Foundations of the theory of probability

§

2.AbsoluteandConditionalMathematicalExpectations 41

E{x)

=

ft.

..

fx(ult

u

2

,...,

u

n

)

P<«i.«*.-.«>

(«*#»)

.

(4)

We

havealready seen that the conditional probability P

B

(A)

possessesallthepropertiesofaprobabilityfunction.Thecorres-

pondingintegral

Eb(x)

=

jx?

B

(dE)

(5)

E

wecalltheconditionalmathematicalexpectation
of

therandom

variablexwithrespecttotheeventB.Since

p

B(

B

)

=

0, JxPB

(dE)=0

weobtainfrom (5) theequation

E

B

(x)

=fxPB

(dE)

=

jxP

B

(dE)

+

jxP

B

(dE)

=JxPB

(dE)

E B

B

B

We

recallthatin

case

AaB,

P

(A\

• 
  

P{AB)

P{A">

wethusobtain

B

From
(6)

andtheequality

(B) P(B)

^B(x)

=

~

]

jxP(dE),

(6)

B

jxP(dE)=P(B)E

B

{x).

(7)

A+B

weobtainatlast

JxP(dE)

=

JxP(dE)

+jxP{dE)

P(A)E

A(*)+

P{B)E

B

(x)

E^W-

—

p-;

1

-—

(8)

and,

inparticular,wehavetheformula

EW

=P(A)E

A

{*)+

P(A)Ei(x). (9)