Foundations of the theory of probability

10 I. ElementaryTheoryofProbability

Amongtherequationsin
(2),

thereareonlyr-r

1

-r

2

• 
  

..

.

-r

n

+

n

• 1 independent equations

8

.

Theorem I. Ifnexperiments

9l

(1

\5l

(2)

,

...

,2i

(M

>aremutu-

allyindependent,

thenany

m

ofthem

(ra<n)

,

9l

(t,)

,$

(

**\

....5(

(

'm)

>

arealsoindependent

9

.

Inthecaseofindependencewethenhavetheequations

:

p«4«••

• 4i

B>

)

=

p

(O

p

C^SW

• ••p

(41-*)

(

g

)

(all

4

mustbedifferent.)

DefinitionII.neventsA

u

A

2 ,

..

.

,A

n

aremutuallyindepen-

dent,ifthedecompositions (trials)

E

=

A

k

+A

k

(k

=

l,2,...,n)

areindependent.

Inthiscaser

x

=

r

2

=

...

=

r

n

=

2,

r

=

2

n

;

therefore,ofthe 2

W

equationsin
(2)

only 2

n

-n-lare independent.Thenecessary

andsufficient
conditionsfortheindependenceoftheeventsA
lt

A

2 ,

..

.

,A

n

arethefollowing 2

n

• n
  
  
  • 1 equations
  
  
  
  

10

:

P(A

{l

A

i2

...A

im)

=P(A

il

)P(A

i2

)...P(A,

im),

(4)

m

—

1,2,

..

.,

n,

i^i

1

<i

2

<--<i

m

<n.

All
of

these

equationsaremutuallyindependent.

Inthecasen

=

2 weobtainfrom

(4)

onlyonecondition

(2

2

-2

• 
  

8

Actually, inthecaseofindependence,onemaychoosearbitrarilyonly

fi

+

r* 2

+

...

+t

n

probabilities

p

U)

=

P

{A

U)

)

soas to complywith the n

conditions

7

"

<i

Therefore,inthegeneralcase,wehaver-1degreesoffreedom,butinthe

caseofindependenceonlyri+r
2

+...+r

n

-n.

9

Toprovethisitis

sufficientto

show

that

fromthe

mutual

independence

ofndecompositionsfollowsthemutualindependence

ofthefirstn-1.Letus

assumethattheequations

(2)

hold.Then

p

(«.

..

<-»,)

=Jp

(«•

• •

<)

Qn

9n

Q.E.D.

10

SeeS.N.Bernstein

[1]pp.

47-57.However,

the

reader

can

easilyprove

thishimself(usingmathematicalinduction).