Foundations of the theory of probability

52

V. ConditionalProbabilitiesandMathematicalExpectations

P

x

(a;B) was defined in

§

1 except on

a set G, which is

suchthatP

(*>

(G)

=

0.Ifwenowregardformula

(3) asthedefi-

nitionof P

x

(a;B) (settingP

x

(a;B)

=

when

thelimit inthe

righthand side of

(3)

fails to exist), then

this new variable

satisfies allrequirements of

§

1.

If,besides,theprobability

densities

f

(x)

(a) and

fg>

(a) exist

andif

f

(x

Ha) >0,

then

formula (3) becomes

P

I

(a;S,=

P(S

)

;|W.

(4)

Moreover,

from formula (3) itfollowsthattheexistenceof a

limit

in (3) andofa probabilitydensity

f

(x)

(a) results inthe

existenceof

/</>

(a).Inthatcase

P(B)12(a)

&#*(*). (5)

IfP(B) >0,thenfrom

(4)

wehave

Incase

f

(x)

(a)

=

0,thenaccordingto

(5)

/<*>

(a)

—

andthere-

fore(6) alsoholds.If,besides,thedistributionofxiscontinuous,

wehave

+

oo

+oo

P(B)=E(P,(B))=j'P

x

(a;B)dFW(a)

=j?x

(a;B)fW(a)da. (7)

—

oo —oo

From

(6)

and

(7)

weobtain

/?(«>=

+

y

d

-*)™

(8)

fPx(a;B)f*{a)da

—oo

Thisequationgivesustheso-calledBayes*Theoremforcontinu-

ousdistributions.Theassumptionsunderwhichthistheoremis

provedarethese: P

X

{B) ismeasurableintheBorelsenseandat

thepointaisdefinedby

formula

(3)

,thedistributionofxiscon-

tinuous, and at the

point

a

there existsa probability density

f

(

*Ha).

§

4. ConditionalMathematicalExpectations

Let

ube

anarbitraryfunctionof

£,

and

y

a,randomvariable.

TherandomvariableE

m

(t/),representableasafunctionofuand

satisfying,

for

any

setAof

$

(M

>

with P

(M

>

(A)

>0,

thecondition