Foundations of the theory of probability

54 V.

ConditionalProbabilitiesandMathematicalExpectations

Ifuandvaretwofunctionsoftheelementaryevent

£,

then

thecouple (u,v) canalwaysberegardedasafunctionofg.

The

followingimportantequationthenholds

:

E

u

E

{UtV)

(y)

=

E

u(y).

(5)

For,Eu(y)

isdennedbytherelation

E{

Mc^}(y)

=

E{ttd}E

M(y)

,

Therefore wemust

show

that E

M

E

(M

,

V)(y)

satisfiesthe equation

E{«cA}(y)

=E

{Mc^

}

E

M

E

(tt>r)(y).

(6)

Fromthedefinition

ofE

(u>v)(y)

itfollowsthat

E{„cA}(y)

=E

{Mc^

}

E

(M>t;)(y)

.

(7)

Fromthedefinitionof E

M

E

(MjV)(y)

itfollows,moreover,that

E{u*a}

E

(W)t,)(y)

• E

{MC^

}

E

m

E

(M>r)

(y)

.

(8)

Equation(6) resultsfromequations

(7)

and

(8)

andthusproves

ourstatement.

Ifweset

y

—

P

U

(B)equaltooneonBandtozerooutsideofB,

then
E

u(y)

=P

u

{B),

E

{UtU)(y)

=P

(UtV)

(B).

Inthiscase,fromformula

(5)

weobtaintheformula

E

M

P(

M

,„)(B)

=

-P

u

(B).

(9)

TheconditionalmathematicalexpectationE

u(y)

mayalsobe

defineddirectlybymeansofthecorrespondingconditionalprob-

abilities.Todothisweconsiderthefollowingsums

:

Sx{u)

=~y

i

°kXP

u

{kX^y<(k

+

\)X}

=TR

k

.

(10)

If E(y) exists,theseries

(10)

almostcertainly* converges. For

wehavefromformula

(3),

of

§ 1

,

E\R

k\

=

\kk\P{kl&y<(k+i)X},

andtheconvergenceoftheseries

^ZMP{U^y<(k+

i)X}=^E\R

k

\

Weuse

almostcertainlyinterchangeablywithalmostsurely.