Foundations of the theory of probability

§3.

Multi-dimensionalDistributionFunctions 25

subsets of spaceR

n

anda probability

function

pfe»»*»•••»•*>

(4')

definedon

gf'.

Thisprobability

function

is

calledthen-dimensional

probabilityfunction
of

therandomvatiablesx

lt

x

2 ,

...

,

x

n

.

Asfollowsdirectlyfromthedefinitionofarandom

variable,

thefield

g'

contains,foreachchoiceofianda

t

(i

=

1,2,

...

,

n)

f

theset

ofallpointsinR

n

forwhichx

{

<

a

{

.Therefore

g'

alsocon-

tainsthe intersection of theabove sets, i.e.the set

L

ai0t

_

aH

of all points of R

n

for

which all the inequalities x

{

<

a

t

hold

(i

=

l,2,...,n)\

Ifwenowdenoteasthen-dimensional

half-openinterval

[tti,a

2 ,

..

.

,a

n

',

Oi,b

2 ,

..

.

,o

n)

;

the setofallpointsinR

n

,

forwhicha

i^^i

<b

i,

thenweseeat

oncethateachsuchintervalbelongstothefield

gf'

since

[a

v

a

t

,

...,a

n;

b

v

b

2

,

..

.,

b

n)

==

^b\bt...b

n

*^o,\b

t

...b

n

^b\a

t

bi...bn

*

*

^b

x

b%...bn-idn

'

TheBorelextensionof

thesystem

of

alln-dimensionalhalf-

openintervalsconsistsofall

Borel

sets

in

R

n

.

Fromthisitfollows

thatinthecase

of

aBorelfield

of

probability

'

7

thefield

5

contains

alltheBorelsetsinthespaceR

n

.

THEOREM:Inthecase

of

aBorel

fieldof

probabilityeachBorel

function

x

=

f(x

lt

x

2

,.

..

,

x

n) of

a

finite

number

of

randomvari-

ables

x

u

x

2 ,

...

,

x

n

isalsoarandomvariable.

Allweneedtoprove thisistopoint out

that

the

set

of

all

points(x

lt

x

2 ,

...

,

x

n)

inR

n

for

which

x

=

f(xu%2,

..

.

,x

n)

<a,

isaBorelset.Inparticular,allfinitesumsandproductsofrandom

variablesarealsorandomvariables.

Definition: Thefunction

is calledthew-dimensionaldistributionfunctionof therandom

variablesx

lf

x

2f

...

,

x

n

.

Asinthe

one-dimensionalcase,

weprovethatthe

n-dimensional

distributionfunction

F

(Xl

'

x Xn)

(a

u

a

2f

...,a

n

)is

non-decreas-

ingandcontinuous ontheleft ineach variable.In analogyto

equations (3) and

(4)

in

§

2,weherehave

1

The

a

f

mayalso assumetheinfinitevalues±

<*>