Foundations of the theory of probability

56 V. ConditionalProbabilitiesandMathematicalExpectations

inthesenseof

§1,

Chap. IV, so that

(12)

is onlya

symbolic

expression.

If

x

is

a

randomvariablethenwecallthe

functionofxanda

Ff(a)

=P

s

(y<a)

theconditionaldistributionfunction
ofy

forknownx.

F

x

{y)

(a) isalmostcertainlydefinedforeverya.Ifa

<

bthen

almostcertainly

Ff(a)^Ff(b).

From (11) and (10) itfollows

4

thatalmostcertainly

E

x(y)

=

lim

k=%

£kX[Ff{{k

+

\)l)

• Ff(kl)]
  . (13)

;.

-+

ok=

• oo

Thisfactcanbeexpressedsymbolicallybytheformula

+ 00

E

x(y)

=

fadFf(a)

(14)

—

oo

Bymeansofthenewdefinitionofmathematicalexpectation

[(10)

and

(11)]

itiseasytoprovethat,forarealfunctionofu,

E«[/My]=/(«)E

M(y)

.

(15)

Cf.footnote3.