Foundations of the theory of probability

40 IV. MathematicalExpectations

Thesecondlineisnothingmorethantheusualdefinitionofthe

Stieltjesintegral

+«>

jadFW(a)=

E(*). (1)

—00

Formula (1) maythereforeserveas adefinitionof themathe-

matical expectation E(x).

Nowletubeafunctionoftheelementaryevent

£,

anda;bea

random variable definedas asingle-valued functionx

—

x(u)

of

u.

Then

P{km^x<(k

+1)m}

=PW{kfn^

x(u)

<

(k

+

\)m},

where

P

(m)

(A)

istheprobability functionofu. It

thenfollows

fromthedefinitionoftheintegralthat

E

£(u)

and, therefore,

E(x)

=Jx{u)PM(dE(«)) (2)

whereE

(u)

denotesthesetofallpossiblevaluesofu.

Inparticular, when uitself is a random variable wehave

+00

E(x)=jxP{dE)=jx(u)

P^idR

1

)

=jx(a)dFW(a).

(3)

E R

l

-00

Whenx(u) iscontinuous,thelastintegralin
(3)

istheordinary

Stieltjesintegral.Wemustnote,however,thattheintegral

jx(a)dF^{a)

canexistevenwhenthemathematicalexpectationE(x)doesnot.

FortheexistenceofE(x),itisnecessaryandsufficientthatthe

integral

f\x(a)\dF(

u

){a)

—00

befinite

4

.

If

u

is

a

point

(u

lf

u

2

,...,u

n

)

ofthe

space

R^thenasaresult

of (2):

4

Cf.V. Glivenko,Surlesvaleursprobablesde fonctions,Rend.

Accad.

Linceiv.

8,

1928,

pp.

480-483.