Foundations of the theory of probability

64 VI. Independence;TheLawofLargeNumbers

Then

s

n

—

E(s„)=z

x

+

z

2

+

• 
  
  
  
  • 
    
    
    
    • 4-
      z
      n,
    
    
    
    
  
  
  
  

E{z

nk)

=EE

3ll

9(

8

...

9lifc

(s

n)

—

EE

?

il9Il...9iJt

_

1

(s

n

)

=

E(s

n

)

• E(s

n

)

=

0.

aM^t)

=

E(4,)

=

^..

Wecaneasilycompute alsothattherandomvariablesz
nk

(k

—

1,2,

...

,

n) areuncorrelated.Forleti

<

k

;

then

5

E^

x9l 8

...

2U_i{

z

ni

z

nk)

=

*n»

E^i,

$(,

....«*_

i(

2

nit)

=

^nttE^M....3lt_

x

(s„)—E

9tl9i

t

...8ifc_ 1

(s

w)]

=

andtherefore

E(z

ni

z

nk)

=

0.

Wethushave

2

(S

H)

=0*(Z

ni)

+

0*(

Zn2

)

+

• ••
  
  
  • O^n)
  
  
  
  

=

ft,

+

#

2

+

• ••
  
  
  • fin
    
    
    • 
      
    
    
    
    
  
  
  
  

Therefore,thecondition

issufficientforthenormalstabilityofthe

variables

s

n

.

§

4. NotesontheConceptofMathematicalExpectation

Wehavedennedthemathematical expectationof arandom

variable xas

E{x)

=

fx?{dE)

=jadF&{a)

,

E

—<x>

wheretheintegralontherightisunderstoodas

+

oo

C

E(x)

=

fadF&{a)

=

lim

fa

dF&(a).

b""*

~

°°

(

1

)

-oo 6

'

Theideasuggestsitselftoconsidertheexpression

E*(x)

=

lim

f

adF&

{a) b

-*

+<x> (2)

-b

ApplicationofFormula (15) in§4,Chap.V.