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............................................................................................... 6 ......................................................................................................... 9 I. ........... 11 1.1. .......................... 11 1.2. , ............................................................................................... 14 1.3. () ............................................................... 18 1.4. ................ 25 1.5. ................................ 26 1.6. . .................................................................................. 33 1.7. ............................................................... 35 1.8. ......................................................... 36 1.9. . ........ 38 1.10. () ....................................................................................... 40 1.11. , ....................... 42 II. ................ 44 2.1. ................................. 44 2.2. ............................ 45 2.2.1. .............................................................. 46 2.2.2. ............ 49 2.3. ....... 51 2.3.1. ......................... 51

2.3.2. .................................................................................... 55 III. ...... 58 3.1. ................................................................................................... 58 3.2. .......................................................................... 61 3.3. ....................................................................... 62 3.4. .................................................................................... 64 3.5. ........................................................ 66 IV. ............................... 72 4.1. . ................................................... 72 4.2. .................................... 77 4.3. ............................................. 81 4.4. ............................................. 85 4.5. .......................................... 88 4.6. ................................................... 91 4.7. ..................................................................................... 93 4.8. ................................................. 97 4.9. ............................................................................................... 99 4.10. .............................................................................. 100 . MATHCAD ............... 104 ............................................................................................. 115 1. .................................................................. 115
4

2. ................... 117 3. ........................................................ 120 4. ........................................................................................................ 122 5. - ................................................................................................. 129 6. ............................................................................. 131 7. .......................... 135 8. ...................................................... 137 9. ...................................................................... 138 ............................................................................................... 142 ................................................................... 145

5

- . , , , , , -, , . , . , , , , . , . . , , , . , .
6

. . - . . , . , . , . , , . , , - . , , , , MathCAD MathLAB. , , , MathCAD. , , , . , . , ,
7

, . , - . . ,

8

. , , . . , x(t ) , . .1. , . , X (t ) = {x(t )} . , . (, x(t ) ) , , , t1 , , X (t1 () X (t ) . , , -, , -, , , . . , . ,
9

)

X 1 . , X 1 = x(t1 ) , x(t )

, .

. .1. .

10

1.1.

I.
1.1.

,
. , . , , , , .

,
, . . , .

(

)

x 1 , x 2 , K , x .
n

P( x1 ), P( x2 ), K , P( x

n

)

. 1.1.1, . , . , .

11

I.

,

, . . , , (

X x

. 1.1.1. () () , .

)

P ,

x , ..

FX ( x ) = P( X x ) .*

(1.1.1)

FX ( x ) : 1) 2)

0 FX ( x ) 1 ,
FX (- ) = 0 ,

- < x < ,

FX ( ) = 1 ,

3) 4)

FX ( x ) x ,
P( x1 < X x2 ) = FX ( x2 ) - FX ( x1 ) .

*

: , , .

. 12

1.1.

,

,

. 1.1.1, . 1.1.2, .

). ( ,

f
X

(x )dx

= P( x < X x + dx ) . (2)
X

(1.1.2) . 1.1.2.

f

(

,

() x )dx () .

,

X x x + dx . f : 1) f 2)
X X

(x )

(x )

0, - < x < , = 1,

- x

f X (x )dx

3)

- x2

f X (u )du

= FX ( x ) ,

(1.1.3)

4)

x1

f X (x )dx

= P( x1 < X x2 ) .
13

I.

. 1.1.2, .

.

,

p(n ) ,

(n - 1 2) (n + 1 2) ( n = 0 , ± 1 , ± 2 , ...), . p(n ) , n .

f

X

(n)

= p(n ) .

(1.1.4)

1.2. ,

X , : X = E[ X ] =

-

xf (x )dx

.*

(1.2.1)

E[X

] ]

X.

E [ X X .

, ,

:

*

f ( x ) , ,

. 14

1.2. ,

1. , E [cX ] = c E [ X ]. 2. , E [ X + Y ] = E [X ] + E [Y ] . 3. ,

E [ X Y ] = E [ X ] E [Y ] .
(1) x :

E [g ( X )] =

-

g (x ) f (x )dx

.

(1.2.2)

g ( x ) = x n . :

X =EX

n

[ ]=
n

x n f ( x )dx .

(1.2.3)

-

E X

[]
n

X (

1-

n = 1 ),

(1), 2- ( n = 2 ), ,

X =EX

2

[ ]=
2

x 2 f ( x )dx .

(1.2.4)

-

, X X . , n - µ n

15

I.

µn = X - X

(

)

n

=E X-X

[(

)

n

]

+

=

-

(x

-X

)

2

f ( x )dx .

(1.2.5)

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

, D[ X ] Var[X ] . ,

µ2 =

2 X

= D[ X ] = Var[ X ] = X - X

(

) = (x
2
-

+

-X

)

2

f ( x )dx . (1.2.6)

-, :
E[X 1 + X 2 + K + X
m

]

= E[X 1 ] + E[X

2

]

+ K + E [ X m ].

,

2 X

= E (X - X =EX

[]
2

[

)2

- 2 E [ X ]X + ( X

] = E[X

2

- 2 XX + X

( )]
2

=

)

2

= X - 2 X X + (X
2

)

2

= X - (X ) .
2 2

(1.2.7)

, . , . : 1. , : D[cX ] = c 2 D[ X ] . 2. : D[ X + Y ] = D[ X ] + D[Y ]. 3. , : D[ X + c ] = D[x ].
16

1.2. ,

4.

D[ X Y ] = E X 2 E Y 2 - (E [ X
X

[][]

X .

Y

,

])2 (E[Y ])2

. . , , . . () . , . , . x p , P{X < x Mo[ X Me[ X

]

X
p

= 1 2 , ..

} = P{X

>x

p

}

, X x p x p , .

]

X

, . Mo[ X

]

()

f

X

(x )

. , Ra[ X

, .

]

max

, Ra[ X ] = x . x
p

-x

min

.

X
17

FX ( x)

I.

FX (x

p

)

= P.

(1.2.8)

x p X , P (X < x
p

)

= P . P , , ; , x0
,2

20%-

. 50%- .
1.3. ()

( ). . , . ()
f (x ) = x- X 1 exp - 2 2 2

(

)

2

, - < x < + ,

(1.3.1)

X , 2 . . 1.3.1, , . , . ,
18

1.3. ()

() . 0,607

f

X

(x )

2 X .

.

. 1.3.1. () () .

. , , :

F (x ) =

-

x

f (u ) du =

1 2

-

x

u-X exp - 2 2

(

)

2

du .

(1.3.3)

,

, , , (.. X = 0 ,

= 1 ). ( x ) :
1 ( x ) = 2

-

x

exp - u 2 2 du .

(

)

(1.3.4)
19

I.

Q( x ) , c ( x ) ,

Q( x ) = 1 - ( x ) .

(1.3.5)

( x ) Q( x ) .
erf ( x ) = ( x ) , erf ( x ) , erfc( x ) = Q( x ) , erfc( x ) . 2 2 erf ( x ) = exp - u du . 0
x

(1.3.6)

(

)

(1.3.7)

. , . 1.3.1, 1.3.2. , , , . .

X

Y , X = exp(Y ) ),

Y = ln X

(,

,

20

1.3. ()

. 1.3.2. x-X . (a) x - X ().

,

Y

2

Y Y . , X

f

X

(

ln x - Y 2 1 exp - , 2 x ) = 2 x 2 Y Y 0,

(

)

x 0, x < 0.

(1.3.8)

. . 1.3.3 . (8) :

f

X

(

1 x) = 2

(ln x - µ )2 exp - , 2 2 x

(1.3.9)

21

I.

. 1.3.3. Y = 1 , a) = 0 , 2 , ) 0 , 3 , ) 0 , 5 .

µ = ln

X
1+
2
X

, = ln 1 +

(

2 X

)

.

, ,

X = exp Y +

2 X

(

2
Y

2,

)

(1.3.10)
(1.3.11)

= [exp

()
2 Y

- 1] exp

2 Y . 2 Y + 2

. , .

22

1.3. ()

(. 1). . , ( ) . , , R = X2 +Y
X

Y ,

(

2 12

)

. X Y

2 , R r 2 exp - r 2 2 f R (r ) = 0,

(

)(

2

)

r 0, r < 0.

(1.3.12)

, 2 . 1.3.4. , , . , , R = rf R (r ) dr =
0

(r
0

2

2 exp - r 2 2 2 dr = ( 2

)(

)

)1

2

, (1.3.13)

23

I.

R = r f R (r ) dr =
2 2 0

(r
0

3

2 exp - r 2 2 2 dr = 2 2 .

)(

)

(1.3.14)

R
2
R

= R 2 - (R

)2 = (2

- 2 ) 2 = 0,429 2 .
2

(1.3.15)

(15)

, . , , , , 2 , .

. 1.3.4. .

, . ,
24

1.4.

r u 2 exp - u 2 2 2 du = 1 - exp - r 2 2 2 , r 0, FR (r ) = 0 0, r < 0.

(

)(

)

(

)

(1.3.16)

1.4.

. n , A , . , p (, q = 1 - p ). X
A . X ,

X . ,
A n , 1 ,

2 ,... n . , X : x1 = 0 , x2 = 1 , x3 = 2 ,... x
k Pn (k ) = Cn p k q n +1

= n .

, :
n-k

,

(1.4.1)

k = 0, 1, 2,K n ,

k Cn =

n! k!(n - k )!

n k . (1) . X = np , D[ X ] = npq . , , A , k , np ,
np = . ,
25

I.

, n , .

Pn (k ) = k e

-

k! .

(1.4.2)

(2) , ( n ) ( p ) . X = D[ X ] = . 2 3.
1.5.

, . . , - .
x1 < x x2 ,

f
X

(x )

1 ( x2 - x1 ), x1 < x x2 , = x x1 , x > x2 . 0,

(1.5.1)

, X = ( x1 + x
2
X

2

)

2, 12 .

(1.5.2) (1.5.3)

= ( x2 - x1

)2

, ,
26

1.5.

0, x x1 , FX ( x ) = ( x - x1 ) ( x2 - x1 ), x1 < x x2 , 1, x > x2 .

(1.5.4)

. , , . , , , 2 . ,

t,

x(t ) = cos( t - ) , , ,

1 2 , 0 < 2, f () = 0, > 2. 0,

(1.5.5)

= ,
2

= 2 3 .

.
, . , , t ,
, t , ,

. , .
27

I.

. 1.5.1. t0 ,

. 1.5.1. .

, t0 + t0 + + t ( ). F () , F ( + t ) - F () . , t , : « t0 t0 + » « t0 + t0 + + t ». 1 - F ( ) , t0 t0 + , t , t ,

F ( + t ) - F () = [1 - F ()](t ) .

(1.5.6)

t t ,
t 0

lim

[F (

+ t ) - F ( )] dF () 1 - F () = = . t d

(1.5.7)

28

1.5.

, F () = 1 - exp - , 0 . (1.5.8)

(

F (0 ) = 0 , .)
, 1 exp - 0, f () = 0, < 0. (1.5.9) (8), .

, . 1.5.2 .

. 1.5.2. . 29

I.

, .
:

( )

P( x ) = p x (1 - p
X = p,

)1-
2

x

, x = 0, 1, 2K ,

X

= p(1 - p ) .

P( x ) = C

m -1 x -1

p mq

x-m

, x = m, m + 1, m + 2K ,

m 1 , 0 < p < 1 , q = 1 - p , X = mp -1 ,
2
X

= mqp -2 .

: -

f

X

(

1 x u -1 x) = B(u , v ) (1 + x )u

+v

=

(u + v ) x (u )(v )

u -1

(1

+x

)-

u +v

,

x > 0, u > 0, v > 0, X= u , v > 1, v -1
2
X

=

u (u + v - 1) , v > 2. (v - 1)2 (v - 2)

f

X

(x )

=

2x a
3

2

2

e

-

x

2 2

2a

,

a > 0,
30

1.5.

X = 2a

2 ,

2 X

=

3 - 8 2 a.

f

X

(x )

=

a2 a + (x - b

[

)2

]

-1

, -< x<,

a > 0, - < b < . .

f
X

(x )

= an x

n -1

exp(- ax ) , x > 0, (n - 1)!

a > 0 , n = 1, 2 , ... , X = na -1 ,
-
2
X

= na -2 .

f

X

(

x) = x ( )

-1

e

- x

, x > 0,

> 0, > 0,
X= ,
2 X

=

. 2

f

X

(x )

a = exp(- a x - b ), - < x < , - < b < , a > 0 , 2 X = b,
2 X

= 2a -2 .

f

X

(x )

= abx

b -1

exp - ax b , x > 0 ,
31

(

)

I.

a > 0, b > 0,

X = (1 a ) 1 + b

(

-1

),

2 X

= (1 a

)2

b

(

(1

+ 2b

-1

) - [(1

+b

-1 2

)]

).

,

. 2 -
f

X

(

n x ) = 2 2

-1

-

n 2

x

n -1 2

x exp - , x > 0 2

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

= 2n .

t -
+ 1 2 2 1 + t f (t ) = 2
- +1 2

, -
, T = 0,
2
T

=

, > 2. -2

F - - f (x ) =
1 u , 2 v 2

B

u x

u 2

u -1 2

u 1 + x

-

u + 2

, x > 0,

u , , X=
, > 2; -2
2
X

=

2 2 (u + - 2 ) , > 4. 2 u ( - 2 ) ( - 4 )

32

1.6. .

1.6. .

, . , . , , . . , . , . ? , ( x = E [ X ]) . . ( , ) . µ 3 . - . , µ 3 3 , .

:

AS =

µ3 . 3

(1.6.1)

33

I.

,

«

»

, , . ( ): « » , (. 1.6.1, ), (. 1.6.1, ).

. 1.6.1. () () .

, .. , , .

,

E= µ
34

(

4

4 - 3 .

)

(1.6.2)

1.7.

µ 4 4 = 3 ; , . , : «»

, , (. 1.6.2, ), , «», (. 1.6.2, ). ,
. 1.6.2. () () . .

.
1.7.

X f
X

(x )

.

u , , (u )

e

jux

.

X .

35

I.
jux

(u ) = e

=

-

e

jux

f

X

(x )dx

.

(1.7.1)

j = - 1 .

f

X

(

1 x) = (u ) e 2 -

- jux

du .

(1.7.2)

, , u = 0 , , ± j , . ,

(0 ) =

(0 ) = j xf
-

-

f X (x )dx
X

= 1, = jX, (1.7.3)

(x )dx

M
X

(r )

(0)

=j

r

-

xr f

(x )dx

= jr X r.

, (3), .
1.8.

, . ? , .. . : X
36

1.8.

, , X . ( ,

, ..), . . X 1 , X 2 , ..., X
n

(
n

),
2 2

X 1 , X 2 , ..., X
2

1 , 2 , ...,

n . , Z , : 1 Z= n

i =1

n

Xi - Xi . i

(1.8.1)

n : 1 lim f Z ( z ) = e n 2
- z2 2

f Z (z

)

.

(1.8.2)

, p q , , i
2 i

> p , E Xi - X

3 i

. ,
37

I.

. , . 4.
1.9. .

. , . X Y . () X Y , X y , .. F ( x, y ) = P ( X x, Y y ) . (1.9.1) , x , Y ,

()

f ( x, y ) , F ( x, y ) .
F ( x, y

)

x

y,

. ,
2 F ( x, y ) , f ( x, y ) = x y

(1.9.2)

.
38

1.9. .

f ( x, y )dxdy = P( x < X x + dx, y < Y y + dy ) .

(1.9.3)

X , M . F (x M ) F (x M ) = P{X x M } = P{X x, M } , P (M ) > 0 P (M ) (1.9.4)

{X x, M } ,
, , X ( ) x M , X (

)

X , , . , , , , f (x M ) = dF (x M ) . dx (1.9.5)

M Y . , M

{Y

y} .

(4), , FX (x Y y ) = P[ X x , M ] F ( x , y ) = . P (M ) FY ( y ) (1.9.6)

M {y1 < Y y 2 } , (4) , FX (x y1 < Y y 2 ) = F ( x, y 2 ) - F ( x, y1 ) . FY ( y 2 ) - FY ( y1 ) (1.9.7)

39

I.

f

X

(

x Y = y) =

F (x Y = y ) f ( x, y ) = . x fY ( y )

(1.9.8)

X Y , fY ( y X = x ) = f ( x, y ) . f X (x ) (1.9.9)

(8) (9), . , f
X

(x y )

fY ( y x ) f (x y ) = f (y x) = f ( x, y ) , fY ( y ) f ( x, y ) , f X (x ) (1.9.10)

(1.9.11)

(10) (11) f ( x, y ) , f ( y x ) = f (x y

) fY ( y )
f
X

(x )

.

(1.9.12)

(12) , .
1.10. ()

() , . Z
X Y , , f
X Z

f

Y

(y)

. f

(x ) (z )

Z = X + Y
40

. 1.10.1.

1.10. ()

. 1.10.1. X + Y = Z z .

Z , FZ ( z ) = P(Z z ) = = P( X + Y z ) , , f ( x, y ) , x + y = z . y x ,

- < x < z - y . ,
z- y

FZ ( z ) =

- -

f (x, y )dxdy

.

(1.10.1)

X Y , . (1)

FZ ( z ) =

z- y - -

f X (x ) fY ( y )dxdy = fY ( y ) f X (x )dxdy
- -

z- y

.

(1.10.2)

Z = X + Y , FZ ( z ) z . ,

dF ( z ) = f Z (z ) = Z dz

-

fY ( y ) f X (

z - y )dy ,

(1.10.3)
41

I.

z . (3) , f Z ( z ) X Y . , FZ ( z ) (1),

FZ ( z ) =

z-x - -

f (x, y )dxdy

.

(1.10.4)

, (3),

FZ ( z ) =

-

f X (x ) fY (

z - x )dx .

(1.10.5)

, , (4) (5).
1.11. ,

.

( ) µ
X

XY

Y

:

µ

XY

= M {[ X - M ( X

)] [Y

- M (Y

)]}

= cov( X , Y ) .

(1.11.1)

, X Y (); , X
Y () .

: 1. , µ
XY

=µ

YX

.

2. ,

µ

XX

=

2

X

,µ

YY

= Y .

2

42

1.11. ,

3. () () , D[ X ± Y ] = D[ X ] + D[Y ] ± 2µ cov(cX , Y ) = c cov( X , Y ) = cov( X , cY ) . 5. , ( ) , cov( X + c, Y ) = cov( X , Y ) =
XY

.

4. ,

= cov( X , Y + c ) = cov( X + c, Y + c ) .
6. ,

µ

XY

X Y .
,

, X Y . , . . .

rXY
:

X

Y

rXY =
rXY

µ XY . X Y

(1.11.2)

,

, .. rXY 1 .

43

II.

II.
2.1.

, . ( , , , ). , . , , {A}

P( A) .

A

u( A) ,

U . U . , U = R + jI , u = r + ji ( j = - 1 ), U R I

FU (u ) = FRI (r , i ) = P{R r , I i},

(2.1.1)

R I

2 FRI (r , i ) f U (u ) = f RI (r , i ) = , ri
44

(2.1.2)

2.2.

n U1 , U 2 , ...,

U n , u1 = r1 + ji1 , u 2 = r2 + ji2 , ...,

FU (u ) = P{R1 r1 , R2 r2 , ..., Rn rn , I1 i1 , I 2 i2 , ..., I n in },

(2 1.3)

, , FU - n

u1 u u= 2 M u n

.

(2.1.4)

FU (u ) 2n {r1 , r2 , ..., rn , i1 , i2 , ..., i
2 n FU (u ) . pU (u ) = r1...rn i1...in
n

}
(2.1.5)

2.2.
, , , "" . , . . , ,
45

II.

, . , .
2.2.1.

N ,

k - a

k

N

k .

a (. 2.2.1):

1 a = ae = N
j

k
k=1

N

e

jk

.

(2.2.1)

. 2.2.1. .

, , , .
46

2.2.

1.

k

N

k

k

, . 2.

k

k

2 . 3. k (- , ) .

r i

r = Re a e

(

j

) )

=

1 N 1 N

k =1 N

N

k cos k ,

(2.2.2)

i = Im a e

(

j

=

k =1

k sin k .

, r i , ,
r

i

r i .

N .

r i :
1 r= N

k =1 N

N

1 k cos k = N
k sin k =

k =1 N

N

k cos k = N cos ,

(2.2.3)

i=

1 N

k =1

1 N

k =1

k sin k = N sin .

, k k k . , , 3,
47

II.

(- , ) , cos = sin = 0 ,

r =i = 0.

(2.2.4)

, , . r
2

2

r

i ,

2

i

2

( r = i = 0 ).

, 1 r= N
2

k =1 n =1 N N

N

N

k n cos k cos n ,

(2.2.5)

i2 =

1 N

k =1 n =1

k n sin k sin n .

,

0, k n, cos k cos n = sin k sin n = 1 2, k = n,
. ,

2 r =i = = 2 . 2
2 2

(2.2.6)

, , ( R I ) , , (. (1.8.1) (1.8.2)), .
48

2.2.

r 2 + i2 1 exp- f RI (r , i ) = f R (r ) f I (i ) = , 2 2 2 2

(2.2.7)

2 . = 2
2

(2.2.8)

2.2.2.

. a :
a = r 2 + i2 , i = arctg . r r = a cos , i = a sin ,
r J = a i a , J det r a i a r = cos - a sin = a . i sin a cos r i . = a cos , i = a sin )a,

(2.2.9)

(2.2.10)

(2.2.11)

, f .. f
A A

(a, )

=f

RI

(r

(2.2.12)

=f

RI

J . (12) (7)
49

II.

f

A

a a2 (a, ) = 2 2 exp- 2 2 0,

, - < 0 , a > 0, (2.2.13) .

(13)

() . ,
f A (a ) =
+ -

f

A

a a2 2 exp- 2 (a, )d = 2 0,

,

> 0, .

(2.2.14)

,

1.3. , , . I (8) (14), f (I ) = I
-1
0

I exp - . I 0

(2.2.15)

I 0 , I 0 = 2 . a= , 2
2

a

= 2 - 2 . 2

(2.2.16)

, (13) a .

50

2.3.

1 a a2 2 exp- 2 f () = 2 2 0 0,

da,

- < , .

(2.2.17)

, . , , (- , ) , , ..
1 , - < , f () = 2 0, . , f
A

(2.2.18)

(a, )

f A (a ) f () . , A .

2.3.
2.3.1.

. , s ( ). . 2.3.1 . 1 r=s+ N i= 1 N

k =1

N

k cos k ,

(2.3.1)

k =1

N

k sin k .

(2.3.2)
51

II.

. 2.3.1. .

, . N R I , , .. (r - s )2 + i 2 1 exp- f RI (r , i ) = . 2 2 2 2 (2.3.3)

, a . A , a (a cos - s )2 + (a sin )2 exp- , a > 0, - < , (2.3.4) f A (a, ) = 2 2 2 2 0, .
52

2.3.

f A (a ) =

A ,

-

a2 + s2 a as exp 2 cos d . exp - f A(a, )d = 2 2 2 2 -

2I 0 as 2 , ,

(

)

I

0

.

a a 2 + s 2 as I 0 , exp - a > 0, (2.3.5) f A (a ) = 2 2 2 2 0, . . . 2.3.2 f A (a ) a k = s .

. 2.3.2. A , ( s ) ( 2 ). k = s .

53

II.

( 2.3.2) , s . , (5).

a=

0

a 2 + s 2 as a2 I0 exp - 2 2 2 2

da

(2.3.6)

a 2 + s 2 as I 0 a = 2 exp - da . 2 2 2 0
2

a

3

(2.3.7)

: a= e 2
-k 2 4

k 2 k 2 k 2 k 2 I1 1 + I 0 + 2 4 2 4
a 2 = 2 2 + k 2 ,

,

(2.3.8)

[

]

(2.3.9)

I 0 I1 , . f () , f () = f
0

A

(a, )da

.

: k 2 sin 2 k cos f () = exp - + (k cos ) , 2 2 2 e
54
-k 2 2

(2.3.10)

2.3.
b

1 (b ) = e 2 - f (

- y2 2

dy .

(2.3.11)

)

k = s

. 2.3.3. k = 0 , k , , = 0 , .. , .

. 2.3.3. f ( ) . k = s .

2.3.2.

, . , s >> ( k >> 1 ). "",

55

II.

. 2.3.4. s " ".

(. 2.3.4). . a , , . , , (a - s )2 1 exp- f A (a ) , s >> . 2 2 2 s >> i tg , s f () a = s ,
2 a

(2.3.12)

(2.3.13)

k 22 k exp- . 2 2

2

(2.3.14)

= 2 , = 0 ,

= 1 k 2 = 2 s 2 .

, ,
56

2.3.

. - ( 5).

57

III.

III.
3.1.
. ( N ). , , . , , . . , , : . , x1 n1 , x2 n2 , xk nk

ni = n .

xi , , , . ni , ni n = Wi . , n , , , 1 x= n

i =1

n

xi ,

(3.1.1)

58

3.1.

xi . , xi X f
X

(x )

.

, - . , , .

) 1 X= n

i =1

n

Xi ,

(3.1.2)

X i f ( x ) , . - . , , X . , . , E
) 1 X = E n

i =1

n

1 Xi = n

i =1

n

1 E[X i ] = n

i =1

n

Xi = X .

(3.1.3)

, , .. . , , .
59

III.

, . ( ) . . , .. n << N . , . N = . ( ) X , ) D ) X , , , X :

D

) X =

E 1 n

( )
2n n i =1 j =1 n 2n i =1 j =1

X i X j - (X
i j

)2
2

=
(3.1.4)

=1

( n ) E[X X ]
X
j

- (X ) .

X

i

, i j . ,

E Xi X

[

j

]

X 2 , i = j, = 2 ( X ) , i j.

(4)

D

) X = (1 n

)2

[n

X 2 + n 2 - n (X

(

) ) ]- (X )
2

2

= X 2 - (X

(

)2

)

n = 2 n ,

(3.1.5)

60

3.2.

2 ( ). ) X , n D . , ,

, ) X n . D
3.2.

, S 2 . , S X 1 , X 2 , ..., X n ,
S = (1 n
2

2

,

)
i =1

n

)2 X - X = (1 n i

)
i =1

n

X i-(1 n

)
j =1

n

X j .

2

(3.2.1)

, , . (1) , 2 (n - 1) , ES = n
2

[]

(3.2.2)

2 . , . , s
2

61

III.
n

1 s = S n (n - 1) = n -1
2 2

i =1

)2 X - X . i

(3.2.3)

( ). (2) (3) . N ,
ES 2 N (n - 1) . n( N - 1)

[]
2

=

(3.2.4)

. ,

s 2 s 2 = S 2 n( N - 1) N (n - 1) .
(3.2.5)

, N (4) (5) (2) (3). , . D S 2 = µ4 -

[](

4

)
)
4

n,

(3.2.6)

µ4 = E X - X

[(

]
- 1) .
2

(3.2.7)

4- . (2) (3) D s2 = n µ4 -

[] (

4

)

(n

(3.2.8)

3.3.

. ,
62

3.3.

. , . , , , , , . , q 100% , q% - . , q . : X- k ) k k XX+ , n n (3.3.1)

, q ) ) ( x ) X . f
X

X + k

q = 100

X - k

f

) X

(x )dx

.

(3.3.2)

3.3.1.

q, % 90 95 99 99,9 99,99

k 1,64 1,96 2,58 3,29 3,89
63

III.

f

) X

(x )

, k q

3.3.1. . q , k , X , , n .
3.4.

,

,

, . (), - . . 3.4.1 d i , i = 1 , 2, ..., n , , X Y .

. 3.4.1. . 64

3.4.

.

d1 + d

2

2 2

+L+ d

2
n

= min .

(3.4.1)

, ,

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

y = a + bx + cx 2 + L + kx j .

(3.4.2)

, (n - 1) - , , , , 0. , , , . . ,
65

III.

. . y = a + bx , (3.4.3)

a b , (1). (1)

[
i =1

n

yi - (a + bxi

)]2

= min .

(3.4.4)

(4), a b .

i =1

n

yi = an + b

i =1

n

xi ,

i =1

n

xi y i = a

i =1

n

xi + bn

i =1

n

xi ,

2

, a b :
a=

yi
i =1 i =1

n

n

xi -

2

xi
i =1 i =1

n

n

yi

n

i =1

n

x -
2 i

i =1

n

xi
2

2

=

(3.4.5)

= (1 n

)
i =1

n

yi - (b n

)
i =1

n

xi ,
n

b = n

i =1

n

xi yi -

xi
i =1 i =1

n

yi

n

i =1

n

xi2

-

i =1

n

xi

2

.

(3.4.6)

(5) (6) , a b .

3.5.

, . , :
66

3.5.

1) ; 2) . , . ()

H0 .

() .

H1 ,

, . U Z , , F v 2 -, T , 2 «-». , , , , K . ( ) K , . , , K : F = s1
2

s2 .

2

(3.5.1)

, , -. () . K .

67

III.

. , , . , . ( ) , . , : ,
. 3.5.1. .

.

() k

,

. ( ) . , K > k , k K < k , k
68

(. 3.5.1, ).

,

(. 3.5.1, ).

3.5.

. , K < k1 , K > k 2 , k 2 > k1 (. 3.5.1, ). . . . k

,

, K P (K > k

, k ,

)

=.

(3.5.2)

, , , , . P (K < k

)

=.

(3.5.3)

, P (K > k

) + P(K

)

= .

(3.5.4)

. , , ,

69

III.

() 0 .

2

2 0

. , n S 2 k = n - 1 . , , ,
2

,

0 .

H0 : M S

()
2

= 0 .

2

(3.5.5)

, , . , , . (n - 1)S 2 . ,
2 2

0 . ,

2

k = n - 1 ,

2 = (n - 1)S

2

0 .
0

2

(3.5.6)
2

H

2 = 0 ,

( ) , , , , : P 2 >
70

[

2

(; k )]

= .

(3.5.7)

3.5.

2

(; k )

2 , . , 2 > 2 <
2

2

,

.

. , , 6.

71

IV.

IV.
4.1. .
. . , {x(t )} , x(t ) , , . X (t1 ) X 1 . . N . n1 , t
1

X (t ) .

x(t ) t1

,

x1 . N n1 ( x1, t1 ) , x 1 , , .. . , t = t1 X (t1 ) x 1 P{ X (t1 ) x1} . , n1 , , .. t1 x1 : F1 ( x1 , t1 ) = P{ X (t1 ) x1} .
72

(4.1.1)

4.1. .

F1 ( x1 , t1
x1 :

)

. F1 ( x1 , t1 ) = f1 ( x1 , t1 ) , x1

(4.1.2)

.

F1 ( x1, t1

)

f1 ( x1 , t1

)

. , . t1 t2 . N n2 , t1 x1 , t2 - x2 . , N n2 ( x1 , t1 , x2 , t
2

)

N

, t = t1 x1 t = t 2 x2 , , ..

. , t = t1 x1

t = t 2 - x2 .
P{ X (t1 ) x1 , X (t 2 ) x2 } x1 , x2 , t1 , t 2 : F2 ( x1 , x2 , t1 , t2 ) = P{ X (t1 ) x1, X (t2 ) x2 } , (4.1.3)
73

.

IV.

F2 ( x1 , x2 , t1 , t

2

)

2

2 F2 ( x1 , x2 , t1 , t x1x2

)

= f 2 ( x1 , x2 , t1 , t2 ) ,

(4.1.4)

. , X (t ) n t1 , t2 , ..., t n x1 , x2 , ..., xn :
P{X (t1 ) x1 , X (t 2 ) x2 , ..., X (t n ) xn } = = F ( x1 , x2 , ..., xn , t1 , t 2 , ..., t n ).

(4.1.5) n-

.

2n

F ( x1 , x2 , ..., xn , t1 , t 2 , ..., t

n

)
)

= f n ( x1 , x2 , ..., xn , t1 , t 2 , ..., t n ) , (4.1.6)

2 F ( x1 , x2 , ..., xn , t1 , t 2 , ..., t x1 x2 ... xn

n

n- . f n ( x1 , x2 , ..., xn , t1 , t 2 , ..., t
n

f1 ( x1 , t1 ) ,

f 2 ( x1 , x2 , t1 , t2 ) , ...,
,

)

. , . , X (t1 ) t1 ,
74

4.1. .

X (t1 ) X (t

2

)

(t

2

- t1 ) .

,

, . "" . . , , . , . , n -

X=

n

-

x f ( x ) dx = lim (1 2T
n T

)

T

-T

n X (t )

dt .

(4.1.7)

, . . . , ( ) F (

f (t ) .

)
(-

, ) ;

, F () = 0 . , 1 f (t ) = F ()e 2 - F (
it

d .

(4.1.8)

)

(-

,

)

, >> :
75

IV.
2 in 2

F () = 1 cn = F ()e 2 -

n = -

cn e

-

,

(4.1.9)

2 in 2

1 d = F ()e 2 -

2 in 2

d =

2 n f . 2

(4.1.10)

(9) 2 , F () (- , ) (± k, (k + 1) ) , k 1 . (9) (8) (10),

f (t ) =

n= -

sin n f t

n n - t-

, .

(4.1.11)

(11) , f 2 2 f (t ) - . 2 = 2

f (t ) =

n = -

n sin t - n f t - n

.

(4.1.12)

(12) .. , f (t ) , ,

, 2 2 , () f (t ) .
76

4.2.

. X (t ) , FX () < . (12) f (t ) F (t ) . , n xn = X , n = 0, ± 1, K (4.1.13)

4.2.
X (t

)

(, , , ), X (t ) , t . , . ( , ). , , . , ( ) . , , ; , , ,
77

IV.

. , , . , ?
) X

{x(t

)}

) 1T X = X (t ) dt . T0

(4.2.1)

) , X ,

, . , ) X . , , . ) , X X , ) X X , . (1) ) X

) 1 T 1T E X = E X (t )dt = E [ X (t )]dt = T 0 T0 = 1 1 X (t )dt = T X t 0 = X . T0
T T

[]

(4.2.2)

) (2) , X , ) . X

78

4.2.

, . . ( , .) (1) , X (t ) . X (t ) , . , X 1 = X (t ) ,

X 2 = X (2t ) , ..., X N = X ( Nt ) , X
)1 X= N

i =1

N

Xi ,

(4.2.3)

(1). ) X - ) 1 E X = E N = 1 N
N i =1

[]

i =1

N

Xi = = 1 N

E[X i ]

i =1

N

(4.2.4) X = X.

, , . ) X ,

) . X

79

IV.

) X 2 = E 1 E 2 N = 1 N2
N

()

i =1 j =1 N

N

N

Xi X j =
i

E [X
i =1 j =1

Xj ,

]

(4.2.5)

. , E Xi X ,
)2 E X = 1 N

[

j

]

X 2 , i = j, = 2 X , i j.

()

() (

2

)[N

X 2 + N2 - N X

(

)( )

2

].

(4.2.6)

, (5) N 2 , N i = j . (6)
) X 2 = (1 N ) X 2 + [1 - (1 N )] X E

()

()

2

=

= (1 N )

2
X

+X .

()

(4.2.7)

2

) X :

) )2 ) D X = E X - E X
= (1 N )
2
X

()

( ) { [ ]}
+ (X

2

=
2
X

)2 - (X )2 = (1 N )

(4.2.8) .

, N . .

80

4.3.

4.3.
. . , ( ) , . . X (t ) , X 1 = X (t1 ) , X 2 = X (t 2 ) , , , R X (t1 , t 2 ) = E [ X 1 X f ( x1 , x
2

]=

-

dx1 x1 x2 f ( x1 , x2 ) dx2 ,
-

(4.3.1)

2

)

.

, . (1) . .. ,

R X (t1 , t 2 ) = RX (t1 + T , t 2 + T ) = E [ X 1 (t1 + T ) X 2 (t 2 + T )] .

(4.3.2)

, T = -t1 ,
81

IV.

RX (t1 , t2 ) = RX (0, t 2 - t1 ) = E [X (0) X (t 2 - t1 )] .

(4.3.3)

, t 2 - t1 . = t 2 - t1

RX (0, t 2 - t1 ) , (1)
R X ( ) = E [ X (t1 ) X (t1 + )] . (4.3.4)

. t1 , (4) R X () = E [ X (t ) X (t + )] . x(t ) , 1 x () = lim T 2T
T

-T

x(t )x(t + )dt
)

= x(t )x(t + ) .

(4.3.5) .

, x(t ) x(t +

x(t ) R x ( ) , ..

x () = Rx () .

(4.3.6)

, , . (4) , =0 R X (0) = E [ X (t1 ) X (t1

)]

. 0 R X () X (t ) X (t + ) .

82

4.3.

X (t ) Y (t ) = X (t ) - X (t + ) . (4.3.7)

, Y (t ) . ( X (t + ) X (t ) .) Y (t ) , : E [Y (t

[

)]2

] = E[[
[

2
Y

X (t ) - X (t +

= E X 2 (t ) - 2X =
2

)]2 ] = (t )X (t + )

+ 2 X 2 (t + ) ,
2
X

]

(4.3.8)

2
X

- 2R X () + 2
2 X

,

d Y = -2 R X () + 2 2 d

= 0,

=

R X (
2
X

)

.

, R X () . . , X (t ) () X (t ) , . + 1 - 1 . = 1 , X (t x (t

)

)

, .. .

= 0 , .. - X (t + ) , X (t ) . = -1
83

IV.

, : X (t + ) X (t ) .
2
X

R X (

)

X (t ) , R X (

)

. , , X (t1 ) X (t1 + ) , , , . , . , , . . :

D X () = ( x - x

)2

=
2

= x 2 - 2 xx + x

=

= 2 - 2 R X ( ) + 2 = 2 2 - R X ( ) . , , . , R X (0 ) = 2 , (4.3.10)

(

)

(4.3.9)

D X ( ) = 2 2 .

(4.3.11)

84

4.4.

. 4.3.1.

. 4.3.1. .

4.4.
, , . . 1. R X (0 ) = X 2 . , X (t

)

,

= 0 .

85

IV.

. X , . 2. R X ( ) = R X (- ) . . , , , . . 3. R X () R X (0 ) .

, , = 0 . , (, X (t ) ), R X () R X (0 ) . 4. X (t

)

, RX () . , X (t ) = A ,

R X () = E[ X (t1 ) X (t1 + )] = E[ AA] = A2 .

(4.4.1)

, X (t ) X N (t

)

, X (t ) = X + N (t ) , R X () = E X + N (t1 ) X + N (t1 + ) =
=E X

[[

][

]]

[(

)

2

+ X N (t1 ) + X N (t1 + ) + N (t1 )N (t1 + ) = =X

]

(4.4.2)

()

2

+ RN ( ) ,

86

4.4.

E[N (t1 )] = E [N (t1 + )] = 0 . , R X (

)

,

X ,

()

2

X (t ) .

, , . X , . 5. X (t

)

, RX (

)

. , . x(t ) X (t

)

, . 6. X (t ) , ,

lim R X ( ) = 0 .

(4.4.3)

, , , , X (t ) X (t + ) . 7. .
87

IV.

[R X ( )] = [R X ()] 0 .

-

RX () exp[-

jt ] dt .

(4.4.4)

,

. 4.5. , , . , . , (4.3.1), f ( x1 , x X (t
2

)

)

RX (t1 , t2 ) , . . , , .

4.5.
, . , , . , .
88

4.5.

, . , - , X (t ) , 0 T x(t ) . () :

) R X () =

1 T -

T -

x(t )x(t + )dt
0

0 << T .

(4.5.1)

x(t

)

. , T - , T , ( ) , x(t ) , x(t + ) . (1), , , x(t ) . , , .. . , t . , x(t ) X (t ) 0, t , 2t , Nt x(t ) x0 , x1 , x2 , ..., x N , (1)
) R X (nt ) =
1 N - n +1
N -n

k =0

Xk X

k +n

n = 0 , 1 , 2 , K , M M << N . (4.5.2)

() x0 , x1 , x2 , ..., x
N

89

IV.

) R X (nt ) . N (

), (2) . , (2), ) R X (nt ) , , . ,
) 1 E R X (nt ) = E N - n +1 1 = N - n +1 = 1 N - n +1

[

]

N -n

k =0

Xk X Xk X

k +n

= = (4.5.3)

N -n k =0

E[
k =0

k +n

]

N -n

R X (nt )

=R X (nt ).

, . , . , , ) 2 D R X (nt ) N

[

]

k =- M

M

R

2 X

(kt )

.

(4.5.4)

, () ,

2M + 1

,

, .

(

M + 1)t

(4), . ,
90

4.6.

) 2 D R X (nt ) T

[

]

-

R

2 X

()d

,

(4.5.7)

T = Nt ().

4.6.
, . X (t

)

Y (t

)

, X 1 = X (t1 ) , Y2 = Y (t1 +

)

R
XY

()

= E [ X 1Y2 ] = dx1 x1 y 2 f ( x1 , y 2 ) dy 2 .
- -

(4.6.1)

, . Y1 = Y (t1 ) , X 2 = X (t1 + ) RYX () = E [Y1 X
2

]=

-

dy1 y1 x2 f ( y1 , x2 )dx2 .
-

(4.6.2)

X (t

)

Y (t

)

, . , , . x(t ) y (t ) X (t ) Y (t ) , ,
91

IV.
T

1 xy () = lim T 2T 1 yx ( ) = lim T 2T

-T T

x(t ) y(t + )

dt ,

(4.6.3)

-T

y(t )x(t + )

dt .

(4.6.4)

, (3) (4) . ,
xy

()
()

=R

XY

()

,

(4.6.5) (4.6.6)

yx

= RYX () .

. 1. X = X (t . 2. . , , RYX () = R
XY

R

XY

(0)

RYX (0

)

)

Y = Y (t ) . , R

XY

(0)

= RYX (0

)

(- )

.

(4.6.7)

, Y (t ) X (t ) . 3. = 0 . , ,
R
XY

() [RX (0)RY (0)]1

/2

.

(4.6.8)

92

4.7.

RYX ( ) . , (8). , . . 7 8.

4.7.
. , , , , . , , , . X k (t
( , , X Tk ) (t

)

X (t ) .

)

,

t

T X (k ) (t ) . 2

( ( ) X Tk ) (t )

Z

T

(k )

()

T2

=

-T 2

( X Tk ) (t )e

- i t

dt .

(4.7.1)

, f = 1 T ,

93

IV.

2( ( GTk ) () = Z Tk ) ( T 2 = T
T2 T2

)

2

=

-T 2 -T 2

X

T

(k )

(t1 )

X

T

(k )

(t 2 )

e

-i(t1 -t

2

)

(4.7.2) dt1dt 2 .

T GT () , , , . GT () FT () = m1{GT ()} =
T2 T2

= 2 /T

-T 2 -T 2

m1 {X T (t1 )X T (t

2

)}e

-i(t1 -t

2

)

dt1dt 2 .

(4.7.3)

m1 . R(t1 , t 2 FT () = T
T2 T2 2

)

X (t ) , FT ()

-T 2 -T 2

R(t1 , t2 )

e

-i(t1 -t

2

)

dt1dt 2 .

(4.7.4)

X (t ) ,

R(t1 , t2 ) = R(t 2 - t1 ) , ,
2 FT () = T
T2 T2

-T 2 -T 2

R(t

2

- t1 )e

-i(t1 -t

2

)

dt1dt 2 .

(4.7.4)

= t1 - t 2 , FT () = 2 1 - R( )e T -T FT () T
T -i

d .

(4.7.5)

94

4.7.

F () = lim FT () = 2 R()e
T -

-i

d .

(4.7.6)

F () , .. T , . , . (6) , ): F () = 2 R()e
- -i

F ()

R(

)

( -

d = 4 R( )cos d ,
0

(4.7.7)

1 R( ) = F ()e 4 - GT ()

i

1 d = F ()cos d . 2 0

(4.7.8)

FT () = m1{GT ()}

,

F () . , (7), F () . , (8) = - = + . 0. ,

1 2 F () 1 2 F (- ) , - F () ,
95

IV.

.

,

, (7) (8) ,

-

R()

d M ,

-

F ()

d N .

(4.7.9)

M

N

.

- , . , F () . (8) , = 0 1 R(0 ) = 4

-

F ()d

,

(4.7.10)

.. . = 0 F (0 ) = 2 R( ) d
-

(4.7.11)

. R () , F ( 0 ) 0 = F (0 ) 4 R(0 ) . (4.7.12)

, 0 ,
96

4.8.

1 = 2F (0

)

0

F ()d =

R(0) . F (0 )

(4.7.13)

, . R(

)

F ()

, , . , F () , R( ) , . , , , .

4.8.
E1 , E2 , K ( ). , Ei , , E j , , . Ei E
j

Ei , , , X (t ) . , t0 X (t ( X

) (t0 ) )

( t > t 0 ) ( X (t ) t < t 0 ) , X (t )

.
97

IV.

F2 ( y, t x0 , t 0 ) , , X (t ) < y , t0 < t X (t0 ) = x0 . , x 0 t0 , t > t0 , t0 . f 2 ( y , t x0 , t 0 ) = F2 ( y, t x0 , t y
0

)

(4.8.1)

. f 2 ( y , t x 0 , t 0 ) = f 2 ( y , x0 ) , = t - t 0 . . . . , 1827 . , . 1905 . . . 1918 ., . , . . , (4.8.2)

. X (t ) , (), X (t1 ) , X (t 2 ) , ..., X (t
98
k

)

4.9.

k ti ( i = 1, 2, ..., k ). : . X (t ,
k

)

k

, ..
k -1

X (t 2 ) - X (t1 ) ,

X (t3 ) - X (t 2 ) , ..., X (t

)

- X (t

)

t1 < t 2 < ... < t

.

X (t ) - X (s ) t s . X (t ) - X (s ) t - s , . M X (t ( X (t X ) M [ X (t )] = 0 ,

,

[

)2

]

) (0)

= 0,

2 = t t > 0 .

, .

4.9.
, . F () . R() , .. . F () = 2 N 0 = const , (4.9.1)

, .
99

IV.

, , « ». N R( ) = 0 2

-

e i d = N 0 ( ) ,

(4.9.2)

.. - . 1, = 0, ( ) = 0, 0. (4.9.3)

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

4.10.
- . , x(t ) L : ) Lx(t ) = y (t ) .
100

)

y(t

)

,

(4.10.1)

4.10.

) , L ,

. y (t ) =

-

h(t - t )x(t )dt

.

(4.10.2)

) h(t - t ) = L{(t - t )} - . - H () . Y () X () Y () = H ()X () . (4.10.3)

Fx () Rx ( ) - Fx () = 2 Rx ( )e
- -i

Rx () ,

(4.3.5).

d .

(4.10.4)

.

, R y () = y (t ) y (t + ) . (5) y (t R y () = h (t - )Rx (t ) dt ,
-

(4.10.5)

)

y (t +

)

(2),

(4.10.6)

101

IV.

h (t - ) = h(t - + )h( ) d .
-

(4.10.7)

(6) .
Fy () = H () Fx () .
2

(4.10.8)

R y (0 ) = y 2 (t ) . (5) (7) ,

(4.10.9)

y

2

=

-

1 h (t )Rx (t )dt = 2

-

H () Fx ()

d .

(4.10.10)

, H () (8) (10), . .

Rxy = x(t ) y (t + ) .
R
xy

(4.10.11)

()

= h( - t )Rx (t )dt .
-

(4.10.12)

,
102

4.10.

Fxy () = H ()Fx () ,

(4.10.13)

H () . y (t (12): x(t ) y (t ) = R

xy

)

x(t

)

,

(0)

= h(t )Rx (t ) dt =
-

(4.10.14) d .

=

-

H ()Fx ()

E = [ y (t ) - x(t

)]2

.

(4.10.15)

. . , , . , , . , , , . , , . 9.
103

. MathCAD

. MathCAD
, , . . , MathCAD, MATLAB, STATISTICA SPSS. MathCAD.

MathCAD

, , , . X , , rnorm(n, X , ) , n , X , . . .1 xi n = 1024 , X = 1 , = 0,5 . , n (. 3.1).

(i

= 1, 2, ..., n

)

104

. .1

,

( 4.1). x , i .

. .1. .

dnorm(x, X , ) . . .2.

. .2. . X = 1 = 0 , 5 ; X = 0 , = 0 , 2 .

FX ( x ) pnorm(x, X , ) . . .3.
105

. MathCAD

. .3. . X = 1 = 0 , 5 ; X = 0 , = 0 , 2 .

x

p

X qnorm( p, X , ) . , 98%
X = 1,
= 0,5

qnorm( p, X , ) = 2,027 . , ,

rlnorm(n, X , ) . . .4 n = 1024 , X = 1 , = 0,5 .

. .4. .

106

rlnorm(x, X , ) .

. .5.

. .5. . X = 1 = 0 , 8 ; X = 0 , = 0 , 4 .

plnorm(x, X , ) . . .6.

. .6. . X = 1 = 0 , 5 ; X = 0 , = 0 , 2 .

qlnorm( p, X ,

)

. , MathCAD .
107

. MathCAD

, ( ) .

d * ( x, par ) ;
p * ( x, par ) ; q * ( x, par ) ; r * (M , par ) M ,
;

x ( ); p ; par .
, , , ,
*

unif

par .

a b .
( )

,
, . , hist (int, x ) ( int , x ) , . .7, 50 . MathCAD : skew ( x ) x ; kurt ( x ) x .
108

,

. .5,
1,14, 2,08; skew ( x ) = 3,34 kurt ( x ) = 20,08 .

. .7. .

.

, R . . ,

x

(,

N = 200 ,

= 2 , ). y

y = xR + 1 - R 2 rnorm( N ,0, ) .

(.1)

109

. MathCAD

x y R . .8.

. .8. . R = 0 , 5 , R = 0,9 . 110

MathCAD , ( covar( x, y ) ) ( corr( x, y ) ). 1 covar( x, y ) = N

(
i =0

N -1

x - x )( y - y ) ,

(.2)

corr ( x, y ) =

covar( x, y ) . x y

(.3)

x y ( R = 0,5 ) : covar( x, y ) = 1,928 , corr( x, y ) = 0,493 .

MathCAD . . , -

,

. , , , , . :

111

. MathCAD

k =

41- b
n

[

2 0 1 - b
(

[

2 D-4

1 2 D - 4 )( N +1 2 n =0

] ) ]
1 2N

b(

D - 2 )n

cos 2sb n k + n ,

(

)

(.4)

n - , N , 0

, b , s , k . D . , , (4) . ( ,

N = 1 ).

10 . , . ,
(0 Akm) = e -( k - m

)2
w

2

.

(.5)

w , m , . -
Akm = e
- i - (k - m
k

)2
w

e

2

.

(.6)

.
2

(6).
I
km

=A

k m .

112

m " " k

cm

k

cm

=

k =0 K

K

I km k . I
km

k =0

(.7)

K . ( m ). . .9, -, , w 20 ( , D = 1,5 ). , . . .9,
k

.
km

. .9, I

m , . . .9,

I

cm

m .

, , , , 50 . ( ).

113

. MathCAD

. , , .

a

. .9. () () . , , . w = 20 . ( , .)

114

1.

1.
, , , . , , , . , . . . 560 , 25 , 0 , 63 . . , - , . , . (. .1.1, ), (. .1.1, ). .
115

. .1.1. () () .

, , , . .1.2. ,

. .1.2. ( t < t 0 ) ( t > t 0 ) . 116

2.

( t < t 0 ) , ( . .1 .4).

2.
, . , .

, (. .2.1).

. .2.1. .

A = L2 N , m , n
117

(). , a = l 2 , , , «» , .. p = m N = T , T . A
A L2 N = = 2 = m + n. al

PN (m ) , N m . , N ,

(

N - 1)

(

N - m + 1) m - .

m N N ( N - 1)( N - 2 )K( N - m + 1) =

(

N! , N - m)!

m , m
N Cm =

N! . m ! (N - m) !

m p ,

(

N -m

)

(1

- p ) . ,

, (1.4.1), p =T . N p , , m = pN , ,

118

2.

PN m = pN .

(m )m (m )

e m!

-m

,

, , (. .2.2). ,

. .2.2. .

, n A Pn

(n )n ( A) =

e n!

-n

,

d = n A .

119

3.
, ()

. : 1) , 2) , , 3) . . K , , . . -, , , , , ( ), , . t A

P(1; t , A) = tAI ( x, y; t ) ,

(.3.1)

, I ( x, y; t ) t ( x, y ) . -, , , . (, .)
120

-,

,

,

3.

, . ( «».) , (. 1.4). - , , - , , . (1.4.2) K (t , t + ) P

(K )K (K ) =
K!

e

-K

,

(.3.2)

K , K =

I (
At

t +

x, y; )ddxdy

(.3.3)

( A ). , , W . W=

I (
At

t +

x, y; )ddxdy .

(.3.4)

, , , , . , I
0

( ,

), W
121

W = I 0 A .

(.3.5)

K : P

(K ) = (W )
K!

K

e

- W

.

(.3.6)

, . W , , h , , K = W = h W , h
-34

(.3.7) ),

( 6,626196 10

, ( , , 1). , = . h (.3.8)

4.
. . .

.
«», .
122

4.

()

m () ( m = 1,2,K, n ) :

()

= 1 () 2 ()K n () .

(.4.1)

n , m () ,

()

. , - () , .

.
. , a , b . ( x
2

)

x

2

, ( x1 : ( x2 ) = ( x1 )K ( x2 - x1 )dx1 ,
-a a

)

(.4.2)

:
K ( x ) = (b

)-

1

2

exp - ix 2 b

(

)

(.4.3)

, , . (2) , 2 K ( x ) ( x1 ) , x1 > a . , n -
123

n ( ). , , , . , . , . , . . a b , (2)

, , x1 x1 = ± a , ( x1 ) . n n (3) , , . . .4.1 n = 1 n = 300 . .

.

, n . « ». , , ,

124

4.

. .4.1. n = 1 (1, ) n = 300 (2, ). 3 .

, , , . , .
125

m :

= 1 + 2 + K + n ,

(.4.4)

m = nm hm , nm m - hm . (, , ) ,

, ,

. , f (

)

. f (

, '

)

.
m

(4) .

. m ( m = 1, 2 , K , n ) , , , .

.
. , . N ,
126

4.

. .4.2.
(t ) =

n =1

N

an cos(n t +

n

)

=

n =1

N

an cos n .

(.4.5)

. .4.2. .

, an n , n :

f (

n

)

= 2 , - n ,

(.4.6)

: f (1 , 2 ,K
N

)

=

f ( n )
n =1

N

.

(.4.7)

, N . f ( ) (v ) = e
iv

.
127

(5)
(v ) = expiv = (va

n =1

N

an cos n =
n

n =1

N

exp{ivan cos

}

=

(van ).
n =1

N

n

)

: (an ) = exp{ian cos
n

}

=
n

1 = exp{ian cos(nt + 2 - J 0 ( x

)}d

n = J 0 (an ),

)

. , (v ) = : 1 f ( ) = 2 , N >> 1 f ( ) = 2 1 exp- 2 2 2 2 Na 2 . , = 2

J 0 (van )
n =1

N

.

(.4.8)

f (

)

(8) -

-

(v )

e

-iv

dv .

, .

128

5. -

5. -
, , . ( ), , ,
.

.

.

. .5.1.
. .5.1. - .

- , . , , . , , . (. .5.2) (), .

129

, .

. .5.2. - .

, . - , , . ,

. , , . ,

. , , ,
130

6.

. .
I

. ? , 2.2 . ,

(2.2.15). , . C ,
C= I = 1. I

, .

6.
, .
131

. . , , . , ,
.

, . , , ; . . . , , . , , . . .
. ,

. . , n
132

6.

, . . . . , () , . , , . . , . , , . , , «» , . ( ) - . N , . , , . N n (,
N < 10 n ), N

.

133

. , , , , , . , , , , ... , , . , , , , . n . n n , , ) X D 2 ) . (. (3.1.5)) n = 2 D X .

[]

, . , , . , . , . ,
134

7.

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

: 1. . 2. , . 3. .

7.
, , . , . .
135

p (t

)

T b(t ) .
x(t ) = p(t ) + b(t ) .

(.7.1)

, p(t ,

)

b(t ) , ..
x(t

)

. , , , «» , ( ) .
xx

()
xx

1 = lim T T

T

[ p(t ) + b(t )][ p(t - ) + b(t - )]dt
0

.

(.7.2)

xx

()

=

pp

()

+

bb

()

+

pb

()

+

bp

()

.

(.7.3)

, b(t ) p(t ) .
pb

()

bp

()

(

, ).
bb

()

( 4.4); , (С 1 )
bb

()

. 1 ,

bb

()

, , , : , .
136

8.

,

> 1

:

xx

()

=

pp

()

.

(.7.4)

xx

()

=

pp

()

+ () ,

(.7.5)

( ) , T А . , .

8.
, , x1
x2 .

( x1 , x2 ; ) . , , , :
V = a( ) cos[( ) - 2t ]d .
r
0

, « »:
V (t ) = V (r ) (t ) + iV (i ) (t ) ,

V
(i )

(t )

= a( ) sin[( ) - 2t ]d .
0

137

V (r ) (t ) V (i ) (t ) . :
(x1 , x 2 ; ) = 12 () = V (x1 , t + )V * (x 2 , t ) .

,

12 (0

)

.

, ,
11 (

.

)

.

, , , , . , .

9.
, (, i ( x, y ).
138

)

)

( {, }

{x, y}

, , « »
s ( x - , y - ) ,

9.

(

x, y ) ,

(, ) . , , . ,
i ( x, y ) =

- -

s(

x - , y - )(, )dd .

(.9.1)

r () =

- -

s ( x, y )
- -

e

i x x + y y

(

)

dxdy . (.9.2)

s(x, y )dxdy

x y x y . r I () r I () = I ( x , y ) =

- -

i ( x, y ) e

rr - i r

dxdy .

(.9.3)

r O() . (1), r r r I () = ()O() , (.9.4)
139

(4.10.3) . , 4.10. . -, . -, , . , .

, ,

Rx ( ) = x(t )x(t +

r rr Ro ( ) = o( )o +

)
)
d

R y () = y (t ) y (t +

r Ri = i ( )i r Foi () = 2 Ri = 2

()

( r (

+

) )
r dr

Fxy () = 2 R
-

xy

()

e

- i r

- -

r Roi (r )e

rr - i r

R y () =

-

h (t - )Rx (t )dt
2

r ( ) (
s - -

rr - Ro ( )d

)

Fy () = H () Fx (

)

r r2 r Fi () = () Fo ( Roi =

)

R

xy

()

= h(t + )Rx dt
-

( ) s(
- -

r rr + Ro ( )d

)

,
140

9.

, , .

r s -

(

)

r s - =

(

r ) s(
- -

- + o d .

)( )

r , , ,
.

141

1. . - . .: , 1964, 772 . 2. .. . .: , 1966, 404 . 3. ' . . .: , 1966, 254 . 4. .. . .: , - , 1968, 288 . 5. 6. . . .: , 1970, 296 . .. . .: , - , 1970, 392 . 7. ., . . 1, 2. .: , 1971, 317 . 8. .. . . .: , 1974, 552 . 9. .. . .: , 1975, 128 . 10. .. . 1. . .: , , 1976, 496 .
142

11.

., ., ., ., . . . . / . .. .: , 1976, 335 .

12.

..,

..,

..

. 2. . .: , - , 1978, 464 . 13. .., .. , ..

. .: , - , 1981, 640 . 14. ., ., ., ., ., . . . . / . . .: , 1983, 488 . 15. . : . . 1. .: , 1983, 312 . 16. 17. . . .: , 1988, 528 . ., . . .: , 1989, 376 . 18. .. . .: , 1998, 576 . 19. .. . .: , 2000, 479 . 20. .. . .: , 2001, 295 .

143

21.

.. . .: , 2004, 176 .

22.

.. , 2008, 288 .

,

. .: -

144

. . , 137 , 9 , 64 , 33 , 16 , 60 , 61 , 61 , 62 , 63

, 100 , 98 , 25

, 58 , 58 , 72 , 96 , 58 , 58 , 9

( ), 11 , 26

, 121 , 63

, 17 , 122 . . , 138 , 138 , 131 , 42 , 42 , 43, 83 145

, 58 , 67 , 67 , 67 , 67 , 66 , 14, 108

, 65 . , 67 , 68 , 68 , 68 , 68 , 69 , 68 , 68 , 58 , 134 , 132 , 132 , 134 , 132 , 133 , 134 , 63 , 59 , 61

, 66 , 64

, 14 , 17 , 65 , 17

, 67 , 15 1- , 15 2- , 15 , 42 , 59

, 63

, 101 , 13, 73 n-, 74 , 50 , 38, 74 , 50 (), 18 , 53

. . , 68 , 69 146

, 38 , 39 , 99 , 99 , 97

(), 98 , 99 , 72 , 99 , 74 , 75 , 75

, 31 , 64 , 58

, 11

, 17 F- -, 32 t- , 32 , 30 -, 30 , 25 , 31 -, 31 , 18 , 31 , 31 , 20 , 30 , 18 , 30 , 17 , 25 , 26 , 23 , 17 -- , 32 , 27

, 11 , 11 , 11 -, 57, 129 , 95 , 58 , 60 , 59 , 14 , 17 , 15 . . , 66 , 67 , 84 , 46

-, 95 , 75, 76 , 63 147

, 64

, 35 , 38 , 38

, 45 , 40 , 25 , 120 , 81, 82 , 81, 91 , 81, 89 , 101 . . , 12 , 73, 74 , 53 , 84

, 36 , 15

, 58 , 58 , 120

, 64 , 96

, 34 , 95

148