Foundations of the theory of probability

26 III.

RandomVariables

limF(«

lf

a

2

,

...,a

n

)

=F(a

v

..

.,«,_

lf

-oo,a

i+1

,

..

.,a

n

)

=

0, (7)

limyfo,a,,

..

.,a

n

)

=

F(+<x>, +<x>,..

.,+

oo)

=

1. (8)

O,

—

+00,a

t

— +oo.

...,a

M

->

+oo

Thedistributionfunction F<

x

*

x

*•*•»)

givesdirectlythevalues

ofP

(Xl

'

*

2

Xh)

onlyforthespecialsetsL

flia,

...

a

„

. If
ourfield,how-

ever,isaBorelfield,then

2

?<*"* >*»)

isuniquely

determinedfor

allBorel

sets

in

R

n

byknowledge of the distributionfunction

Ifthere existsthederivative

wecallthisderivativethen-dimensionalprobabilitydensityof

therandomvariables
x
u

x

2

,...,x

n

atthepointa

u

a

2r

.. ,a„.

If

alsoforevery
point (a
11

a

2 ,

...

,

a

n)

p(xux*.

...,*„>

(a

x

a

2

...

an)

=

| f

...jf{a

lt

a

2

a

n

)da,da

2

...da

n

,

—OO—oo —oo

thenthe

distributionofx

lf

x

2

,...,se»

iscalledcontinuous.For

everyBorelsetAc#

M

,wehavetheequality

pfeu......,«.)

(4)-=yj.

.

.jf(a

v

a

%t

..

.,

flji^rffl,.••<**„. (9)

4

Inclosingthissectionweshallmakeonemore

remarkabout

therelationshipsbetween thevarious

probabilityfunctionsand

distributionfunctions.

Giventhesubstitution

s

/i.

2, .... n\

andlet^denotethetransformation

*i

=

x

ik

(k

=

1,2,

...,n)

ofspacei?

w

intoitself.Itisthenobviousthat

pfrv*^.

••-,*»,)

(4)

=

p(*i,*.,...,«w{r-i^)}.

(10)

Now

letx'

=

Pk(x)

bethe"projection" ofthespaceR

n

on

the

spaceR

k

(k<n),sothatthepoint(x

lf

x

2

,

..

.

,x

n)

ismappedonto

thepoint(x

u

x

2t

..

.

,^

fc)

.Then,asaresultofFormula

(2)

in

§

1,

Cf.

§3,

IVintheSecondChapter.