7.

, . . . . .

. . . X , (X = ) = , ( = j) = qj (X = , = j) = , , X ,

. . X , (^1), (2) (^1

 

 

(*1 ,*2)

 

,

(X = , = ) =

0, =

(X = ), = , ,

I) = V = - V log2 .

^ ^

. . . X

H (X) = X = I ).

:

1) I(X, ) ^ 0, I(X, ) = 0 X ;

2) I(^)= I(^);

3) HX = 0 X ;

4) I(X, ) = HX + - (X, ), (X, ) = - £ j ^2 j;

5) I(X, ) ^ I(X, X). I(X, ) = I(X, X), X .

1) -1 ^ ( = 1) 1 ^ 1 21 ^ ^2 .

' (*) = V p,j log2 ^ - V ^!^ =

12 = 12 = 12 = ,

. . I(X, ) = 0 j = ,qj ], . . X . X , j = qj , , 1 , , 0, , I(^) = 0;

2) ;

3) HX = 0, , HX, , , X ;

4)

j

HX = £ , ^2 , = £ log2 , ,

y =^2 ^ log2 qj =^2 ^ log2 qj

j

X + - (X, ) = \ , (12 , - log2 , - log2 ) = /(X, );

5) I(X, ) = + - (,) ^ - (X, ) ^ 0.

- (X, ) = - > , 1og2 , + > , 1og2 = > 1og2 (, /, ),

^- = (X = 4, = ) ^ ^- = ( = ), 1 , , 0, , 0.

= I(X, X) = I(X, ), ^ , 0. 13 = (X = X, = ) = (X = X/ = )( = ) {(^, 0} (X = X/ = ) {0,1}, , X .

. . X , , . . I(X, ) = 0.

.

. . . XI, X2 . XI X2 , 1- 2- , = XI + X2. I(, XI), I(XI ), I(, ).

. . . XI X2 , . . .

XI 112 3 4 5 6

1/6 , .. = 1...6 (^ = (XI = ) = 1/6.

. . . ,

 

 

 

( = ) = (XI + X2 = ), = 2...12,

, XI, X2

(XI = = ) = (XI = ) = ),

 

 

 

 

1/36.

-\- = 1 < , < 6

-\- = 1 < , < 6

, :


 

 

2 \

1

 

1

2

3

4

5

6

 

 

 

 

 

 

1

 

 

2

3

4

5

6

7

 

 

 

 

 

 

2

 

 

3

4

5

6

7

8

 

 

 

 

 

 

3

 

 

4

5

6

7

8

9

 

 

 

 

 

 

4

 

 

5

6

7

8

9

10

 

 

 

 

 

 

5

 

 

6

7

8

9

10

11

 

 

 

 

 

 

6

 

 

7

8

9

10

11

12,

 

 

 

 

= + 2

2

3

4

 

5

 

6

7

8

9

10

11

12

V

/

/

/ /

/

~~5

/

/ /

5/ /

/

/

 

/3,

.. = 2...12, V* = ( = ) = (6 - |7 - |)/36. . . . X 1

^ = ( = , XI = 3) = ( = /1 = 3 ) (1 = 3),

,

( = 2, 1 = 1) = ( = 2/1 = 1) ( 1 = 1) =

= (2 = 1) (1 = 1) = 1/36.

 

= (

=

,1

= )

= 1 /36,

0,

1 ^ .

3 ^ 6,

\

1 \

2

3

4

5

6 7

89

10

11 12

1

1/

1/

1/

1/

1/ 1/

00

0

00

2

0

1/

1/

1/

1/ 1/

11/ 10

0

00

3

0

0

1/

1/

1/ 1/

1/ 1/

0

00

4

0

0

0

1/

1/ 1/

1/ 1/

1/

00

5

0

0

0

0

1/ 1/

1/ 1/

1/

11/ 10

6

0

0

0

0

0 1/

1/ 1/

1/

1/ 1/

I(,1 ) = ] ] ^ log2

3=1 1^-

1 ֠ ֠ 102 1

36

3 = 1 -

1

6v

11

12

= 36( £1082 6^+^1082 6V7+■ ■ ■+£1082 6V7+£1082 6^) =

=2 = = =7

= ^((102 +1<^2 | + +1<^2 6) + + (1<^2 ^7 +l0g2 5 + +log2 )) =

1 3 6 = (2]^2 + 4 log2 3 + 6 log2 2 + 8 log2 3 + 10 log2 6 + 6 ^2 1) = 3 - 2 5 -

= (2 + 2 log2 3 + 4 log2 3 + 6 + 8 log2 3 - 8 + 10 log2 3 + 10 - 10 log2 5)/36 = = (10 + 24 log2 3 - 10 log2 5)/36 0.69 /.

1(XI, 1) = 1(2,2) = - - = 1 l0g2 = l0g2 6 = 1 +l0g2 3

2. 58 /.

1 ( ) = - =2 l0g2 =

1 36

= (2 log2 36 +4 log2 18 + 6 log2 12 + 8 log2 9 + 10 log2 + 6 log2 6) = 36 5

= (4+4 log2 3+4+8 log2 3+12+6 log2 3+16 log2 3+20+20 log2 3-10 log2 5+ + 6 + 6 log2 3)/36 = (46 + 60 log2 3 - 10 log2 5)/36 3.27 /.

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

2log2 6 = 1 (1 ,1 )/18 1 (1,) 36, = 2 = 12, 1. , = 7, 1, 6 ^2 1 = 0.

, 4- , .

(,1) = -£,砠 l0g2 = log2 36 = 2(1 + log2 3) = 21

5. 17 /.

1 (,1) = 1 + - (1,) = - 1 3.27 - 2.58 = 0.69 /.

4- , , .

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

. . . .

I 1 2 3 4 5 6 I 0 1

~\ 1/6 | 1/2

, % = 1...6 = ( = ) = 1/6 , , = 0...1 = ( = ) = 1/2.

. . .

X

1 3

5 2 4 6

1 3 5

2

4

6

0 0

0 111

111

0

0

0

V

 

1/6

0

 

 

 

/0, + , , = ( = , = 3) = | ^ .

I(X, ) = <,5 V,- ^2 =66 l0g2 2 = 1 /.

I(, ) = - 3=0 4 log2 ^ =22 log2 2 = 1 /.

, . . 1 . I(X, ) = I(, ) = 1 / 3- , X , , .. I(X, ) = I(X, X) = 1 3 ~ 2.58 /. -

, X, X .

(X, ) = - ,3- ^2 = ^2 6=1 + log2 3 = X,

I(X, ) = X + - X = = 1 /.

► 5

. . . X,

XI 1234 56 78 V | 0.1 0.2 0.1 0.05 0.1 0.05 0.3 0.1.

► 6

. . . XI X2 , ... "", . XI ?

► 7

XI . . . Z = (XI + 1)2 X2, . . . XI X2 0, 1? Xl ^. XI Z ?

► 8

. . . XI, X2 .. . . I(XI, X2), XI X2

XI

0

0

1

1

 

0

1

0

1

V

1/3

1/6

1/6

1/3.

► 9

. . . XI X2 , 1 4. . . . , , . . = X1 + X2. I^, ), X1 .

► 10

XI . . . Z = XI *X2, 堠 . . . . XI X2 .

► 11

. . . XI 1, 0 1 . . . . X2 0, 1 2. X1 X2 . = X]2 + X2. I^, ), I^2, ), Xl, X2, .

► 12

. . . X, , Z , Z = X + . X

XI 0 13 4 I 2 2

V | 1/8 1/8 1/4 1/2 V | 3/8 5/8.