Foundations of the theory of probability

§5.

Independence 9

Bernsteinisactuallydedicatedtothefundamental investigation

of series

of independent random variables. Though the latest

dissertations (Markov,

Bernsteinandothers) frequentlyfailto

assume complete

independence, they nevertheless reveal the

necessityofintroducinganalogous,

weaker,conditions,inorder

toobtainsufficiently

significant

results (see

inthischapter

§6,

Markovchains)

.

Wethussee,intheconceptofindependence,atleastthegerm

ofthe peculiar type of problem inprobability theory. Inthis

book, however, we shall not stressthat fact, for here we are

interested mainly inthe logical foundationfor the specialized

investigationsofthetheoryofprobability.

Inconsequence, oneof themost important problemsinthe

philosophy of thenatural sciences is—inaddition tothe well-

knownoneregardingthe essence oftheconcept ofprobability

itself—to

makeprecisethepremiseswhichwouldmakeit

possible

toregard anygiven realevents asindependent. Thisquestion,

however,isbeyondthescopeofthisbook.

Letusturntothedefinitionofindependence.Givennexperi-

ments
5l

(1)

,5l

(2)

,

...

,5l

U)

,

thatis,ndecompositions

E

=

Af

+

A$

]

+

h

A

1

*}

i=\,2,...,n

ofthebasicsetE.Itisthenpossibletoassignr

=

r

1

r

2

...r

n

proba-

bilities (inthegeneralcase)

P^...qn

=P(A

(

q\

)

A%;..

A

{

q

n

J)^0

whichareentirelyarbitraryexceptforthesinglecondition

7

that

2 Ah<? 8 ...«»

= 1

• 
  

(!)

Definition I. nexperiments 3i

(1)

,

5l

(2)

,

...
,

3l

(n

>

arecalled

mutually independent, iffor any
q

l9

q

2 ,

...

, q

n

thefollowing

equationholdstrue

:

p(4>4?•••

O

=

p

«>)p

(4?)

• • p(4:') • (2)

7

Onemayconstructafieldofprobabilitywitharbitraryprobabilitiessub-

jectonlytotheabove-mentionedconditions,asfollows:Eiscomposedofr

elements
£«,qt

.

..

q

n

. Let the corresponding elementary probabilities be

PqiQt...in>

andfinallylet A

q

i]

bethesetofall

£

f/l9, tm

.

9m

forwhich

<7t

=

q-