Foundations of the theory of probability

§

2. DefinitionofRandomVariablesandofDistributionFunctions 23

Since

g

(

*>

isafield,

thenalongwiththeintervals (-oo,«a) it

containsallpossible

finitesumsofhalf-openintervals [a,-b).If

ourfieldofprobabilityisa

Borelfield,then

$

and

5

U)

areBorel

fields
;therefore,in

this

case

%

(x)

containsallBorelsets

ofR

1

,

Theprobabilityfunctionofa

randomvariable

we

shalldenote

inthefuturebyP<*>(A').Itisdefinedfor

all

setsofthe

field

ft<*>.

In particular, for the most important case, the

Borel

field of

probability, P

(x)

isdefinedforallBorel setsofR

1

.

Definition.Thefunction

F<*Ha)

=P<*>

(-*>',

a)

=p

{x<a},

where

• ooand

4-

ooareallowablevaluesofa,iscalledthedistri-

bution
functionof

therandomvariablex.

Fromthe

definition

it

follows

at

once that

FW(-oo) =0,FW(

+

oo) =

1

.

(1)

Theprobabilityofthe realizationofboth inequalitiesa^x<b,

isobviouslygivenbytheformula

?{x

c

[a;

b)}=F&{b)

• F&(a) (2)

Fromthis,wehave,fora
<

b,

FW(a)§FW(5)

which
meansthatF

(x)

(a) isanon-decreasingfunction.Nowlet

fli<a
2

<...<a

n

< ...<b

;

then

^{xa[a

n

;b)}=

n

Therefore, inaccordancewith thecontinuityaxiom,

FV(b)-F(*)(a

n)

=

P{xcz[a

n>

b)}

approacheszeroas«->

• 
  

oo.

FromthisitisclearthatF

(x)

(a) is

continuousonthe
left.

Inananalogous

waywecan

prove

theformulae:

limFW

(a)

=

FW(.-oo)

=

0, a

-+

• 
  

oo

, (3)

limFW

(a)

=

F«

(

+

oo

)

=

1, a

• +oo-
  (4)

Ifthefieldofprobability

(5,

P) isaBorelfield,thevaluesof

theprobabilityfunctionP<*>(A)
for all BorelsetsA of i^

1

are

uniquelydetermined

byknowledgeofthe distributionfunction