, . . , .
, , :
xi | x 1 | x 2 | xn | ||
pi | p 1 | p 2 | pn |
, , , , ,
, (xi, pi).
x 1 x 2 x 3 x 4 x 5
- :
() = 1 1 + 2 2 + + . (7.1)
, , .
1. , .
2. , -.
3. - () . , - .
.
1) :
() = . (7.2)
. , = 1, () = ·1 = .
2) :
() = (). (7.3)
.
xi | x 1 | x 2 | xn | |
pi | p 1 | p 2 | pn |
:
xi | x 1 | x 2 | xn | |
pi | p 1 | p 2 | pn |
() = 1 1 + 2 2 + + = ( 1 1 + 2 2 + + ) = ().
, , . .
Y XY, Y, - .
1) :
M (XY) = M (X) M (Y). (7.4)
|
|
. , Y :
xi | x 1 | x 2 |
pi | p 1 | p 2 |
i | 1 | 2 |
gi | g 1 | g 2 |
XY :
Y | x 1 y 1 | x 2 y 1 | x 1 y 2 | x 2 y 2 |
p | p 1 g 1 | p 2 g 1 | p 1 g 2 | p 2 g 2 |
, M (XY) = x 1 y 1· p 1 g 1 + x 2 y 1· p 2 g 1 + x 1 y 2· p 1 g 2 + x 2 y 2· p 2 g 2 = y 1 g 1(x 1 p 1 + x 2 p 2) + + y 2 g 2(x 1 p 1 + x 2 p 2) = (y 1 g 1 + y 2 g 2) (x 1 p 1 + x 2 p 2) = M (X)· M (Y).
1. .
2. 3 , .
Y + Y, Y; ( ).
4) ( -) :
M (X + Y) = M (X) + M (Y). (7.5)
.
, , - 3. X + Y 1 + 1, 1 + 2, 2 + 1, 2 + 2. 11, 12, 21 22. ( + Y) = (x 1 + y 1) p 11 + (x 1 + y 2) p 12 + (x 2 + y 1) p 21 + (x 2 + y 2) p 22 =
= x 1(p 11 + p 12) + x 2(p 21 + p 22) + y 1(p 11 + p 21) + y 2(p 12 + p 22).
, 11 + 22 = 1. , , , X + Y 1 + 1 1 + 2 11 + 22, , , = 1 ( 1). -, p 21 + p 22 = 2, p 11 + p 21 = g 1, p 12 + p 22 = g 2. ,
M (X + Y) = x 1 p 1 + x 2 p 2 + y 1 g 1 + y 2 g 2 = M (X) + M (Y).
. 4 , .