Foundations of the theory of probability

§

3. ConditionalProbabilities

withRespecttoaKandomVariable 51

oppositepoints

forourpoles,

so

thateachmeridiancirclewillbe

uniquely

defined bythelongitude

v,

^

ip

<n

.Since

y>

varies

from only

to^r,—inotherwords,weareconsideringcomplete

meridian

circles (andnotmerelysemicircles)—thelatitude

mustvaryfrom

—

nto -\-n(andnot

from

—-

to

+

^

)

.Borelset

thefollowing

problem: Required todetermine "theconditional

probability distribution" of latitude
t

—7i<0<+tz, for

a

givenlongitude^.

Itiseasytocalculatethat

e

%

P

y>{0x

=g

<G

2

}

=

if\cosG\

d0.

Theprobabilitydistributionof foragiven V isnot

uniform.

If

weassumethattheconditionalprobabilitydistributionof

"withthehypothesisthat

$

liesonthegivenmeridiancircle"

mustbeuniform, thenwehavearrivedatacontradiction.

Thisshowsthattheconceptofaconditionalprobabilitywith

regardtoanisolatedgivenhypothesis whoseprobabilityequals

is inadmissible. For we can obtain a probability distribution

for onthemeridiancircleonlyifweregardthiscircle
as

an

elementofthedecompositionoftheentirespherical
surfaceinto

meridiancircleswiththegivenpoles.

§

3. ConditionalProbabilitieswithRespect
   toaRandomVariable

Ifa? is

a

randomvariable

and

P

X

(B)

asa

function

of x is

measurable
inthe

Borel

sense,then

P

X

(B) canbe definedinan

elementary
way.Forwecanrewriteformula (2) in
§

1,tolook

asfollows

:

P(£)PJ»(ii)=/P,(B)

Pl*)(dE)

. (1)

A

Inthiscaseweobtainfrom
(1)

atoncethat

a

P{B)Ff(a)=JPu

(a;BydFW(a).

(2)

—oo

Inaccordance
withatheoremof
Lebesgue

2

itfollowsfrom

(2)

that

P^BJ-PWllmgg+j^gg

^o

(3)

whichis alwaystrueexceptfor
asetHofpointsa forwhich

P<*>(H) =

.

2

Lebesgue,

I.c,1928,

pp.

301-302.