
,
i ,n
,
i ,n
1
i ,n f i
i ,n f i ,n
1, n
i,n
fi
1, n
, n
0,
(4.5.5)
a( x; t ), b( x; t ) .
, (4.5.5) f 1 ( x, t )
. ' x
'
.
'
t
,
, ,
, '
.
, (
.
'
).
b( x; t )
,
. .
0
t
x2 . b( x; t ) '
. .
, .
4.6. S, l .
l
1, L .
X (t ) , t T, -
T S t0
t1 .
1. , F1 ( x N ; t N / x0 ,..., x N 1 ; t 0 ,..., t N 1 ) = F1 ( x N ; t N / x N 1 ; t N 1 ) . . , P{x N ; t N / x 0 ,..., x N 1 ; t 0 ,..., t N 1 } , X (t N )
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xN x0 ,
S, , X (t 0 ), , X (t N 1 ) , xN 1 S , P{x N ; t N / x0 ,..., x N 1 ; t 0 ,..., t N 1 } = P{x N ; t N / x N 1 ; t N 1 } .
1. Pk (t ) = P{ X (t ) t (k
k
}
k
1, L)
kl
(t , t ' )
l
Pl (t ' ) = P{ X (t ' )
t'.
l
:
t'
l
}
L
Pl (t ' ) =
Pk (t )
kl
(t , t ' ) .
(4.6.1)
k 1
. 4.6.1. 1 4
1.
,
1, 2
4
. 2. ,
,
,
,
t
. 4.6.1
/ 1
'
kl
(t , t ' )
0
L
2
kl
(t , t ' ) 1, k
kl
(t 0 , t 0
1, L
l 1
3
lim t
0
t)
kl
*)
L kl (t 0 , t ' ) =
4 *)
.
kq
(t 0 , t )
ql
(t , t ' ) ,
q 1
t0
t
t' :
kl
k l; k l.
1 0
kl
2. , : L
t a ql (t )
t1
ql
kl
(t 0 , t )
kq
(t 0 , t )a ql (t ), k , l
1, L ,
q 1
(t , t1 ) t1 t .
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(4.6.2)
aql (t ) ,
2. (4.6.2), .
(
)
'
2. L
all (t )
a ql (t ) .
(4.6.3)
q 1 q l
ql
(4.6.3).
2 1, L)
(t , t ' ) (q, l
t.
t'
,
t' t t ) 1 all (t ) t ;
ll
(t , t
ql
(t , t
t ) aql (t ) t ( q (4.6.4), (4.6.5)
t
(4.6.4)
l ).
(4.6.5) ,
, t
,
,
,
(4.6.2). (4.6.4), (4.6.5)
(4.6.3)
. 3. : L kl
t0
(t 0 , t )
a kq (t 0 )
ql
(t 0 , t ), k , l
1, L, t
3
(4.6.6)
2. Pl (t )
4. t
t0 .
q 1
l
q 1, L
' t
: d Pl (t ) dt
L
Pq (t )a ql (t ), l
1, L .
(4.6.7)
q 1
(4.6.2)
. k.
Pk (t 0 )
(4.6.1),
(4.6.7).
L
(4.6.7)
.
,
Pl (t )
'
aql (t ) . ,
3. kl
(t , t ' ) kl (t , t ' ) =
t , t' : kl
( ),
t' t .
1. : a kl
(4.6.4), (4.6.5)
,
const . 2.
(4.6.2), (4.6.6)
:
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L kl
( )
kq
1, L ,
( )a ql , k , l
q 1 L
a kq
kl ( )
ql
1, L .
( ), k , l
q 1
,
4.
P(t ) = {Pl (t ), l P
1, L} ,
lim P(t )
,
t
. . . 5.
P(t )
, : P(t ) = P . 1. .
.
2.
4.7. '
,
, ,
,
, ,
,
.
'
,
,
.
'
, .
-
. ,
,
,
,
, ,
. '
.
,
, . . 4.7.1 .
, 4.7.1 ,
F1 ( x N / x0 ,..., x N 1 ) = F1 ( x N / x N 1 ), N F1 ( x N / x 0 ,..., x N 1 ) = F1 ( x N / x N
M
1, 2,...
,..., x N 1 ), N
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1,2,...
. 4.7.1 F1 ( x N ; t N / x 0 ,..., x N 1 ; t 0 ,..., t N 1 ) = F1 ( x N ; t N / x N 1 ; t N 1 ) F1 ( x N ; t N / x1 ,..., x N 1 ; t1 ,..., t N 1 ) = F1 ( x N ; t N / x N 1 ; t N 1 ) m
x0 , 2
R( )
m(t ) R (t1 , t 2 )
(
2
e
(1 e
2 t
e t
x0 e
,
t2
),
t1 ; t
min(t1 , t 2 )
0
-
)
4.8. ,
T
S (l
l
t0
X (t ) t T,
S
1, L) ,
t1 ... 1.
X (t ) , (k
( j 1, L)
j
k
1, L) jk
T jk
,
j
k
F (t )
f jk (t ) . 1.
F (t )
,
i
t0 . , ,
2. F (t ) . 3. f jk (t ) (t Tc ) ), t
Tc (
0, Tc , 2Tc ,
, . 4. ,
f jk (t ) ( j , k
, 1, L) ,
, j
f jk (t )
e
t
k
,
.
t0 ,
1.
i
,
F (t )
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,
( j, k
1, L) .
wt (t )
(i 1, L)
i
L
wi (t )
f ik (t ),
ik
.
(4.8.1)
k 1
. . .
(4.8.1)
mTi
,
i
(k
k
1, L)
L
mTi
ik
mTik ,
(4.8.2)
k 1
t f ik (t ) dt .
mTik 0
2.
jk
f jk (t ) ( j, k
1, L) .
ij
t0 t
L i (t )
ij
i
f ik ( )
ik k 1
(t )
0
,
ij (t )
i
t
,
,
j i
(t )
kj
(t
(1 i, j L) , ,
P Ti t
)d
,
0
(4.8.3)
t
: i
wi (t ) dt ,
(t ) t
wt (t )
i
.
ij
(t ) -
.
j
. (4.8.3)
1.
t
, (
.
= i ),
,
,
t
i
( j
2.
k
( , t]
i ),
. '
(4.8.3) , t
. '
(4.8.3)
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Pj mT j
ij (t )
L
,
(4.8.4)
Pk mTk k 1
Pk (k
1, L)
k k
(4.8.2). .
'
,
t
(4.8.4)
, mT j
i
,
t0 . 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
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5. 5.1. ,
1. . .
. :
1.
; ;
. . (
2. )
,
,
. ,
3. ,
,
,
,
. ,
4. ,
,
.
'
2.
, '
,
, '
.
. . 1. (
. 5.1.1):
Pk ...k (t1 , t1
1
M
, ,
,
N-
t1; ...; t M , t M tM ) [t1 , t1 t1 ] ,
, [tM , t M tM ] , , k1 ,..., kM ; f N (t1 ,..., t N ) t1 ,..., t N
;
N1
f N ( 1 ,..., t1
0;
2
t2
t1 ; ... ;
N
tN
tN
N
) .
1
. 5.1.1.
,
5.
Pk1 ...k M (t1
, t1
t1
;
; tM
, tM
tM
)
k1
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kM
,
t1
M Pk1 ...k M (t1
, t1
t1
, t1 t1 , ; ; tM
Pk1 ...k M (ti , ti
;tM
;
, tM
ti ; ; t M , tM
, tM
tM
)
tM ) . ,
6.
Pk (t , t (k
tM
[t , t
t) P1 (t , t
2) : P1 (t , t
t)
Pk (t , t
t ), k
t] t)
2,3,....
. . (
7. ), , ,
-
,
,
, , M
Pk1 ...k M (t1 , t1
t1 ;...; t M , t M
Pk (tm , t m
tM ) =
m
tm ) .
m 1
. 5.1.2.
. 5.1.2.
1. : N
f N ( 1 ,...,
f1( n ) ( n ) .
N)= n 1
, (
.
5.4). 2.
. 3. 1.
, ,
, ,
2.
. ,
,
,
, .
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,
3.
,
,
,
:
. '
. .
'
T,
. 2. ,
, ,
m(t1 , t2 )
(
. 5.1.3):
D(t1 , t2 )
[t 1 , t 2 ] ; m
D
n
n
n;
n 1
(t ) ,
; ,
(t ) ,
m(t , t T ) m(t , t T ) = m(T ) , D(t , t T ) = D(T ) . (t ) m(t ) m(0, t )
.
t,
D (t , t T ) 8.
t [0, t ] : (t )
. 5.1.3.
.
lim t
m(t
t ) m(t ) . t
0
(
)
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(
9.
(t )
)
Pk (t , t
t)
k 1
t t
0:
Pk (t , t (t )
lim t
t)
k 1
.
t
0
,
1.
(t )
,
(t )
;
.
(t )
,
2.
(t ) . (t )
3.
,
(t )
.
5.2. 1. . .
1.
Pk (T )
[0, T ]
,
k
(
. 5.2.1),
(
. 5.2.2):
: ( T )k e k!
Pk (T )
. 5.2.1.
T
K
.
[0, T ] .
2. 1) 2)
; 0
; 3)
n
tn
(t n
1
n
tn 1,
)
tn ,
,
[t n , t n
, f1 ( ),
. 5.2.2. ; 0 ;
n
:
f 1 ( 0 ),
f1 (
f1 ( )
e
n
)
:
.
-
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n
]
,
,
.
5.2.1. 5.2.1
T T ,
0
,
0
,
n
T
D(T )
T
m
m0
m
D
D0
D
n
,
m(T )
1 n
1 2
n
.
3. :
. ,
.
5.3. M
,
1.
M m (m
1, M ),
m
.
m 1
,
1. .
,
M
,
,
M
. ,
2. 5 10
.
1. .
(t ) .
2.
Pk (t 1 , t 2 )
[t 1 , t 2 ]
,
,
k
: Pk (t1 , t 2 )
a k (t1 , t 2 ) e k!
a ( t1 ,t 2 )
,
t2
a (t1 , t 2 )
(t ) dt . t1
[t 1 , t 2 ]
3.
[t 1 , t 2 ] ).
(t )
(
t
[t1 , t 2 ] : f1 ( )
cp
cp
(t )
e
cp
, ,
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.
. f1 ( )
. . 5.3.1. 5.3.1
[t 1 , t 2 ] (
)
[t 1 , t 2 ] [t 1 , t 2 ]
m(t1 , t2 )
a (t1 , t2 )
D (t1 , t2 )
a (t1 , t2 ) 1
m
cp
1
D
2 cp
5.4. ,
. n
(n = 1,2,...)
, N
f N ( 1 ,...,
f 1( n ) ( n ) .
N)= n 1
,
. .
. . .
. 5.4.1. 1. , f1( N ) ( )
f1( 2 ) ( )
,
f1 ( ) . :
1. 1
,
,
N
,
. (
2. 1
3
.
5.3). 5.4
1. ,
. ,
. 2. ,
N
.
L
, 115
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,
.
L1 ,..., LN
.
1
. 3.
N
-
N
f N ( 1 ,...,
f 1(1) ( 1 )
N)
f1 ( n ) , n 2
f 1(1) ( 1 )
( 0,
t
), f1 ( ) ( . 5.4.1). f 1(1) ( 1 ) , .
4.
. 5.4.1.
:
t=0 ,
t1
,
f1 ( )
,
1
t1
, -
; ;
; 0
n
.
f1(1) ( 1 )
1.
f1 ( )
'
: 1
f1(1) ( 1 )
f1 ( )d ] ,
[1 0
1 . m 2.
0
f1(1) ( 0 ) , . . 5.4.1). 1.
(
tn
n
1
[t n , t n
n
] (
.
, . 5.4.1),
~
,
f1(1) ( 1 ) ,
f1 ( n ) . f1 ( ) .
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. . 3
(
). ,
. .
,
(
).
,
. ,
. ,
,
. 5.4.2. k-
1 .
, ,
k (
. 5.4.2).
. 5.4.2.
(
3n
;
k
1 .
):
,
k
,
, :
k n
;
n 1
f1 ( n ) e 1 ,1
1. 2. 3.
n
,
0.
n
. .
k
:
k,
k k
. .
,
1
.
k-
1. fk ( e )
( e )k 1 e (k 1)!
e
,
. 2.
l
0,
e
Pl , k (T )
k[0, T ] ( l 1) k 1
Pl , k (T ) n lk
( T )n e n!
T
.
. 5.4.1. 117 Create PDF files without this message by purchasing novaPDF printer (http://www.novapdf.com)
5.4.1
k
k
m e ,k
m
1
,k n 1
k
D e ,k
D
2
n
k
1
n 1
n
,k
2
k
k
n 1
1
1
n
. .
k , .
k m
m2 D
,
1 , m
;
D
. 5.4.3.
k
1.
,
k , n
, n = 1, k ,
f1 ( n )
n
e
n n
.
k
1.
k,
const .
n
k
1.
k
fk (
)
n n 1
e
k
n
.
k n 1
(
i
n
)
n ,i 1 n i
. 5.4.1.
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5.4.4. ,
. :
(
. 5.4.3).
. 5.4.3.
'
1. '
,
. , .
«
»
. :
2. 1)
: 0,
f1 ( )
; (
e
)
,
,
[0, a] :
2) 0, f1 ( )
,
1 , a
a
; a.
. 5.4.2
. 5.4.2
1
m 1
D T
2
1
m a2 D 12 2 T a 2
, . 5.4.4.
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a 2
Pk (t , t
t)
Pk ( t )
Pk (t1 , t 2 )
M
Pk1 ...k N
f N ( 1 ,...,
N
Pk1 ...k N
)
N
Pk1 ...Pk N
Pk1 ...Pk N
f1( n ) ( n ) n 1
f1( n ) ( n ) f1 ( )
i
k
e
1-
-
k-
. 5.4.4.
5.5. '
,
. , . '
,
, ,
INTERNET, ,
,
,
,
,
,
. ,
, ,
,
:
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.
.
. .
.
. ' ,
,
,
. , . 5
. 5.5.1. 5.5.1 5
24, 40, 47, 49 16, 32 20, 23, 88, 89, 105, 109, 113 38, 79, 83
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6. -
1. (
).
2.
,
, .
6.1. 6.1.1. 1. (
). ,
1.
.
(
)
.
X
.
xn , x n ( t ),
X (t )
n 1,2,... . 2.
, ,
,
. ,
. . x1 ,..., x N ,
2. 3. ,
N
. 4. . ,
N
5. .
F1 ( x) ,
x1 ,..., x N
,
6.
,
F1 ( x) . N,
7. ,
N-
x1(1) ... xN(1) = x1( 2 ) ... xN( 2 ) = ( X 1 , ... ... ...
, XN),
(
). ,
1. ,
x,
.
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( X1,
2.
,XN)
X n ( n 1, N ) F1 ( xn ) ,
. FN (x )
3. N -
F1 ( x n )
xn
: N
F1 ( xn ) .
FN (x ) = n 1
Lx
8. .
Lx
,
.
f1 ( x n ) Xn
: N
Lx =
f1 ( x n ) .
(6.1.1)
n 1
f1 ( xn )
, Xn .
.
X
. . 9.
Y
Y ( X 1 ,..., X N )
X.
Y ( X 1 ,..., X N )
.
Lx f1 ( x)
. ,
,
. (
10.
),
. 1. , . 2. ,
,
F1 ( x) ,
D.
m,
,
*
F1 ( x) ,
,
D* .
m* , ,
3.
, ,
. .
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6.1.2.
.
. (
1.
)
, 1, N ).
xn ( n 2.
j
Nj :
xj ,
N
Nj j
N
.
1.
j
,
,
(
), . 2.
pj 3. * j
1)
xj .
X
,
:
j
=1;
j
2)
N
j
pj.
3.
xj
j
: (x j ,
j
),
j
1, J . ,
4. ( xj ,
* j
), j = 1, J
'
.
(6; 3; 5; 2; 3; 2; 5; 2; 5; 5). , : (2; 2; 2; 3; 3; 5; 5; 5; 5; 6).
1. ,
(2;
3 1 4 1 ), (3; ), (5; ), ( 6; ). 10 5 10 10
. 6.1.1.
. 6.1.1.
.
J
,
xj .
, ,
, ,
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.
5. X,
. X
1. , .
, 10 20. 2.
,
. ,
, : . ,
3.
,
,
. * i
p ,
6.
i-
Ni
,
i-
,
: pi* =
N
,
Ni . N
7. ,
.
fi
8.
*
pi* xi : f i = . xi
* i
*
p i,
1.
, ,
,
. 2.
f i* . ,
9. .
, * i
p .
i. . .
100
2. .
(
6.1.1),
(
. 6.1.2)
(
.
. 6.1.3). 6.1.1
p *j
[-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]
0,012
0,050
0,144
0,266
0,240
0,176
0,092
0,020
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. 6.1.2.
. 6.1.3.
10. *
F ( x) ,
x
X
x.
X
F * ( x)
1. F (x) , .
F * ( x)
2.
x.
F * ( x)
3. . 11.
F * ( x) , X
x. . 6.1.4.
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,
. 6.1.4. (
2)
6.1.3. 1. ( X 1 ,..., X N )
X
(
),
m* = X
m*
N
1 N
N
Xn .
(6.1.2)
n 1
2. ( X 1 ,..., X N ) (
)
m *x =
1 N
N
Xn .
(6.1.3)
n 1
3.
X
( X 1 ,..., X N )
N *
=
1 N
N
(X n
m *x ) .
(6.1.4)
n 1
X
4.
D x* =
1 N
N
m *x ) 2 .
(X n
( X 1 ,..., X N )
(6.1.5)
n 1
6.2. .
N
. . ,
. .
6.2.1. 127 Create PDF files without this message by purchasing novaPDF printer (http://www.novapdf.com)
.
,
.
X
.
mx <
Dx < .
,
X mx Dx 2
-
,
,
P X
Dx
mx
.
2
(6.2.1)
X
.
F (x) . ( x m x ) 2 dF ( x) .
Dx
, x mx < .
,
( x m x ) 2 dF ( x) .
Dx x mx
( x mx )
2
2
Dx
2
. 2
dF ( x)
P( x m x
),
x mx
(6.2.1). 6.2.2. X 1 ,..., X N
.
m x ,..., m x 1
D x ,...,D x ( D x 1
n
C
, n 1, N ).
-
YN
X 1 ,..., X N my
m x ,..., m x : 1
lim
YN
m yN
0. .
.
my
(6.2.2)
Dy
YN =
1 N
N
Xn n 1
: my D yN
D xn
C,
Dy
1 N mx ; N n1 n 1 N Dx . N2 n 1 n
C . N
YN
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,
YN
D yN
m yN
C . N 2 (6.2.2).
2
N
X
. .
X 1 ,..., X N
. X,
.
(6.2.2)
: lim
N
1 N
N
Xn
0.
mx
n 1
. . 6.2.3. ,
. '
.
'
pa .
N
Na N
pa : lim
N
Na N
0.
pa
(6.2.3)
X 1 ,..., X N ,
. .
pa 0
:1
q a =(1 p a ). m x = pa ,
Xn
n
D x =(1 p a ) 2 p a +(0 p a ) 2 q a = p a q a n
.
X 1 ,..., X N Na N
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
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7. 7.1. , . :
(
) ,
,
(
.
F (x /
1
)
.
,...,
K
X 1 ,..., X N = X ( )
. X F ( x; 1 ,...,
)
1
x1 ,..., x N
K
)
,...,
K
.
,
X 1 ,..., X N , 1.
1
F (x /
1
,...,
K
) 1
F ( x; 1 ,...,
K
)
N
,
,...,
K
,..., ,
K
.
,
. ,
2.
, .
.
( 1,
,
K
)
(
*
), .
,..., F ( x; 1 ,..., K ) 1
F (x / 1
,...,
1
,...,
1
( X 1 ,..., X N );
K
)
K
,
X 1 ,..., X N :
K
1
.......... .......... .......... .. K
K
( X 1 ,..., X N ).
X,
1.
,...,
1
K
;
,
1
,...,
x K
.
2. .
,
,
. 3. ,
7.5.
7.2. 7.2.1. ,
1. / -
,
:
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M[
/ ]
.
(7.2.1) .
(
) M[
0
. ,
.
/ ]
X.
, 1. 2
2
M[
/ ]. :
2
2
2 0
2
*
M 2
M ,
2
*
,
(7.2.2) .
2
,
, 2 0
.
2.
X. ,
. ,
3. , .
,
,
,
, ,
. .
7.2.1. 7.2.1 ,
x
(
* x
X) R xy
R *xy
N 1n 1
kN
1
Xn
mx
n 1
N
1
D x*
Dx
N
1 N
m *x
mx
(X n
m *x ) 2
D x*
1 N
N 1n
(X 1
n
m* )2 x
* x
N
N 1n
(Xn
m *x )(Yn
1
m *y )
R*xy
1 N
1 N kN
1
N
mx ) 2
(X n n 1
1 N
N
(X n 1
n
mx )2
N
(X n n 1
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m x )(Yn
my )
kN
. N-1 2 , N 2
N 1 2
kN
.
( )
-
(
kN
2).
. 7.2.2.
7.2.2 kN kN
N
N
kN
N
kN
N
kN
N
kN
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
2. ,
/ : (7.2.3)
,
N lim M [
/ ]
N
.
7.2.2. ,
. : lim P{
/ } 0,
N
N
'
0
,
(7.2.4) . ,
1. (
).
(7.2.4)
lim M
2
*
N
/
0. ,
2. .
N
7.2.3. . -
.
. * e
1. , * e
N * i
-
M [(
e
,
:
) 2 / ] < M [(
) 2 / ] , i =1, 2,.
i
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(7.2.5)
1. *
D [
M [(
)2
/ ] < D[
i
]
].
. 2.
D[
e
/ ] , i =1, 2,.
(7.2.6)
,
3. .
, X
,
,
,
. l,
2. * e *
: l
M [( M [(
e
)2 / ] . )2 / ]
(7.2.7) [0,1].
. , l =1.
, *
3.
,
N .
l'(
4.
1
2
: M [(
l'( )
M [(
1
)2 / ]
2
)2 / ]
.
(7.2.8)
l' (
.
, . D ,
*
*
/
. x
1.
*
X
N . f N (x / ) x .
/
,
2
1 D
*
/
=M
*
M
*
/
2
0
/ M
ln f N ( X / )
,
2
/
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(7.2.9)
;
(
M *
),
D
,
,
X;
0
. . *
M
*
*
/
( x ) f N ( x / ) dx .
( x ) f N ( x / ) dx
0
(7.2.11) *
f N (x / )
.
(7.2.11)
:
f N (x / )
( x)
(7.2.10)
dx
0
1
f N (x / )
.
(7.2.12)
ln f N ( x / )
(7.2.12)
.
(7.2.13)
: *
M
ln f N ( x / )
(x)
*
ln f N ( X / )
f N ( x / ) dx 1
/
0
1
0
.
f N ( x / ) dx 1 . (7.2.15) : f N (x / ) dx
(7.2.15)
0.
(7.2.16)
(7.2.13), (7.2.16) ln f N ( x / ) f N ( x / ) dx
ln f N ( X / )
M
(7.2.17) M M
*
/
M
*
/
(7.2.14)
/
0,
0.
(7.2.17)
:
ln f N ( X / )
/
0 .
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(7.2.18)
(7.2.14) *
M
*
M
(7.2.18): ln f N ( X / )
/
/
0
=1
.
(7.2.19)
,
(7.2.9). 5. JN
2
ln f N ( X / )
M
/
.
(7.2.20) :
1. 2
JN
M
ln f N ( X / ) 2
. 2
fN (x / )
f N (x / )
2
2
ln f N ( x / )
f N (x / )
M
2
f N ( x / ) dx
2
2
M
ln f N ( x / ) 2
ln f N ( x / )
.
.
2
(7.2.21). (7.2.20)
2.
.
(7.2.16), ln f N ( x / ) f N ( x / ) dx , 2
2
(7.2.20),
2
f N (x / )
2
ln f N ( x / )
ln f N ( x / )
2
x.
ln f N ( x / )
(7.2.21) (7.2.13):
(7.2.13): ln f N ( x / ) f N (x / )
f N (x / )
.
,
2
2
/
: 2 0
1 D
*
/
JN
.
(7.2.22)
3. D
*
1 . JN
/
(7.2.23)
4. M
*
2
*
M
/
/
,
: 2 0
1 M
*
2
/
D
*
/
JN
.
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(7.2.24)
*
, *
. (7.2.24)
5.
,
0
(-1).
, .
6. : 2 0
1 M
2
*
J1
*
D
/
/
,
N J1
(7.2.25)
: 2
J1 (7.2.25) 7. 1) 6 .
M
ln f 1 ( X / )
.
2
(7.2.21), (7.2.24). (
.
, . ,
e 2 0
1 M 2.
2
*
/
.
JN
(7.2.26) 6
e
,
: ln f N ( x / )
1 JN
* e
.
(7.2.27)
2, 6 .
,
6 .
,
e *
M M
* e
e 2
/
/
, D
* e
1 JN
/ 1
1. 2. (7.2.24), (7.2.26). 3.
(7.2.28)
6
. 6
1 6
(7.2.24) .
,
, 4. ,
, 5.
6 ,
, , , ,
.
,
.
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2
, 0
,
(
.
2
(7.2.2)) .
6
6. ln f N ( x / )
, (7.2.27),
f N (x / ) : f N (x / ) = exp k1 ( ) k1 ( ) k 2 ( ) 7.
k 2 ( ) h( x ) , h(x ) , ,
e
,
.
x.
,
, ,
,
,
. .
. . *
K
.
I
7. I ij
ln f N ( X / ) ln f N ( X / )
M
i
( i, j
1, K ) ,
(7.2.29)
j
K-
N-
fN (X / ) N -
X. ,
1.
ln f N ( X / ) ln f N ( X / )
M
i 2
M
j
ln f N ( X / ) i
(i, j
1, K ) .
j
(7.2.29)
: 2
I ij
M
ln f N ( X / ) i
(i, j
1, K ) .
(7.2.30)
j
X
N-
2.
S( ) R, ST( )
I ij
R
1
i
R
*
S( )
(i, j 1, K ) .
(7.2.31)
)T / ]
(7.2.32)
j
M [(
)(
,
1.
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X
x
3.
( 1 ,...,
K
D
).
K K d ij
1
0i
( ) (i, j
1, K ) ,
j 0i
( ) (i 1, K )
0
U T R 1U . (7.2.33)
U K-
1.
( ).
U T D I D 1U ,
(7.2.33)
. (7.2.33) UT IU .
2. T
1
U R U (7.2.34) UTR 1 U
3.
(7.2.34) , *
1 T
U IU
1,
. *
8 .
1e
,
*
,
Ke
*
,
e
,
R U T R 1U
,
1. ,
N
* 1e
,
UT IU . (7.2.35)
,
1 e
(7.2.35)
*
,
Ke
. ,
2.
(7.2.35),
. *
(
e
* 1e
,
,
* Ke
)
:
I 1 bT ,
*
ln f N ( x / )
b
(7.2.36)
(i 1, K ) .
i
3. . . , (7.2.35),
4. . 8 .
* 1e
,
,
,
* Ke
*
,
e
, * 1i
,
,
N
*
*
Ki
i T
1
U R U
U T R 1 U (i 1, 2 , ) .
e
(7.2.37)
i
. 4. ,
x
N f1 ( ) .
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( N 1)
f N 1 (x, ) 2
f N 1 ( x; )
,
f N 1 (x ; )
,
x
2
lim
(
M( ,
JN JN
*
) f N 1 ( x ; ) dx *
)2
J N1 ,
(7.2.38) : 2
ln f N 1 ( X ; )
M 2
=
ln f N 1 ( X ; )
= M
,
2
1.
(7.2.39)
M
(7.2.38) , (7.2.39) (7.2.38)
*
0 .
X
.
,
*
.
K
(7.2.38), (7.2.39)
2.
. U T R *1U
R
*
UT IU ,
(7.2.40)
K K
:
R
*
=M (
*
)(
)T ,
*
(7.2.41)
I I ij
ln f N
M
K
(X; )
ln f N
i
K
(X ; ) ,
j
(7.2.42) *
(7.2.41) , 3.
(7.2.42)
X
. (
8 ,
1
8
.
8 ).
4. 8
f K ( ; x) . 5.
, . 7.2.4.
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,
. N-
f N x1 X
*
xN
. ,
1. *
fN x
, , ,
. 1 N
m*x
. m
,
N
N
xn n 1
N 1
1
* x
N
1n
xn
.
1
,
2.
'
.
. .
3. 4.
,
.
,
'
,
.
.
( 1,
,
f N ( x1 ,
)
, xN /
f N x1 f1
K
:
xN
f1
/ ) xN
h( x1
/ )h( x1
xN ) ,
(7.2.43)
,
x,
,
. :
.
fK ( / x
,
(x ) .
x,
. 7.2.3. 7.2.3 X
,
-
, -
(
-
)
m *x mx
Dx* Dx
D x*
1 N
N
Xn
+
+
+
+
+
( l1 )
+
+
+
n 1
1 N ( X n m *x ) 2 N n1 1 N ( X n m *x ) 2 N 1n 1
(l 2
l1 )
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+
NA N
p*
+
+
+
+ (l 2
.
l1 ) .
7.3. 7.3.1. .
f 1 ( x;
X
,
. X. -
, ,
,
: m
x f ( x; )dx .
m ' 1
( ). , 1 (m ) ,
(7.3.1)
: (7.3.1)
,
(7.3.2)
.
mv
*
(7.3.2). : *
N
1 N
1 v
X nv . n 1
1. . *
2.
X ,..., m1 ,..., m K
, 1
,...,
1
. m1
1
(
1
,...,
K
K
);
................................. mK
K
(
1
,...,
(7.3.3)
).
'
, m ,..., m* ,
,
,
* 1
m1 ,..., m * 1
1
(m1* ,..., m * );
............................... * K
3.
K
(m1* ,..., m * ).
,
, .
,
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,
1
,
. 7.3.2.
N-
. (
LX ( )
X 1 ,..., X N ,
) ,
*
. ,
. *
.
X, . . , LX ( ) : LX (
f1 ( x n / ) .
)
(7.3.4)
n 1
(7.3.4). N
ln L X (
ln f 1 ( x n / ) .
)
(7.3.5)
n 1
.
(7.3.4)
(7.3.5)
.
*
N
ln f 1 ( x n / )
n 1
1
0;
................................. N ln f1 ( x n / ) 0. n 1
(7.3.6)
K
(7.3.6)
1.
'
'
,
.
.
'
'
,
(
). 2. .
(7.3.6) ,
. :
. 1) 2)
;
N ;
3) ;
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,
4)
.
N
. . 7.3.3. ,
,
,
. . ,
,
,
. ,
( )
. ,
1. , . 2. ,
, . ,
.
,
f ( ).
,
Lx ( ) .
f (x, ) . f (x, )
f ( ) Lx ( )
f ( / x) f ( x) . , f ( )Lx ( ) . f (x) (7.3.4) (7.3.7),
f ( / x)
, (7.3.7)
, .
,
(7.3.6): N
ln f ( )
ln f 1 ( x n / ) n 1
1
0;
1
.........................................................
(7.3.8)
N
ln f ( )
ln f 1 ( x n / ) n 1
K
0.
K
,
1. ,
(7.3.6), .
2. (
), ,
. , (
).
:
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1)
,
(
,
X , ,
);
,
2)
,
(
,
,
, ).
7.4. ,
,
.
. )2 (
M( D[
P(|
] ),
|
)
,
*
.
R (
).
.
.
,
X .
.
X mx ,
mx
2 x
2 x
.
,
.
N
: 1 N
mx
N
Xn . n 1
f1 (m *x ) ,
:
D[mx ]
P( mx
mx
). .
.
mx
, Xn ,
.
M [mx ]
mx . : D[m x ]
Dx . N P( m x
,
Xn
mx
)
: P( m x
mx
) = 2F
F (x)
N Dx
1, (
7).
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,
, ,
2
. 7.4.1. (t-
N
-
).
5 6. 2
P 2(
2
P(T
)
t )
S (t , N )dt ,
,
t
8 9. ,
. ,
,
.
, M [(
*
)2 ],
P(|
|
).
,
,
.
, . 7.4.1
X
N ,
N
1 N
1) m x
Xn n 1
N Dx
Dx
mx
3) M m x
N Dx
4) D m x
N Dx
5) P (m x
Dx
Dx
1
N Dx
= 2F
1
mx ) 2
(Xn n 1
Dx N Dx
3) M
N Dx
N
2)
mx
mx
mx ) 1 N
1) D x
Dx N
N Dx
mx
2)
mx
,
Dx N
2
N
-
N
Dx
4) D
Dx N
2N
Dx
5) P
Dx N Dx
P 2(
N 2
=P 2( 2 2)
,
2 1,2
2
2 1 )
(N
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)
mx
N
1
1) D x
N 1n
m x , Dx
Dx
1
N
1 N
Xn n 1
Dx N Dx
2) Dx N
mx )2 ,
(X n
2
-
(N 1) 3) M
Dx N
N 1
Dx
4) D
Dx N
2( N 1)
Dx
. 7.4.1 5) P
Dx N Dx 2
P 2(
mx
t
mx
N
Dx
=P 2( 2 1, 2
,
2
(( N 1)
2 1 )
)
N
Xn n 1 N
1
N 1n
(X n
mx )2
1
t ) (N-1) 3) M [t ] 0 N 1 4) D[t ] N 3 2)
m x , Dx
2 2)
1 N
1) m x Dx
(N 1)
5) P( t *
(t-
) 1 2 S (t , N 1)dt
7.5. 7.5.1. .
. P(
)
.
,
: P( I
(
,
).
(7.5.1)
).
I
. .
,
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I
.
,
I
(
. 7.5.1).
. 7.5.1.
(
1.
)
,
I
, ,
2
: I
[
].
,
(7.5.2)
. ,
,
2 .
7.5.3. 2.
(
). (
3.
)
,
(7.5.2)
,
I :
,
P(
I ).
7.5.2. .
. ,
I P(
'
)=
(7.5.3)
. '
.
, ,
.
,
. . ,
, (
, ( ),
, ). .
. mx
X,
.
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mx
mx :
,
Xn . n 1
,
N
mx M [mx ]
N
1 N
mx
,
. Dx D[m x ] . N Dx
mx ,
1
Dx
X
:
N
N 1n
mx )2 .
(X n 1
F (x) ,
, (7.5.3) N Dx
2F
arg F (x)
1
,
Dx* 1 arg F N 2 F (x) . 1 t = arg F 2
,
,
,
: m *x
I , Dx
D x* ; m *x N
t
D x* . N
t
,
X,
,
rxy
X
Y.
. 7.5.1. 7.5.1 , (
)
, mx
t
Dx
0.8 N 1.2 Dx N ( N 1)
t
rxy .
t
t
Dx N
1 rxy* N 1
.
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7.5.3.
X
,
. . (
.
*
*
)
, (
*
, (
) .
(
*
,
*
),
,
)
,
. . . mx
X
, 1 N X
mx Dx
N
Xn , n 1
.
(m x
.
mx )
Dx .
(m x (
Dx
*
) = (m x
mx ) N , Dx
mx )
: 1
Dx
N
N 1n 1 ( *
. 7.4.1,
(X n
mx )2 .
)
.
N 1
P(
t ) 1 2 S (t , N 1)dt
,
t
S (t , N 1)
. S (t , N 1)dt (
9),
t
t mx
t
Dx N
1 2 mx
mx
t
Dx N
.
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(7.5.4)
(7.5.4)
.
, , .
.
rxy . , :
rxy * 1 1 rxy ln , 2 1 rxy*
Z rxy
thZ . , Z
N
1 1 rxy ln 2 1 rxy
mz 1
Dz
N
3
, rxy 2( N 1)
rxy .
,
. 7.5.2. 7.5.2 (
)
,
/
mx
1
m *x
I
Dx
I
2
D x* , m *x N
t
D x* N 2 2
mx Dx
I
4 : 1.
1 (
t
th ( Z
N
my
3
t
), th ( Z
N
9 2
2
1
t 1
2.
2 1
2
mx
mx
D x* N
D x* ( N 1) D x* ( N 1) , 2 2
I
3
,
D x* N
t
). 2 1, 2
2
1
,
1 2
2
N
.
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3
)
3.
2 1, 2
3
2
1
, 4.
1 2
4
N 1
.
2
t
.
7.6. . : 1 q 1 p.
p
N
X, 0 .
xn
n-
( n 1, N )
.
.
xn
N
(
p).
1 (
P*
). :
p
1 N
P* P*
N
Xn . n 1
,
p.
:
m P* 2 P*
p,
(7.6.1)
p(1 p) . N
(7.6.2)
2 (
). P*
.
, ,
,
(7.6.1) (7.6.2).
I P1
P
*
, P2
P
*
[ P1 , P2 ]
(7.6.3)
,
,
: 2F
1 P
F (x)
( (7.6.4)
F
1 2
arg F
1 2
(7.6.4)
.
7).
, P*
t
,
*
t ,
(7.6.5)
,
. (7.6.1) (7.6.3), (7.6.5) p (1 p) . P* p t N
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(7.6.6)
(7.6.6)
,
p,
t2
P*
t
2N
P1, 2
1
t2
P * (1 P * ) N 2 t
4N 2
,
(7.6.7)
N
I . (
.
7.6.1).
(1' 5' ) (1' ' 5' ' ) ( P2 , P1 ) 10 (2' ,2' ' ) , N 15 (3' ,3' ' ) , N 20 (4' ,4' ' )
. 7.6.1.
(1' ,1' ' ) , N
N
N 5 25 (5' ,5' ' )
P*
1. (7.6.7). ,
X
. 2. , P1, 2
P
*
P * (1 P * ) , N t2 . N
t
(7.6.7)
(7.6.8)
3. (
)
(7.6.7)
,
. [16]
. . 7.6.1.
7
7.6.1 7 11, 24, 40, 47, 49, 81, 93, 108
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13, 16, 32, 42, 75, 80, 84, 91 5, 21, 22, 45, 46, 50, 59, 90, 101 2, 28, 43, 52, 53, 55, 56, 62, 63, 65, 76, 85, 97, 100, 102, 106, 107
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8. 8.1.
,
.
,
,
,
.
:
,
, 1.
( ). ,
,
. .
1. ,
.
2. ,
,
,
,
,
. ,
,
, ,
. (
1. )
, 1.
, ,
.
, .
,
, ,
,
.
. ,
.
' ( ). 3.
.
(
)
, ,
,
:
.
,
,
,
. 4. .
,
,
.
2. (
), ,
Y
3. X
.
X, Y y( X ) ,
-
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Y
X (
Y
. 8.1.1).
y( X ) y( X ) .
y( X ) 4.
,
Y= y( X ) + h( X , Y ) , , h( X , Y ) y( X ) , h( X , Y ) . ( y x) y.
X ,Y . . 8.1.1).
(
(8.1.1) .
h( X , Y )
y
y = y( x)
( x n , y n ) , n 1, N ,
. 8.1.1.
x
( x n , y n ) , n 1, N , ,
5.
y = y( x)
h( X , Y ) :
m y (x)
Y/x
1. . , .
.
. ' (
. Y
)
. ,
2. 1,
,
.
. :
2. (
).
.
y( x) ,
Y X. X.
xn
Y
,
(n
1, N )
xn
y( xn ) ( n 1, N ) .
y( xn ) Z ( xn ) ,
,
Y Z ( xn )
xn .
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Z ( xn ) h( x n , Y ) . .
1.
Z ( xn )
,
,
, .
,
. 2. . . 3.
y
x: ( x)
y ( x )) 2 ]
M y [(Y / x
h( x, Y ) ),
y x)
( (
(8.1.2) Y/x
; M y[ ]
Y. . , y( xn ) Z ( xn )
Y/x (
. . 9.5).
. (
) y ( x)
, m y (x)
y( x) = m y (x) .
Y/x,
(8.1.2), : ( x)
Dy ( x) (my ( x )
m y (x) =M Y / x
D y (x) =D Y / x
y ( x )) 2 ,
(8.1.3)
Y/x (
. 8.1.2).
m y (x)
. 8.1.2.
y( x) ,
,
Y/x,
m y (x)
(x)
y( x) m y (x) y( x)
4.
X
Y.
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,
y( x) X Y.
5. ( y1 ,..., y N )
x1 ,..., x N y( x) ,
.
, y( x) = m y (x) .
1.
,
,
,
.
N
y( x)
, N.
,
2.
. :
1
,
,
K
. y( x) ,
k-
,
y( x) (k
, y k 0 ( x ) ,
1, K )
.
,
. ,
3. ,
,
X
Y.
, ,
.
,
,
-
.
Y
6. y ( x ) Mx
(X )
y 0 ( x )
,
M x [M y [ Y / X
M x [ Dy ( X )
X
X
my( X )
2
y ( X ) ]] (8.1.4)
2
y ( X ) ],
Y
h ( X ,Y ) .
7.
0
Y
X
(
). : M x[
1.
M x D y (X )
2
( X )]
m y (X ) y0 ( X ) . 2.
y 0 ( x )
my (x) ( 3.
(8.1.4)).
my (x)
,
y 0 ( x ) = my (x) , .
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2
M x[
( X )] = 0
0
:
M x [ D y ( X )] .
0
Y
4.
Y
X
X
.
8.2. y 0 ( x )
1. , y 0 ( x ) = a 0 b0 x ,
(8.2.1) .
a 0 , b0 (8.1.4),
. ( a, b )
a, b 2
( a, b)
M x [ D y ( X ) m y ( X ) a bX ]. a b , .
,
a
b,
a0
my
rxy
y
m x ; b0
rxy
x
x
= Dx ;
y
R xy
= D y ; rxy =
x
y
,
x
. y
(8.2.1), y 0 ( x )
my
y
rxy
(8.2.2)
( x m x ).
x 0
0
b0
2.
= rxy
2 y
(1 rxy2 ) .
(8.2.3)
y
,
x
(8.2.2),
,
Y
X. 1.
Y
(8.2.2), (8.2.3)
X X 2.
,
Y. (8.2.3)
,
0
X (
Y. rxy 1 ).
, .
X
.
Y
Y
X
.
0
D y (x) .
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. f1 ( y / x ) .
: 1
f1 ( y / x) =
2 (1 rxy2 )
y
1 y my 2 2 y (1 rxy2 )
exp
y
m y (x) = m y + rxy
2
rxy
y
( x mx )
.
x
( x m x ),
(8.2.4)
x
y 0 ( x ) m y ( x ) ,
D y (x) = y2 (1 rxy2 ) . (8.2.4), (8.2.2) (8.2.5), (8.2.3), D y ( x) .
0
(8.2.5)
X
. D y (x) Y (8.1.3)) -
x (
Y
-
x ),
(x)
x (
.
x, x.
8.3. m x , m y , D x , D y , R xy
. , L,
( y1 ,..., y L )
, a0
,
( x1 ,..., x L ) ,
b0 :
mx , m y , R xy* = rxy*
a0*
* x
D=
* y
*2 x
, D=
*2 y
.
b0* a0
my
rxy
y
mx ;
(8.3.1)
x
b0* = rxy*
mx =
Dx =
1 L 1l
1 L
L
.
(8.3.2)
1 L yl ; Ll1 1 L ( yl Dy = L 1l 1
l 1
m *x ) 2 ;
1
(8.3.3) m *y ) 2 ;
(8.3.4)
L
1
Rxy*
* x
xl ; m y =
L
( xl
* y
L 1l
( xl
m *x )( y l
m *y ) .
(8.3.5)
1
a0
.
b0
(8.3.3) (8.3.5), .
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,
,
.
. (a 0* , b0* )
D *y (1 rxy*2 ) . .
8.4. 1. y( X 2 ,..., X N ) , X 2 ,..., X N .
X1
X 2 ,..., X N
Y
X1
X1
y( X 2 ,..., X N ) h1 ( X 1 , X 2 ,..., X N ) : X 1 = y( X 2 ,..., X N ) + h1 ( X 1 , X 2 ,..., X N ) , h1 ( X 1 , X 2 ,..., X N ) . 1. ( x2 ,..., xN )
M x1 [( X 1 / x2 ,..., x N y ( x2 ,..., xN ))2 ] .
1. )
(
,
y( x2 ,..., x N ) , m x1 ( x2 ,..., x N )
, X 1 / x 2 ,..., x N ,
y( x2 ,..., x N ) = m x1 ( x2 ,..., x N ) .
(8.4.1)
2.
X1
X 2 ,..., X N
y0 ( X 2 ,..., X N )
,
M x2 ,..., xN [ ( X 2 ,..., X N )]
0
y ( X 2 ,..., X N ))2 ],
M x2 ,..., x N [ Dx1 ( X 2 ,..., X N ) ( mx1 ( X 2 ,..., X N )
Dx1 ( x2 ,..., x N )
X1.
3.
0
2.
M x2 ,..., xN
( X 2 ,..., X N )
X 2,..., X N
X1
1.
(8.4.2)
. (8.4.2) , y 0 ( x2 ,..., x N ) (8.4.1)
,
mx1 ( x2 ,..., x N ) . ,
(8.4.2) ,
y 0 ( x2 ,..., x N )
. 0
4.
M x2 ,..., x N [ Dx1 ( X 2 ,..., X N )]. ,
X1
X 2 ,..., X N ,
X1
X 2 ,..., X N .
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5.
X 2,..., X N
X1 1n
(n
1n
2, N ) ,
11 in
, R( xi , x n ) 2.
Xn
Xi
(i, n 1, N ). X1 X 2,..., X N
N
y 0 ( x2 ,..., x N )
y
mx1
1n
( xn
m xn ).
n 2
(
6.
) r ( x1 , y0 )
1
1
2 x1
r ( x1, y 0 )
. X1
. 11
X 2,..., X N .
, X 2,..., X N (
X1
)
. 3. 0
2 x1
(1 r 2 ( x1 , y 0 )).
2. X 2 ,..., X N
X1
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H0 , H 0i ( i
H0
1, I ). H 0i .
4. : 1) 2)
; ;
3) ,
, . 9.5.2. Z
1.
K ( z) 1 2
1
( 1)
e
2
2 2
z
,
z
0
(9.5.2)
1
. N
. *
Fh ( x)
F ( x)
D: D
max F * ( x) Fh ( x) .
x (
N
Z
Fh ( x)
ND .
,
2.
(9.5.3)
, )
(9.5.3),
.
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(
10). (9.5.1)
,
.
, K ( z ) 1 2e
2z2
.
(9.5.4) 1 K (z) ,
, (9.5.4) 1 2 ln , 2N
d
(9.5.5)
. : N
1) (9.5.5)
d ;
x1 ,..., x N
2) *
F ( x) ; Fh (x ) ,
3)
H0 ,
F * ( x) , max F * ( x) Fh ( x) ;
d
x
4)
d
d
d
( d
(
d )
d )
H0 .
D
,
1. *
F ( x)
Fh (x)
, «
». ,
2. . 27.005-97 N
3. , 100, 30.
9.5.3.
-
. 2
F * ( x) (*)
,
,
( F * ( x) Fh ( x)) 2 ( Fh ( x ))dFh ( x),
(9.5.6)
Fh ( x)
, . N
1. 2
,
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Fh ( x)
(9.5.6), ,
-
. ( Fh ( x ))
2.
[ Fh ( x )(1 Fh ( x))]
1
N 2
,
(9.5.6), ,
Fh ( x) .
1.
, . : 1 12 N 2
2
N 2
N
2 n 1
1 N
N
2n 1 2N
Fh ( x n ) n 1
2n 1 ln Fh ( x n ) 2N
2
,
(9.5.7)
2n 1 ln 1 Fh ( x n ) 2N
1
-
(9.5.8) -
.
2. (9.5.8). N 50 .
-
11 12. 27.005-97
,
9.5.4.
-
X
1.
(
X
I
. 9.5.2).
Pi ,
i-
N,
i-
li .
,
N
Pi I 2
(l i
i 1
( I 1)
-
NPi ) 2 NPi
(9.5.9) .
X
. 9.5.2.
1,
2.
Pi Pi (
I
X * 1
,...,
* K
K
), * 1
,...,
* K
,
. N
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I 2
[l i
NPi (
i 1
(I
-
*
NPi (
1 *
,...,
1
*
,...,
K * K
)]2
)
,
K 1) ,
1.
(9.5.10) .
-
(9.5.10),
(9.5.9) ).
( -
1. 27.005-97: N 100; I 7 25 ( I = 15 18 100 N 200 ; I = 18 20 N 400 ; I = 35 40 N 1000 );
1) 2)
I = 25 30 3)
NPi
,
N
, 200 ;
i 1, I .
5,
d ,
2. ,
(9.5.9)
f(
2
2
, I 1) d
(9.5.11)
d
(9.5.10)
f(
2
,I
K 1)d
2
.
(9.5.12)
d
9.6. ( X1 ,..., X M )
,
. , , Xm
.
FM ( x1 ,..., x M )
(m 1, M ) : M
FM ( x1 ,..., x M )
F1 ( x m ) . m 1
' X
Y.
( X ,Y ) ( x1 , y1 ),...,( xN , yN )
. F ( x, y ) N. Y,
,
F ( x, y ) I J
1. F ( x )F * ( y ) . (
*
H0 F ( x, y ) F ( x ) F ( y ) . -
*
,
X
X
' '
Y
). (9.5.9),
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I 2
l ij
J
N
2
li l j N
,
li l j
i 1 j 1
(9.6.1)
l ij
( X ,Y ) , (i -
ij -
X li
Y );
lix
lij
j-
j
i-
X,
; lj
l jy
lij i
Y, (9.6.1)
, N
j-
.
( I 1)( J 1)
.
, (
. 9.6.1). 9.6.1
X
Y
x1
x2
xI
lj
y1
l11
l12
l1I
l1 y
y2
l 21
l 22
l 2I
l2y
yJ
l J1
lJ 2
l JI
l Jy
li
l1x
l2x
l Ix
N
d ,
. ,
f(
2
, ( I 1)( J 1))d
2
.
d
-
2. 2
( F * ( x, y ) F * ( x) F * ( y )) 2 dF * ( x, y ) .
(9.6.2)
(9.6.2), 2
1 N5
N
(l1i l3i
l 2 i l 4i ) 2 ,
(9.6.3)
i 1
l1i , l 2 i , l 3i , l 4 i ,
,
, ( xi , y j ) (
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. 9.6.1).
. 9.6.1.
( xi , y j )
F ( b)
B
4
2
N
2
.
13.
9.7. 1. ( x11 ,..., x 1N1 ), ( x12 ,..., x N2 2 ),..., ( x1M ,..., x NMM )
M
,
(F(x )
F1 ( x )
N 1 , N 2 ,..., N M
X 1 , X 2 ,..., X M F2 ( x ) ... FM ( x )) ,
X1
X (X
F (x) X2
X M ).
...
.
-
1.
M
( F * ( x m ) F * ( x )) .
'
m 1
X 1 , X 2 ,..., X M ,
I
(
. 9.7.1). (9.5.10), M
2
I
N m 1 i 1
N m li N
l im
N m li
2
,
lim
(9.7.1) Xm,
i-
; li
X m ( m 1, M ) ,
i-
; N
:
M
N
Nm . m 1
, ( M 1)( I 1)
(9.7.1), .
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-
X 1 , X 2 ,..., X M
. 9.7.1.
I (M
2. max F1* ( x ) F2* ( x ) .
D
x
N1
, z
D
2)
N1 N 2 N1N 2
N2 (9.5.2).
(9.5.5), N1N 2 2 ln . 2( N 1 N 2 )
d
. 9.7.1. , ,
. (
,
,
),
.. 9.7.1
N
D
1 2 ln 2N
d
max F * ( x ) Fh ( x) x
2
( F * ( x)
(
) 1
2
1 N
12 N 2 2
(
)
(
10)
(
11)
(
12)
Fh ( x )) 2 dFh ( x )
N
Fh ( x n ) n 1
2n 1 2N
2
( F * ( x ) Fh ( x )) 2 ( Fh ( x))dFh ( x) 2
1
(1
2 N
N
( n 1
2n 1 ln Fh ( x n ) 2N
2n 1 ) ln 1 Fh ( x n ) ) 2N
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. 9.7.1 I
(
)
NPi
i 1
(
(I 1) I
[l i
2
(
NPi ) 2
(l i
2
NPi (
(I
(
-
* 1 *
1
,...,
N
K
(
8)
(
8)
)] 2
)
2
N li l j
( I 1)( J 1) )
(
2
K *
lil j
l ij
J
i 1 j 1
-
*
,...,
K 1) )
I 2
8)
)
NPi (
i 1
)
(
( F * ( x, y ) F * ( x) F * ( y )) 2 dF * ( x, y )
4
B 1
2
N5 M
-
2
N
(
l 2i l 4i ) 2
(l1i l 3i
2 13)
N
2
i 1
I
l im
N m 1 i 1
N m li N N m li
2
(
8)
( M 1)( I 1) )
(
D
N1 N 2 2 ln 2( N 1 N 2 )
d
max F1* ( x ) F2 * ( x) x
(
9
10)
. 9.7.2. 9.7.2 9 9, 11, 24, 47, 49, 68, 81, 93, 108 4, 13, 32, 42, 69, 75, 84, 91, 100 12, 18, 21, 44, 50, 58, 90, 101 2, 8, 14, 19, 26, 27, 34, 36, 51, 52, 53, 55, 56, 57, 63, 64, 66, 76, 85, 92, 96, 97, 102, 103, 106
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10. 10.1. X (t )
1. x n (t ) (n 1,2,...)
. ,
1.
.
(
)
. ( xn (t ) (n 1,2,...) .
2.
X (t )
)
xn (t )
3. N
( n 1, N ) , x (t ) ( x1 (t ),..., xN (t ) ), . .
,
.
X (t )
N
T.
xn (t )
N,
4. N-
, X (t ) ( X 1 (t ),..., X N (t ) ),
.
X (t )
1.
x (t ) ,
X (t ) . ,
2. .
,
. ,
.
.
,
,
, . . . :
x1 (t ),..., xN (t ) , ( ,
1. X (t ) , ,
).
X (t ) . t1 ,..., t I (t1
2. T
0;
tI
T) X (t )
I X (t1 ), X (t 2 ),..., X (t I ) .
3.
, ,
,
(10.1.1) . .
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, (10.1.1). 4. (t1 ,..., t M ) .
10.2. 10.2.1. xn (t ) ( n 1, N )
X (t ) (
N
m*x (t )
. 10.2.1).
xn (t ) : 1 N
m*x (t )
N
xn ( t ) .
xn (t ) ( n 1, N )
. 10.2.1.
(10.2.1)
n 1
m x (t )
X (t ) X (t ) ,
(10.2.1)
. (10.2.1)
1.
. ,
M m *x (t )
1 N
N
M X n (t )
m x (t ) ,
n 1
X (t ) .
m x (t ) (10.2.1)
2.
. ,
1) 2)
1 D x (t ) , N lim D m*x (t ) 0 (
D m *x (t )
X (t ) ;
D x (t )
N
m *x (t )
,
m x (t )
); 3) : lim P m *x (t ) m x (t )
N
3.
'
(10.2.1)
0. .
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.
,
X (t )
, * x
D m (t )
. t. 10.2.2.
x(t ) ( (t
,
x (t )
. 10.2.2.
. 10.2.2) 0, T ).
mx
X (t ) m*x : T
m*x
1 x (t )dt . T 0
(10.2.2)
X (t ) R x (t1 , t 2 )
R x (t 2 1.
t1 )
, (10.2.2)
. . T
M m *x
,
(10.2.2)
2.
1 M X (t ) dt T 0 X (t ) . .
mx ,
mx
, 1) D m
* x
T T
1 T2
Rx (t1 , t 2 ) dt1 dt 2 0 0
2 T
T
1 0
T
2 T
R x ( )d ,
(10.2.3)
X (t ) ;
Rx ( ) 2) lim
T
T
1 0
T
Rx ( )dt
0
;
3)
: * x
lim P m (t ) m x (t )
N
3.
0.
(10.2.2)
, T
m
* x
1 g (t ) x(t )dt . T 0
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(10.2.2)
. (10.2.2) 2Dx T 1 r ( )d , T 0 T , . 10.2.3).
4. D m *x
r( )
. (
D m *x Dx
. 10.2.3.
T
m *x r( )
T(
e
)
10.2.3.
X (t ) , , t1
(
0; t 2
t , ..., t I
( I 1) t
x (ti ) ( i 1, I )
. 10.2.4).
X (t )
x (t i )
. 10.2.4.
m
* x
m*x
, x ( ti ) : m *x
1 I
I
x (t i ) . i 1
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(10.2.4)
.
' .
X (t ) R x (t 2
t1 ) 1.
, (10.2.4) 1 M m*x I (10.2.4)
2.
(10.2.4) ,
.
I
M X (ti )
mx .
i 1
. :
1)
(10.2.3)
:
T
2 1 R x ( )d , I 0 T
D m *x I
Rx ( )
D x rx ( )
(t i
X (t ) , rx ( )
) , Dx
i 1
X (t ) , X (t ) ,
D m *x * x
2) lim D m I
I Dx 1 2 I i
1
1 1
(t )
, Rx ( )
-
i r x (i t ) ; I
0
;
3) 3.
T . (10, 100) I
.
I
15 20.
I, e
D m*x (t )
D[m x ] Dx
. 10.2.5.
r( )
. )
( ( I 1) t I. ( . 10.2.5),
(
)
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10.2.4.
X (t ) , xn (t ) (t [0, T ], n 1, N )
, (
N
. 10.2.6).
m*x X (t )
t
: m*x
m
* xn
1 N
N
m*xn ,
(10.2.5)
n 1
1T x n (t )dt . T 0
m*x
x n (t ) ( n 1, N )
. 10.2.6.
X (t ) X (t ) R x (t 2
t1 ) 1.
, (10.2.5) , 1 M m *x NT
(10.2.5) .
.
N T
M X (t ) dt
mx ,
n 1 0
X (t ) .
mx (10.2.5)
2.
. :
T
1)
2)
2 1 Rx ( ) d , NT 0 T X (t ) ; * lim D mx 0 ;
D m*x
Rx ( )
T
3)
. 3. .
(10.2.5) ,
Tn N
m*x
1 N
n
m*xn
n 1 N
, n
n 1
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. ,
n
1 , Dxn Dxn
D m*xn
m*xn . 1
D[m x ]
.
N n n 1
10.2.5.
X (t ) , ,
2T0
: m x (t ) m x (t )(
mx ( )
t) ,
t T0
t T0 . T
(10.2.6) 2T0
x(t ) . m*x (t )
,
x(t )
2T0 m*x (t )
1 2T0
X (t ) :
t T0
x ( )d .
(10.2.7)
t T0
. .
. .
.
X (t ) D m*x (t )
. Rx (t1 , t 2 )
,
* x
m (t ) * x
* x
D m (t )
M m (t ) m(t )
2
1 4T02
t T0 t T0
R x ( , )d d
. (10.2.8)
t T0 t T0
10.2.1
m *x
m*x
m *x
( )
1 N
1 T
1 I
1) 2) 3)
N
x n (t ) n 1
-
'
1) 2) 3)
T
x (t )dt 0
1) 2) 3) 15 20
I
x (t i ) i 1
I
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.10.2.1 1 N
m *x
(
m*xn
)
m*x (t )
( )
N
m *xn ,
1) 2)
n 1 T
1 T
x n (t )dt 0
1 2T0
D m*x (t )
t T0
t T0 t T0
1 4T02 t
x ( )d t T0
Rx ( , ) d d
, mx ( )
m x (t )
m x (t )(
T0 t T0
t
t ),
T0
t
T0
. 10.2.1.
10.3. 10.3.1. N
X (t )
xn (t ) mx (t ) .
: D x* (t )
1 N
N
x n (t ) m *x (t ) 2 ,
D x* (t )
n 1
* x
m (t )
1
N
N 1n
x n (t ) m *x (t ) 2 , 1
. ,
1. ,
.
.
2. : D D x* (t )
2 2 D x (t ) , N
D D x* (t )
2 N 1
D x2 (t ) ,
X (t ) .
Dx (t ) 10.3.2.
x(t )
X (t ) . mx
: Dx*
1 T
T
2
x (t ) m*x dt ,
(10.3.1)
0
m*x
.
X (t ) , 1.
.
D[m *x ] .
(10.3.1)
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(10.3.1) Dx*
1 T
T
x(t ) mx 2dt
mx*
2
mx
,
(10.3.2)
0
(10.3.2).
X (t ) .
(10.3.1)
2. 3.
X (t )
DD
* x
4 T
T
1
Rx2 ( )d .
T
0
(10.3.3)
(
D[ Dx* ]
T
, . 10.3.1).
D D x* D x2
. 10.3.1.
D*x r( )
(
e
)
10.3.3.
X (t ) , , x ( ti )
t1
0; t 2
t , ..., t I
( I 1) t (
. 10.2.4).
: D x*
1 I
2
I
x (t i ) m x , i 1
mx
. ,
. 1.
(10.3.4)
.
.
, * x
m , 2.
.
(10.3.4)
.
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(10.3.4)
(10.3.4)
3. D[ D x ]
I 2 D x2 1 2 I i
1
i 2 rx (i t ) , I
1 1
(10.3.3). (
1.
T
D[ Dx* ]
( I 1) t ) I (
. 10.3.2).
D Dx* Dx2
. 10.3.2.
Dx*
I,
r( )
(
, ,
2.
e
)
(10, 100),
I I
.
20
. . 10.3.1. 10.3.1
D x* (t )
x n (t ) m *x (t )
1 T
(
2
N
N 1
* x
2
n 1
1
D x* (t )
D
N
1 N
x n (t ) m *x (t ) n 1
T * x
2
( x(t ) m ) dt
1) 2)
-
1) 2)
-
1) 2)
D m x*
1) 2)
D m x*
0
)
D x*
1 I
I
( x (t i ) m *x ) 2 i 1
3) 20
( )
I
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10.4. 10.4.1. N
X (t ) .
xn (t ) m x (t )
R*x (t1 , t2 )
: N
1 N
R x* (t1 , t 2 )
x n (t1 ) m x (t1 ) x n (t 2 ) m x (t 2 ) , n 1
m x (t ) N
1
R x* (t1 , t 2 )
N 1n
x n (t1 ) m *x (t1 ) x n (t 2 ) m *x (t 2 ) , 1
* x
m (t )
. .
.
10.4.2.
x(t )
X (t ) ( t
0, T ).
. . mx
1.
. :
* x1
R ( )
1 T
T
0
R ( )
T
)dt , 0
T,
(10.4.1)
0
1
* x2
0
x (t ) x (t T
0
0
x(t ) x(t
)dt , 0
(10.4.1)
(10.4.2)
T,
(10.4.2)
0
0
x (t )
x ( t ) mx . . .
X (t ) Rx ( )
,
(10.4.1) (10.4.2) (10.4.1)
1. ( (10.4.2)
. T
Rx ( ) .
T
(10.4.1)
).
.
X (t )
2. .
X (t )
3. (10.4.1) (10.4.2)
,
,
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D[ R x*2 ( )]
T
1 T
D[ Rx*1 ( )]
t
1 (T
[ R x2 (t ) R x (t
T
) T
1 2
2
T (1 T
t
1 )
(T
[ Rx2 (t )
T
)
) Rx (t R x (t
) R x (t
(10.4.3)
)]dt. (10.4.4)
D[ Rx*1 ( )]
T
.
)]dt ,
D[ Rx*2 ( )]
T. D[ R x*1 ( )]
T
4.
D[ Rx*2 ( )]
, (
. 10.4.1).
Rx*1 ( )
. 10.4.1.
Rx ( )
rx ( )
(
.
Rx*2 ( ) e
,
T
4 .
(10.4.2)
(10.4.1). 2.
mx
* x
R ( )
1 T
5/ 2 )
.
T
x(t ) m *x
x(t
) m *x dt ,
(10.4.5)
0
m*x
.
X (t )
. ,
(10.4.5) ,
.
(10.4.5)
. (10.4.1). .
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10.4.3.
X (t ) , , x (t i ) (t i
(i 1) t , i 1, I ) . mx 1 I
R x* (l t )
I l
x((i 1) t ) m *x x ((i l 1) t ) m *x , (l
0, I 1) ,
(10.4.6)
i 1
m*x
.
X (t ) ,
(10.4.6) (10.4.6) . (10.4.6) (10.4.6)
1. 2. 3.
. .
X (t )
I D R x* (l t )
1 Ii
2
R x* ((i 1) t ) R x* ((i l 1) t ) R x* ((i l 1) t ) .
(10.4.7)
(10.4.7)
I
, . 0
mx
x * (t )
. 10.4.1,
x (t ) m *x . 10.4.1
R x* (t 1 , t 2 ) 1 N
N
0
0
x n* (t 1 ) x n* (t 2 )
R (t 1 , t 2 ) N
1
0
0
x n* (t 1 ) x n* (t 2 )
(
1 T
)
0
( )
I 1
0 * n
-
0
1) 2)
0
x * (t ) x * (t
)d
0
R x* (l t ) 1 I
1) 2)
n 1
R x* (t 1 , t 2 ) T
-
n 1
* x
N 1
1) 2)
0 * n
x ((i 1) t ) x ((i l 1) t )
1) 2)
i 1
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10.5. x(t ) X (t ) ( t
0, T )
. T
A( )
x (t )e
j t
dt .
0
.
x(t ) ,
ST ( )
T: 2 1 A( ) . T ST ( ) '
ST ( )
. * x
R( )
' : T
Rx* ( )e
ST ( )
j
d .
(10.5.1)
T *
1.
(t )
(t ) .
(t )
(t ) ,
F
. .
*
F
'
'
(10.5.1) Sx ( ) . ST ( )
,
T .
S x* ( )
. Sx ( )
. : .
1)
N
Tn .
n-
STn ( )
: S x* ( )
2)
1 N
N
ST n ( ) ;
(10.5.2)
n 1
.
ST ( ) .
: S x* ( )
W( )
1 2
W(
1
)S T (
1
)d
, 1 2
W( ) d
1
,
(10.5.3) :
1,
W( ) W(
);
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(10.5.4)
3)
. * x
R ( ).
w( ) , ' : T
S x* ( )
j
w( ) R x* ( )e
d .
(10.5.5)
T
: w(0) 1 , w( ) w(
w( )
T . N
0
w( )
.
);
W( )
'
' : 1 W( )e j d , W( ) 2
w( )
w( ) e
j
d .
X (t ) , Rx ( )
Sx ( )
, , 1.
. (10.5.2)
. (10.5.3)
,
(10.5.5)
. Sx ( ) K
1 4
K Sx ( ) ,
2
W ( )d . (10.5.3)
2.
X (t )
T D Sx* ( ) 1 T
KD
DS ( ) , (10.5.4), (10.5.8), (10.5.9)).
(10.5.7) (10.5.5)
K D S x2 ( ) ,
(10.5.8)
1 2 T
(10.5.9)
w 2 ( )d
W 2 ( )d .
. * x
(10.5.6)
W( ) ( , 10
.
Sx ( ) (10.5.7)), W( ) (
.
. 10.5.1. 10.5.1 10 24, 47, 49 32, 91 50, 101 2, 14, 26, 51, 52, 53, 54, 55, 66, 76, 84, 85, 86, 92, 96, 97, 100, 106
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11. 11.1. 11.1.1.
. (
,
)
.
. .
9
,
. , . .
'
. . M
. 1 1
1 N1
2 1
2 N2
M 1
( x ,..., x ), ( x ,..., x ),..., ( x ,..., x
M NM
N 1 , N 2 ,..., N M
)
(
. 11.1.1).
, X 1 , X 2 ,..., X M
: D1 , D2 ,..., D M .
m1 , m 2 ,..., m M
. 11.1.1.
,
(
). m-
.
m m Dm*
1 Nm
* m
1
Nm
* m
Dm* :
x nm ;
(11.1.1)
n 1 Nm
Nm 1 n
( x nm
mm* ) 2 .
(11.1.2)
1 M
M
N
Nm m 1
m*
D*
: m*
1 N
M
Nm
x nm m 1
n 1
1 N
M
N m mm* ; m 1
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(11.1.3)
N 1m
*
D
Nm
M
1
D*
1
( x nm
1
m* ) 2
N 1
n 1
[( N
M )D*
( M 1) D0* ] , (11.1.4)
,
: 1
D*
D0*
M
N
( N m 1)Dm* ,
M
(11.1.5)
m 1
,
( * m
m ): 1
D0*
M
M
1m
N m (mm*
m* ) 2 .
(11.1.6)
1
,
1. . 11.1.1.
, 11.1.1
, N
M D
2
D*
2
M 1 * D0 D (mn*
m k* ) (m n
mk )
Nn Nk Nn Nk
mk )
Nn Nk Nn Nk
D (mn*
m *k ) (mn D
*
Dn* Dk* Dn* Dk*
ln
D0* D* ln
D0* D*
(N
-
M)
(M 1)
-
t-
Nn
Nk
v2 -
m1
N n 1, m2
Nk
1
z-
m1
N n 1, m2
Nk
1
(
2
)
v2 -
m1
M
1 , m2
N
M
z-
m1 (
M
1 , m2 )
N
M
5 9, 14.
.
,
2. . 11.1.1,
, .
. 11.1.2. ,
3. ,
.
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4. (
(
1
,...
) ).
L
. . . ,
. . .
11.1.2 , ,
,
Dn
Dn* ln Dk*
Dk m1
( mn
mk
Dn
(
mn* (
m1
(
m2
D1
mk* Nk
DM )
...
(
m1
2
Nk
Nn Nk Nn Nk
*
1)
t 1
2
2
)
D0* ln D*
mM
...
D2
Nn
1
N n 1, m2
D
Dk )
z
z 1
2
N 1 , m2
N
M)
11.1.2. ,
(
1
,...,
L
),
,
. . ,
. 1
.
,...,
L
,
,
,
, . . . 1
2
M2
.
1
M1
,
. 1
2
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2
x11 , x12 , ..., x1M 2 , x 21 , x 22 , ..., x 2 M 2 ,
(11.1.7)
......................... x M 11 , x M 1 2 , ..., x M 1M 2
M
M1 M 2 (
. 11.1.2). 1
2
.
. m *m2
mm* 1
(11.1.7) m* :
m *m2
1 M2
m m* 1
1 M 1M 2
m*
. 11.1.2.
1 M1
M1
x m1m2 ;
(11.1.8)
x m1m2 ;
(11.1.9)
m1 1 M2 m2 1 M1
M2
x m1m2 .
(11.1.10)
m1 1 m 2 1
,
1
2
, D*
M1 1 M 1 M 2 1 m1 1
1 ( M 1 1)( M 2 1) D * M 1M 2 1
M2
m* ) 2
( x m1m2 m2 1
(11.1.11) * 1
( M 1 1)M 2 D
* 2
M 1 ( M 2 1) D ,
: D*
1 ( M 1 1)(M 2 1)
M1
M2
( x m1m2
m*
m * m1
m*
m2
)2 ,
m1 1 m2 1
,
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(11.1.12)
1
D1*
M1 1
M1
(mm* 1
m* ) 2 ,
(11.1.13)
m1 1
, 1
D2*
M2 1
M2
(m*
m* )2 ,
m2
(11.1.14)
m2 1
. ,
1. D , D1* , D2* *
-
( M 1 1)( M 2 1) , ( M 1 1)
D1* D*
2. m2
( M 1 1)( M 2 1) ,
(
m2
.
.
v2 -
m1
M 1 1,
D1* D*
ln
.
. 11.1.1 11.1.2). D2* 3. D*
v2 -
M 2 1,
m1
D2* D*
ln
( M 1 1)( M 2 1) ,
( M 2 1)
.
, .
L
.
M M L.
, . .
.
1. .
, , . 2
M ,
3
M ,
. ,
2. 1
,...,
L
l
,
, :
l
(
,
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),
.
l
L
L l 1.
11.2. (
. 11.2.1).
. 11.2.1.
. . ,
1. . 2. . 3.
K (
) mk ,
, Rk ( k
Dk
1, K ).
4. ,
, .
. K (
*
mk* ,
,
Rk* ,
k
Dk*
1, K * ).
.
'
.
, (
).
.
f (x ) N -
X. (
K
L
2 ,
L
, )
I
.
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.
:
mk* ,
Rk* ( k
Dk*
L
.
1, K ).
.
X
,
.
, * 1
R11* .
a
b1n* ( n 1, N )
N* 1
.
R11*
N-
, :
* 1
a ,
, b1n* ,
,
, * 1
(
).
* b1m , * 1m
a1*
. * b1m
,
. , ,
. 2
L K
2
L
l 1
l-
.
I
(
). ,
.
,
,
. , .
11.3. 11.3.1. 1. , ,
,
. ,
. . 1. ( (
,
),
.). 1. m *x
1 N
N
xn n 1
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(11.3.1)
X,
( x1 , x 2 ,..., x N ) , N
1 N
D x*
( xn
(11.3.2)
1
(11.3.3)
n 1
1 N
F * ( x) (
m *x ) 2 ,
xn x
xn
n,
f * ( x)
1 N
x)
N
( x xn ) .
(11.3.4)
n 1
2. m *x Dx*
1 k 1 k
N
xn f ( xn ) ,
(11.3.5)
n 1 N
( xn
m *x ) 2 f ( x n )
(11.3.6)
n 1
X, f (x)
( x1 , x 2 ,..., x N ) N
k
,
f ( xn ) . n 1
X
3.
*
.
, .
, (
1. (
f (x, ) ,
,
),
( ).
(
), ),
. . . 11.3.1 X.
4.
, . ,
,
1,
0,2.
, 2 ,
2,7.
, ,
, ,
.
. 11.3.1.
,
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2. ,
. ,
, . ,
1. ,
,
:
. ,
; (
) .
,
;
(
)
; .
. 2. .
,
,
, )
(
. ,
,
,
, .
,
,
,
,
,
. .
,
,
, , . ,
. ,
. ,
. ,
3.
, ,
,
,
.
. ,
, ,
,
,
. .
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T*
2.
t,
,
, .
1. . . 5. 1 N
xc xn , c,
yn
| x n | c, c | x n | c,
yn ,
(11.3.7)
n
,
,
,
c.
,
c.
. , , . ,
2. ,
. ,
3 . . . . T
3 .
*
t,
X,
,
F (x) F0 ( x) ( ( F ( x ), F0 ( x)) F ( x)
( (
F ( x)
(t * )
F0 ( x )
(t * ))
(t * )
F0 ( x )
0) (t * )
0 ).
1. , (
F ( x)
(t * )
F0 ( x )
(t * ))
0.
2.
,
( )
(
. 11.3.2).
3.
. ,
, .
. (
6.
)
X :
m *x
med ( x1 ,..., x N ) ,
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(11.3.8)
1 N 2N
x
xn ,
(11.3.9)
n
, N
N
.
. 11.3.2.
:
f (x)
f 0 ( x) ,
7. X,
:
s
* x
* x
med (| x1
m |,..., | x N
* x
m |) .
(11.3.10)
1. .
, ,
. 6, 7,
.
. . ,
2.
, .
. 11.3.2. 9.2
. H0
x
H1
( x1 ,..., x N ) .
, l( x )
L( x / H1 ) L( x / H 0 )
ln l ( x ) ln l ( x )
c:
c.
(11.3.11) H0 . H1 .
,
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(11.3.11) , ,
, , N
ln l ( x )
ln l1 ( xn )
c,
(11.3.12)
n 1
L( xn / H1 ) L ( xn / H 0 )
l1 ( xn )
H1
xn . L( x n / H 0 )
L( x n / H 1 )
.
H0 ,
,
, . ,
,
,
. .
. x
( x1 ,..., x N ) : H0
H0 ~ L ( xn / H 0 ) ,
H1 .
x 0, ~ L ( xn / H 1 ) , 0
H1
,
, 1
.
1
~ L ( x n / H 0 ) (1 ~ L ( x n / H 1 ) (1 0
,
1
.
0 1
) L( x n / H 0 )
) L( x n / H 1 )
0 1
:
w0 ( x n ) ,
(11.3.13)
w1 ( x n ) ,
(11.3.14)
,
( L( x n / H 0 )
L( x n / H 1 ) ), w0 ( x n )
w1 ( x n )
.
. q0 ( x)
, q1 ( x ) ,
: (1
q0 ( x)
(1 (1 (1
q1 ( x) c'
c' '
) L( x / H 0 ), l1 ( x ) c ' ' , L( x / H 1 ) , l1 ( x ) c ' ' ; 0) c' ' 0
) L( x / H 1 ), l1 ( x) c' , l1 ( x ) c ' , 1 )c' L ( x / H 0 ), 1
( c ' c' ' ), q 0 ( x) q 0 ( x)
q1 ( x ) .
l q (x)
q1 ( x)
: lq ( x )
bc '', l1 ( x) c '', bl1 ( x ), c ' l1 ( x ) c '', bc ', l1 ( x ) c ',
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(11.3.15)
b
(1 (1
) . ) 0 1
l q ( xn )
xn ,
l1 ( x n ) .
(11.3.12) 1. . 2. 1,
0
. 11.3.3. . (
)
60. .
.
(
)
.
, . ,
,
,
, , .
.
A
. .
~ A, r
A
A
rI ,
, I
, .
, .
r
1. .
~ A
r, 10 100
. 2.
.
, . 11
. 11.3.1. 11.3.1 11
24, 40, 47, 49, 81, 93, 108 16, 17, 32, 75, 84 21, 46, 50, 90, 101, 110 43, 52, 53, 55, 56, 62, 63, 65, 76, 85, 97, 100, 102
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1 -
.
,
: (t t 0 )
0,
t
t0 ;
,
t
t0 . . 1.
-
1 e j 2 f df
( ) (t t 0 )
lim 0
2
1 2
( )
t0
1 x (t0 ) 2 x ( t0 )
x (t ) (t t0 )dt a
x (t )
e
2
2
ej d
0 b
( t t0 ) 2
1
a; t 0
t0
b (a
b);
a a
t0 t0
b;
b, t
,
t0
t0
(t t 0 )dt 1 ,
0
t0
A (r )
( t t0 )
( )
( 1) r r!
(t )
e
j t0
(t ) , tr
r=0, 1, 2,
2
-
(z )
-
e t t z 1 dt (Re z
(z)
0) ,
0
( z 1)
' z
z N
z ( z) , (1) 1. . 2.1.
:
(N )
( N 1)! ( N
0,1, ) .
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.
2.1.
-
(0,5)
,
, N
(2n 1) (N
0,5)
( N
0,5)
n 1
,
2N ( 2N )
.
N
(2n 1) n 1
3 , . { X 1 ,..., X N }
1.
X . F1 ( x ) ,..., FN (x) F (x) . 1) X , FN ( x ) F ( x) 0
4) P{ X N
x,
, ;
N
,
P{ X N
X
,
M[ X N
}
F( x) -
0
;
N
X
3) N
:
X
2)
X
X 1 ,..., X N
l.i.m. X N
.
X N
( .
' lim X N
N
1.
2
]
0
X ;
N
X} 1
X
), X.
.
. .
, ,
;
, .
. 3.1. 218
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.
3.1.
2. . 3.
X (t ) ( t
T ).
(
2.
)
X (t ) ( t T ), T : M [ X 2 (t )] .
t
. ,
. 3. t, l.i.m. X (t t
lim M [ X (t t
X (t ) 2 ]
t)
0
t)
0
X (t ) ,
0.
X (t )
1. t
,
m x (t )
t t
t1
R x ( t1 , t 2 )
t2 .
X (t )
4. X ' (t )
l.i.m. t
X (t
t) t
0
t, X (t )
.
X ' (t ) X (t ) . X (t )
2. t m x (t )
t
, t1
t2
t
2
t1 t 2
R x ( t1 , t 2 )
R x (t1 , t 2 ) .
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X (t )
5. tn
N
T( ), tn 1
tn Y( )
T( ) tn
l.i.m.
max t n
X (t n ) t n
0
X (t )dt .
n
T( )
Y( ) X (t ) . X (t )
,
.
m x (t )
R x ( t1 , t 2 ) m x (t )dt ,
,
T( )
Mx
X (t )dt = T( )
Mx
m x (t )dt , T( )
X (t1 ) X (t 2 )dt1 dt 2 = T( )T( )
R x (t1 , t 2 )dt1 dt 2 . T( )T( )
Rx (t1 , t 2 )dt1 dt 2 T( )T( )
[ m x (t )dt ]2 . T( )
4
s(t )
1.
,
s (t )
' 1
s (t )
H [s (t )] ,
s ( ) d t ,
1
s (t )
s( ) d t
H
H [ s (t )] . ,
.
1. 2.
.
,
, s (t ) s (t )dt
,
0. (
3.
s(t )
t, s (t t 0 ) ) s (t t 0 ) .
1.
Fs ( )
4.
s(t )
s (t )
sign ( )
s(t )
Fs ( )
' 1 1 1. s (t )
Fs ( ) = jFs ( )sign( ) , 0, 0. | Fs ( ) |
| Fs ( ) |
:
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) t0 (
| Fs ( ) | = | Fs ( ) | . arg Fs ( )
2.
s(t )
s (t )
arg Fs ( )
'
arg Fs ( ) = arg Fs ( )
2
sign ( ) .
5.
.
s(t )
,
, ,
s (t )
. 6. s (t )
cos(
0t
)
0
s (t ) sin(
0
t
).
7.
s(t ) h(t )
1 , t
K( )
j, j,
0, . 0
. /2.
.
.
4.1. 4.1
s(t )
s (t )
1 t sin t cos t 1 cos t t t 1 t2 1 t T ln t T
(t ) cos t sin t sin t t 1 1 t2 1, | t | T , 0, | t | T 2. s (t )
js (t ) .
s (t )
3. .
S (t )
s (t ) ,
.
(t )
arg s (t ) .
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.
S (t )
1.
(t )
s (t )
s(t )
'
s (t ) :
s 2 (t ) s 2 (t ) ,
S (t ) (t )
arctg
s (t ) . s (t )
.
s (t )
2. A
S (t )
'
(t ) .
S (t )e j
s (t )
(t)
.
.
s (t )
1. A e
<<
:
0 .
s (t )
S (t )e j (
0t
( t ))
.
4. .
S (t )
S (t )e j
(t)
.
2. ,
.
X (t )
, X (t ) ,
( X (t ) = H [ X (t )] ). m x (t ) m x (t )
1.
X (t ) : 2. R xx
X (t ) , X (t )
m x (t ) = H [m x (t )] . R x ( t1 , t 2 ) , (t1 , t 2 ) , Rx x (t1 , t 2 )
Rx (t1 , t 2 ) :
1
Rx (t1 , t 2 ) = R x ( t1 , t 2 ) =
Rxx (t 2 , ) d , t1
1
Rxx (t1 , t 2 ) = R x x ( t1 , t 2 ) =
R xx ( , t 2 ) d , t1
1
R x (t 2 , ) d , t1
1
R x (t 2 , ) d . t1
X (t )
3. ,
X (t )
.
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X (t )
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F ( x) x 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9
x
F (x ) 0,5000 0,4602 0,4207 0,3821 0,3446 0,3085 0,2743 0,2420 0,2119 0,1841 0,1587 0,1357 0,1151 0,0968 0,0808 0,0668 0,0548 0,0446 0,0359 0,0288
x
1
e
2
0,80 0,81 0,82 0,83 0,84 0,85
0,86 0,87 0,88 0,89 0,90
1,475 1,513 1,554 1,597 1,643
2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9
F (x) 0,9772 0,9821 0,9861 0,9893 0,9918 0,9938 0,9953 0,9965 0,9974 0,9981 0,9986 0,9990 0,9993 0,9995 0,9997 0,9998 0,9998 0,9999 0,9999 1,0000
1
arg F
2 t
t
1,282 1,310 1,340 1,371 1,404 1,439
x
F (x) 0,5000 0,5398 0,5793 0,6179 0,6554 0,6915 0,7257 0,7580 0,7881 0,8159 0,8413 0,8643 0,8849 0,9032 0,9192 0,9332 0,9452 0,9554 0,9641 0,9713
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9
t
t
dt
x
F (x ) 0,0228 0,0179 0,0139 0,0107 0,0082 0,0062 0,0047 0,0035 0,0026 0,0019 0,0014 0,0010 0,0007 0,0005 0,0003 0,0002 0,0002 0,0001 0,0001 0,0000
2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9
1 2 t 2
0,91 0,92 0,93 0,94 0,95 0,96
t
1,694 1,750 1,810 1,880 1,960 2,053
0,97 0,98 0,99 0,9973 0,999
2,169 2,325 2,576 3,000 3,290
8 2
2
P(
2
2
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) x2
2
n 2
n 2
n 2
e
2
d
2
,
2
0
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n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0,999
0,995
0,99
0,98
0,975
0,05157
0,0 4393
0,0 3157
0,0 3628
0,0 3982
0,002 0,0243 0,0908 0,210 0,381 0,598 0,857 1,152 1,479 1,834 2,214 2,617 3,041 3,483 3,942 4,416 4,905 5,407 5,921 6,447 6,983 7,529 8,085 8,649 9,222 9,803 10,391 10,986 11,588
0,0100 0,0717 0,207 0,412 0,676 0,989 1,344 1,735 2,156 2,603 3,074 3,565 4,075 6,501 5,142 5,697 6,265 6,844 7,434 8,034 8,643 9,260 9,886 10,520 11,160 11,808 12,461 13,121 13,787
0,0201 0,115 0,297 0,554 0,872 1,239 1,646 2,088 2,558 3,053 3,571 4,107 4,660 5,229 5,812 6,408 7,015 7,633 8,260 8,897 9,542 10,196 10,856 11,524 12,198 12,879 13,565 14,256 14,953
0,0404 0,185 0,429 0,752 1,134 1,564 2,032 2,532 3,059 3,609 4,178 4,765 5,368 5,985 6,614 7,255 7,906 8,567 9,237 9,915 10,600 11,293 11,992 12,697 13,409 14,125 14,847 15,574 16,306
0,0506 0,216 0,484 0,831 1,237 1,690 2,180 2,700 3,247 3,816 4,404 5,009 5,629 6,262 6,908 7,564 8,231 8,907 9,591 10,283 10,982 11,688 12,401 13,120 13,844 14,573 15,308 16,047 16,791
0,95 0,0039 0,103 0,352 0,711 1,145 1,635 2,167 2,733 3,325 3,940 4,575 5,226 5,892 6,571 7,261 7,962 8,672 9,390 10,117 10,851 11,591 12,338 13,091 13,848 14,611 15,379 16,151 16,928 17,708 18,493
0,9 0,0158 0,211 0,584 1,064 1,610 2,204 2,833 3,490 4,168 4,865 5,578 6,304 7,042 7,790 8,547 9,312 10,085 10,865 11,651 12,443 13,240 14,041 14,848 15,659 16,473 17,292 18,114 18,939 19,768 20,599
0,8 0,0642 0,446 1,005 1,649 2,343 3,070 3,822 4,594 5,380 6,179 6,989 7,807 8,634 9,467 10,307 11,152 12,002 12,857 13,716 14,578 15,445 16,314 17,187 18,062 18,940 19,820 20,703 21,588 22,475 23,364
0,75 0,102 0,575 1,213 1,923 2,675 3,455 4,255 5,071 6,899 6,737 7,584 8,438 9,299 10,165 11,036 11,912 12,792 13,675 14,562 15,452 16,344 17,240 18,137 19,037 19,939 20,843 21,749 22,675 23,567 24,478
0,70 0,148 0,713 1,424 2,195 3,000 3,828 4,671 5,527 6,393 7,267 8,148 9,034 9,926 10,821 11,721 12,624 13,531 14,440 15,352 16,266 17,128 18,101 19,021 19,943 20,867 21,792 22,719 23,647 24,577 25,508
. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0,50 0,455 1,386 2,366 3,357 4,351 5,348 6,346 7,344 8,343 9,342 10,341 11,340 12,340 13,339 14,338 15,338 16,338 17,338 18,338 19,337
0,30 1,074 2,408 3,665 4,878 6,064 7,231 8,383 9,524 10,656 11,781 12,899 14,011 15,119 16,222 17,322 18,418 19,511 20,601 21,689 22,775
0,20 1,642 3,219 4,642 5,989 7,289 8,558 9,803 11,030 12,242 13,442 14,631 15,812 16,985 18,151 19,311 20,465 21,615 22,760 23,900 25,038
0,10 2,706 4,605 6,251 7,779 9,236 10,645 12,017 13,362 14,684 15,987 17,275 18,549 19,812 21,064 22,307 23,542 24,769 25,989 27,204 28,412
0,05 3,841 5,991 7,815 9,488 11,070 12,592 14,067 15,507 16,919 18,307 19,675 21,026 22,362 23,685 24,996 26,296 27,587 28,869 30,141 31,410
0,025 5,024 7,378 9,348 11,143 12,832 14,449 16,013 17,535 19,023 20,483 21,920 23,337 24,736 26,119 27,488 28,845 30,191 31,526 32,852 34,170
0,02 5,412 7,824 9,837 11,668 13,388 15,033 16,622 18,168 19,679 21,161 22,618 24,054 25,472 26,873 28,259 29,633 10,995 32,346 33,687 35,020
0,01 6,635 9,210 11,345 13,277 15,086 16,812 18,475 20,090 21,666 23,209 24,725 26,217 27,688 29,141 30,578 32,000 33,409 34,805 36,191 37,566
0,005 7,879 10,597 12,838 14,860 16,750 18,548 20,278 21,955 23,589 25,188 26,757 28,300 29,819 31,319 32,801 34,267 35,718 37,156 38,582 39,997
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0,001 10,827 13,815 16,268 18,465 20,517 22,457 24,322 26,125 27,877 29,588 31,264 32,909 34,528 36,123 37,697 39,252 40,790 42,321 43,820 45,315
. n
0,50 20,337 21,337 22,337 23,337 24,337 25,336 26,336 27,336 28,336 29,336
21 22 23 24 25 26 27 28 29 30
0,30 23,858 24,939 26,018 27,096 28,172 29,246 30,319 31,391 32,461 33,530
0,20 26,171 27,301 28,429 29,553 30,675 31,795 32,912 34,027 35,139 36,250
0,10 29,615 30,813 32,007 33,196 34,382 35,563 36,741 37,916 39,087 40,256
0,05 32,671 33,924 35,172 36,415 37,652 38,885 40,113 41,337 42,557 43,773
0,025 35,479 36,781 38,076 39,364 40,646 41,923 43,194 44,161 45,722 46,979
0,02 36,343 37,659 38,968 40,270 41,566 42,856 44,140 45,419 46,693 47,962
0,01 38,932 40,289 41,638 42,980 44,314 45,642 46,963 48,278 49,588 50,892
0,005 41,401 42,796 44,181 45,558 46,928 48,290 49,645 50,993 52,336 53,672
0,001 46,697 48,268 49,728 51,179 52,620 54,052 55,476 56,893 58,302 59,703
9 (t -
P (T
t ) t
N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0,45 0,158 0,142 0,137 0,134 0,132 0,131 0,130 0,130 0,129 0,129 0,128 0,128 0,128 0,127 0,127 0,127 0,126
0,4 0,325 0,289 0,277 0,271 0,267 0,265 0,263 0,262 0,261 0,260 0,259 0,258 0,258 0,257 0,257 0,256 0,256
)
N 1 ( ) t2 2 (1 ) N N ( N / 2) 0,35 0,510 0,445 0,424 0,414 0,408 0,404 0,402 0,399 0,398 0,397 0,395 0,393 0,392 0,393 0,391 0,390 0,389
0,3 0,727 0,617 0,584 0,569 0,559 0,553 0,549 0,546 0,543 0,542 0,539 0,537 0,535 0,534 0,533 0,531 0,530
N 1 2
0,25 1,000 0,816 0,765 0,741 0,727 0,718 0,711 0,706 0,703 0,700 0,695 0,692 0,690 0,688 0,687 0,684 0,683
dt
0,2 1,376 1,061 0,978 0,941 0,920 0,906 0,896 0,889 0,883 0,879 0,873 0,868 0,865 0,862 0,860 0,856 0,854
0,15 1,963 1,336 1,250 1,190 1,156 1,134 1,119 1,108 1,100 1,093 1,083 1,076 1,071 1,067 1,064 1,058 1,055
. N 1 2 3 4 5 6 7 8 9 10
0,1 3,078 1,886 1,638 1,533 1,476 1,440 1,415 1,397 1,383 1,372
0,05 6,314 2,920 2,353 2,132 2,015 1,943 1,895 1,860 1,833 1,812
0,025 12,706 4,303 3,181 2,776 2,571 2,447 2,365 2,306 2,262 2,228
0,01 31,821 6,965 4,541 3,747 3,365 3,143 2,998 2,896 2,821 2,764
0,005 63,657 9,965 5,841 4,604 4,032 3,707 3,499 3,355 3,250 3,169
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0,0005 636,619 31,598 12,940 8,610 6,869 5,959 5,405 5,041 4,781 4,587
. 12 14 16 18 20 25 30
1,356 1,345 1,337 1,330 1,325 1,316 1,310
1,782 1,761 1,746 1,734 1,725 1,708 1,697
2,179 2,145 2,120 2,101 2,086 2,060 2,042
2,681 2,624 2,583 2,552 2,528 2,485 2,457
3,055 2,977 2,921 2,878 2,845 2,787 2,750
4,318 4,140 4,015 3,922 3,850 3,725 3,646
10 P( Z
z)
K ( z) 1 2
1
( 1)
e
2
2 2
z
,
z
0
1
z
K (z )
z
K (z )
z
K (z )
z
K (z )
0,1 0,2 0,3 0,4 0,5
0,000 0,000 0,000 0,003 0,036
0,6 0,7 0,8 0,9 1,00
0,136 0,289 0,456 0,607 0,730
1,1 1,2 1,3 1,4 1,5
0,822 0,888 0,932 0,960 0,978
1,6 1,7 1,8 1,9 2,0
0,988 0,994 0,997 0,998 0,999
11 P( 2
F(
0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,11 0,12 0,13 0,14 0,15 0,16 0,17 0,18 0,19
2
2
)
0,000 0,001 0,003 0,025 0,067 0,1237 0,1860 0,2484 0,3081 0,3638 0,4151 0,4619 0,5045 0,5432 0,5784 0,6104 0,6395 0,6660 0,6902 0,7123
0,20 0,21 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30 0,31 0,32 0,33 0,34 0,35 0,36 0,37 0,38 0,39
F(
2
2
2
2
)
0,7325 0,7511 0,7681 0,7838 0,7983 0,8116 0,8239 0,8353 0,8459 0,8557 0,8648 0,8733 0,8811 0,8884 0,8953 0,9016 0,9076 0,9131 0,9183 0,9232
2 F( )
) F(
0,40 0,41 0,42 0,43 0,44 0,45 0,46 0,47 0,48 0,49 0,50 0,51 0,52 0,53 0,54 0,55 0,56 0,57 0,58 0,59
2
2
)
0,9277 0,9320 0,9360 0,9397 0,9432 0,9465 0,9496 0,9525 0,9552 0,9577 0,9602 0,9624 0,9645 0,9665 0,9684 0,9702 0,9719 0,9734 0,9749 0,9763
F(
0,60 0,61 0,62 0,63 0,64 0,65 0,66 0,67 0,68 0,69 0,70 0,71 0,72 0,73 0,74 0,75 0,76 0,77 0,78 0,79
2
2
)
0,9776 0,9788 0,9800 0,9811 0,9821 0,9831 0,9840 0,9845 0,9857 0,9865 0,9872 0,9879 0,9886 0,9892 0,9898 0,9904 0,9909 0,9914 0,9918 0,9923
F(
0,80 0,81 0,82 0,83 0,84 0,85 0,86 0,87 0,88 0,89 0,90 0,91 0,92 0,93 0,94 0,95 0,96 0,97 0,98 0,99
2
)
0,9927 0,9931 0,9934 0,9938 0,9941 0,9944 0,9947 0,9950 0,9952 0,9955 0,9957 0,9959 0,9962 0,9964 0,9966 0,9967 0,9969 0,9971 0,9972 0,9974
12 P( 2
0,00 0,1 0,2
2
2 2
)
F(
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0,0000 0,0000 0,0096
1,0 1,1 1,2
F(
2
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)( F(
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0,6427 0,6912 0,7324
2
2,0 2,1 2,2
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0,9081 0,9190 0,9285
3,0 3,1 3,2
F(
F(
2
2
)
0,9726 0,9756 0,9783
4 5 6
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F(
2
)
0,9913 0,9971 0,9990
. 0,3 0,4 0,5 0,6 0,7 0,8 0,9
0,0618 0,1512 0,2531 0,3520 0,4411 0,5189 0,5857
1,3 1,4 1,5 1,6 1,7 1,8 1,9
0,7676 0,7977 0,8235 0,8457 0,8648 0,8813 0,8957
2,3 2,4 2,5 2,6 2,7 2,8 2,9
0,9368 0,9440 0,9505 0,9560 0,9610 0,9654 0,9692
3,3 3,4 3,5 3,6 3,7 3,8 3,9
0,9806 0,9828 0,9846 0,9863 0,9877 0,9891 0,9902
7 8 9
0,9996 0,9999 0,9999
13
B4
P b 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,1 1,2
F (b) 0,00000 0,00086 0,01158 0,4867 0,11594 0,20293 0,29652 0,38730 0,47027 0,54354
b 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0 2,1 2,2
2
F (b) 0,60697 0,66131 0,70763 0,74704 0,78060 0,80922 0,83369 0,85469 0,87275 0,88335
2
N
b
F (b)
F (b) 0,90185 0,91357 0,92377 0,93268 0,94047 0,94730 0,95329 0,95857 0,96322 0,96732
b 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2
F (b) 0,97094 0,97414 0,97698 0,97949 0,98172 0,98370 0,98515 0,98702 0,98841 0,98965
b 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4,0 4,1 4,2
F (b ) 0,99075 0,99174 0,99261 0,99339 0,99409 0,99471 0,99627 0,99576 0,99857 0,99952
b 4,3 4,4 4,5 4,6 4,7 4,8 4,9 5,0 6,0 7,0
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2
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.
1.4.1
(
1.1.4 1.1.4 1.1.3 1.1.2 ) 1.1.3 3.5.1 11.1.1 11.1.1 9.4 8.1 11.1.1
9.5.1 2.4.1 ,
9.2 9.2
9.1
4.2 1.2.1 1.2.1 ( ) 1.2.2 , 1.2.3, 1.3.3 , 1.2.3 1.2.1 , ( ) 1.2.3, 1.3.3 , 1.2.3 1.2.3 1.2.3 1.3.3 , 1.2.3, 1.3.3 , 1.2.3 , 1.2.3 , 1.2.3 1.2.3, 1.3.3 1.2.3, 1.3.3 1.2.3, 1.3.3 1.2.3 1.2.3, 1.3.3 , 1.2.3 , ( . ) , ( 1.2.3, 1.3.3 1.2.1
9.1 9.5.1 9.1 9.7 9.1 9.5.1 9.1 9.1 6.1.2 1.1.4
1.1.5
1.1.1
1.1.5 1.2.3, 1.3.3, 2.2.2 6.1.3 8.1, 8.4 2.2.3 2.2.3 2.2
( ) 6.1.1, 10.1 1.2.3, 1.3.3 1.3.3 ) (
6.1.1
1.3.2
1.3.2 ( ) 1.3.2
,
( ( 1.3.3
6.1.3 3.4 3.4
) 1.3.3 ) 1.3.3 .3 7.5.1
1.5.1 1.1.1
6.1.1 6.1.1 6.1.1, 10.1 6.1.1 6.1.1
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6.1.1 2.4.1 8.5
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3.4 3.6.1
9.2
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1.1.4 4.8 (
) 4.6 1.1.1
4.2
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1.1.1
1.1.1
1.2.3 1.2.3 11.2
,
1.2.3
1.2.3 7.5.1 4.3 1.2.3 4.3 1.3.3 8.4 8.4
( 8.6
8.2, 8.4 9.2 9.5.1 9.5.3 9.5.4
9.2 9.2 9.2
9.2 9.4 9.1
9.1 9.7 1.2.3, 1.3.3
( 4.8 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2
1.2.3 ( ) 2.2.2 7.2.3 6.2.1 1.1.1
9.1
9.1 3.2.3, .4 3.2.3
1.4.1 ,
8.8.1 7.2.3 1.3.3 8.8.1 4.2 1.2.3
2.4.1 2.4.1 2.4.1 ( ) 3.6.1 2.4.1 2.4.3 3.6.1 2.4.1 2.4.1 3.6.1 2.4.3 3.6.1 3.6.1
7.5.1
1.1.2 10.2.5
9.5.1 1.1.3
) 4.2
4.4.1
) 1.1.2
8.7 8.7 6.1.3 1.3.3 1.3.3 1.2.3, 6.1.3 1.2.3 1.2.3 1.2.3, 1.3.3, 6.1.3
9.5.2
1.1.4 1.1.3 1.1.3 1.1.2 1.1.3 1.1.2 1.1.2 1.1.2 1.1.2 ( 1.1.2 1.1.2 1.1.2 1.1.2 9.5.1 ) 1.2.3
1.2.3
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