Foundations of the theory of probability

12 I. ElementaryTheoryofProbability

five probabilities

P(A^

}

)

is that the conditional probability

of

the resultsA

q

w

of

experiments

3t

(i

'>

under the hypothesis

that

several other tests 2l

(il)

,

9l

(i,)

,...,W

ik)

have hod

definite

results

A&\AM,A

i

**>,...,A

{

£

)

is equal

to the absolute probability

Onthebasisofformulas (4) wecanproveinananalogous

mannerthefollowingtheorem

:

Theorem III. //allprobabilitiesP(A

k)

arepositive, then

a

necessary and

sufficient

condition

for

mutual

independence

of

theeventsA

lt

A

2i

...,

A

n

is

the

satisfactionoftheequations

P,

iA

...^(A)

=PW

(7)

for

anypairwisedifferenti

lt

i

2 ,

. ..
,

i

k,

i-

Inthecasen

—

2 theconditions

(7)

reducetotwo

equations:

P

Al

(A

2

)

=

P(A

2 )

f

|

P

AAA

l

)

=

P(A

1

). J

Itiseasytoseethatthefirstequationin

(8)

aloneisanecessary

andsufficientconditionfortheindependenceofA

x

andA

2

pro-

videdP(A

1 ) >

0.

§

6. ConditionalProbabilitiesasRandomVariables,

MarkovChains

Let 51 beadecompositionofthefundamentalsetE:

E

=

A*+A

2

+...+A

r,

andxarealfunctionoftheelementaryevent

£T

whichforevery

setA

q

isequalto

a

correspondingconstanta

q

.x

is

thencalleda

random

variable,andthesum

E(x) -2a

Q

P(A

5

)

Q

is called themathematical expectation of the variable x. The

theoryofrandomvariableswillbedevelopedinChaps.IllandIV.

Weshallnotlimitourselvestheremerelytothoserandomvari-

ableswhichcanassumeonlyafinitenumberofdifferent

values.

ArandomvariablewhichforeverysetA

q

assumesthevalue

PA

qi

(B),weshallcall theconditionalprobability

of

theeventB

afterthegivenexperiment%andshalldesignateitbyP^(B).Two

experiments

5l

(1)

and

3l

(2)

areindependentif, andonlyif,