Foundations of the theory of probability

8 I. ElementaryTheoryofProbability

A

lt

A

2 ,

...

,

A

n

areoften called "hypotheses"andformula

(14) is consideredas the probability

P*(A

{)

of thehypothesis

Ai

after the occurrence of event X. [P(A*) then denotes the

aprioriprobabilityofA*.]

Proof:

From

(12)

we

have

PWP^(X)

Px(Ai)

P(X)

To

obtaintheformula(14) itonlyremainstosubstituteforthe

probability P(X) itsvalue derivedfrom (13) by applyingthe

theoremontotalprobability.

§

5. Independence

Theconceptofmutualindependence oftwo ormoreexperi-

mentsholds,in

a

certainsense,

a

central

position

inthe

theoryof

probability. Indeed, as we have already seen, the theory of

probabilitycanberegardedfromthemathematicalpointofview

asaspecialapplicationofthegeneraltheoryofadditivesetfunc-

tions. Onenaturally asks,howdid ithappenthatthetheoryof

probabilitydeveloped intoalargeindividual sciencepossessing

itsownmethods?

Inorder

to

answerthisquestion,

wemustpointoutthespe-

cializationundergonebygeneralproblemsinthetheoryofaddi-

tive set functions when they are proposed in the theory of

probability.

Thefactthatouradditivesetfunction P(A) isnon-negative

andsatisfiestheconditionP(E)

=

1,

doesnotinitselfcausenew

difficulties.

Random variables (see Chap.

Ill) from

a

mathe-

maticalpointofviewrepresentmerelyfunctionsmeasurablewith

respect to P(A), while their mathematical expectations are

abstract Lebesgueintegrals. (Thisanalogywasexplainedfully

forthefirsttimeintheworkofFrechet

6

.)

Themereintroduction

oftheabove

concepts,therefore,would notbesufficienttopro-

duceabasis

for

the

developmentof

a

largenewtheory.

Historically, the independence

of experiments and random

variablesrepresentstheverymathematicalconcept

thathasgiven

thetheoryof probabilityitspeculiarstamp.Theclassical

work

orLaPlace,Poisson,Tchebychev,Markov,Liapounov,Mises,and

•SeeFrechet

[1]

and

[2].