Foundations of the theory of probability

ChapterV

CONDITIONAL PROBABILITIES AND

MATHEMATICAL EXPECTATIONS

§

1. ConditionalProbabilities

In

§

6,ChapterI,wedennedtheconditionalprobability,P^

(B)

,

oftheeventBwithrespecttotrial%.Itwasthereassumedthat%

allows ofonlyafinitenumberofdifferentpossibleresults. We

can,however,defineP%(B)alsoforthecaseofan
%

withaninfinite

setofpossibleresults,i.e.thecaseinwhichthesetEispartitioned

into
an

infinite

number

of

non-intersectingsubsets.Inparticular,

weobtain
sucha

partitioning

ifweconsideranarbitraryfunction

uof£anddefineaselementsofthepartition9l„thesetsu

=

con-

stant.TheconditionalprobabilityP%
U

{B)

wealsodenotebyP

U

(B).

Anypartitioning 51 ofthesetEcanbedennedasthepartitioning

5i
M

whichis"induced"byafunctionuof

£,

ifoneassignstoevery

$,

asu(£),thatsetofthepartitioning
51

ofEwhichcontains

|.

Twofunctionsuandu'of

£

determinethesamepartitioning

5l
M

=

9l

M

'Of

thesetE

if

andonlyifthereexistsaone-to-one

cor-

respondence

u'

=

f(u) betweentheirdomains

$

U)

and

5

(M,)

such

thatv!
(£)

isidenticalwith

fu(£)

.Thereadercaneasilyshowthat

therandomvariablesP
M

(Z?) andP

M

*(B),definedbelow,areinthis

casethesame.Theyarethusdetermined,infact,bythepartition

9L

=

^itself,

TodefineP

U

(B) wemayusethefollowingequation:

P{u

C

a}(B)

=

E

{ucA}

P

u

(B). (1)

Itiseasy

toprovethat

ifthe

setE

(u)

ofallpossiblevaluesofuis

finite,equation

(1) holdstrueforany

choiceofA (whenP

U

(B)

is

defined

asin

§

6,Chap.I).Inthegeneralcase(in

whichP

U

(B)

isnot

yetdefined) weshallprovethatthere always

exists one

andonly

onerandomvariableP

U

(B) (exceptforthematter

of

equivalence)whichisdefinedasafunctionofuandwhichsatis-

fies

equation (1) for every choice of A from

5

(m)

sucn that

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