Foundations of the theory of probability

ChapterVI

INDEPENDENCE; THE LAW OF LARGE NUMBERS

§

1. Independence

Definition 1 :Twofunctions,uandvof

|,

aremutuallyinde-

pendentifforanytwosets,Aof
$

(w)

,

andBof

%

(v)

,thefollow-

ing equationholds:

P(ucA,vczB)=P{uczA)P{vc:B)

=

PW(A)P«(B). (1)

If

thesets

E

(u)

and

E

{v)

consistofonlyafinitenumberofelements,

£(«)

=

%

+u

2

+

• ••
  +u
  n,

#*>

=

»!+.

w,

+

• ••
  
  
  • 
    
  
  
  
  

v

m

,

thenourdefinitionofindependenceofuandvis identicalwith

thedefinitionofindependenceofthepartitions

k

E=^{v

=

v

k

}

k

asin
§5,

Chap.I.

Forthe independenceofuandv,the followingcondition is

necessary and sufficient. For
any

choice of

set

A

in

$

(w)

the

following
equationholds almostcertainly:

P

v

(uczA)

=

P{uczA)t (2)

InthecaseP

(v

>(£)

=

0,bothequations

(1)

and

(2)

aresatisfied,

andthereforeweneed onlyprovetheir equivalence
inthecase

P

(v)

(B)

>

0.Inthiscase

(1) isequivalenttotherelation

P

{vc

b}(uczA)

=

P{uc:A) (3)

andthereforetotherelation

E

{vcB}

P

v

{uciA)=

P(«c2).

(4)

Ontheotherhand,itisobviousthatequation (4) followsfrom

57