Foundations of the theory of probability

§

4. NotesontheConceptofMathematical

Expectation

65

asageneralizedmathematical expectation.Welosein thiscase,

ofcourse,severalsimplepropertiesofmathematicalexpectation.

Forexample, inthiscasetheformula

E(s+

y)

=

E(x) +E(y)

is not always true. In this form the generalization is hardly

admissible. We mayadd however that, with some restrictive

supplementaryconditions,definition
(2)

becomesentirelynatural

anduseful.

Wecandiscusstheproblemasfollows.Let

X\

t

X21•••

j

X

n,

• 
  

  •

  
  

beasequenceofmutuallyindependentvariables,havingthesame

distributionfunction F

(x

^(a)

=F

(Xn)

(a), (n

=

1, 2, ...

)

asx.

Letfurther

*1+*2H

1"

*n

We nowask whether there exists
a

constant E*

(x)

such that

for
everye>

limP(|s

n

-E*(*)|

><0=O, w^+cx). (3)

Theansweris :

//

suchaconstantE*(x)exists,itisexpressedby

Formula
(2).

The

necessaryandsufficientconditionthat

Formula

(3) holdconsistsintheexistenceof limit (2) andtherelation

P(|*|>n)-o(±). (4)

To prove this we applythe theoremthat condition

(4)

is

necessaryandsufficientforthestabilityofthearithmeticmeans

s„, where,inthecaseofstability, wemayset

6

+n

d

n

=jadF(

x

)(a).

—

n

Ifthereexistsamathematicalexpectationintheformersense

(Formula

(1)),

thencondition

(4)

isalwaysfulfilled

7

. Sincein

thiscase£(x)

=

E*(x),thecondition

(3)

actuallydoesdefinea

generalization oftheconcept ofmathematical expectation. For

the generalized mathematical

expectation,

Properties I-VII

8

Cf.

A.

Kolmogorov,Bemerkungenzumeiner

Arbeit,

"UberdieSummen

zufdlligerGrossen"Math.Ann.v.102,1929,pp.484-488,TheoremXII.

7

Ibid,TheoremXIII.