Foundations of the theory of probability

60 VI. Independence;TheLawofLargeNumbers

InordertoshowthisitisenoughtotakeR

M

forthebasicsetE

andB%

M

forthefield

g,

andtodefinethedistributionfunctions

F/hp*.../** (

seeChap.Ill,

§4)

byequation

(3).

Letusalsonotethatfromthemutualindependenceofeach

finitegroupofvariables x^ (equation

(3))

therefollows,aswe

haveseenabove,themutualindependenceofall x^onB%

M

.In

more inclusive fields

of probability this propertymay

be lost.

Toconcludethissection,

we

shallgiveafewmorecriteriafor

theindependenceoftworandomvariables.

Iftwo randomvariables xand

y

aremutuallyindependent

andif E(x)andE(y) arefinitethenalmostcertainly

E,(y)

=E(y)

(5)

E

y

(x)=E(x).

These formulas represent an immediate consequence of the

seconddefinitionofconditionalmathematicalexpectation (For-

mulas
(10)

and

(11)

ofChap.V,

§

4).Therefore,inthecaseof

independenceboth

E[y-E,(y)J»

and

2

=

E[*-E,(*)]»

1

o

2

(y)

S

o

2

(#)

areequaltozero (providedv

2

(x) > andv

2

(y)

>0).Thenum-

ber
f

2

iscalledthecorrelationratioof

y

withrespecttox,and

g

2

thesameforxwithrespectto
y

(Pearson)

.

From

(5)

itfurtherfollowsthat

E(xy)

=

E(x) E(y)

.

(6)

ToprovethisweapplyFormula(15) of
§

4, Chap.V:

E(xy)=EE

x

{xy)=E[xE

x(y)]

=E[xE(y)]=E(y)E(x).

Therefore, inthecase ofindependence

r

=

E(*,y)-E(x)E(y)

o (x)a

(y)

isalsoequaltozero;r,asis wellknown, isthecorrelation co-

efficient

of

x

and

y.

Iftworandom

variables

x

and

y

satisfyequation

(6),

then

theyarecalleduncorrected. Forthesum

S

—

x*+x

2

+...-fx

n