Foundations of the theory of probability

24

III. Random

Variables

F

(x)

(a) (cf.

§3,

IIIinChap. II).Sinceourmaininterestliesin

thesevaluesof P

(x)

(A), thedistributionfunctionplays

a

most

significantroleinallourfuturework.

IfthedistributionfunctionF

(x)

(a) isdifferentiate,then

we

callitsderivativewithrespecttoa,

the

probabilitydensityofxatthepointa.

a

IfalsoF

(x)

(a)

=

j

f

ix)

(a) daforeacha, thenwemayex-

—oo

press

the

probabilityfunction?

(x)

(A)

for

each

Borel setA

in

terms

of

f

(x)

(a) in

the

followingmanner:

Pto(A)=ff(*){a)da.

(5)

A

Inthiscasewecallthedistribution

of

xcontinuous.And

in

the

general

case,wewrite,analogously

PW(A)-=

fdFW\a).

(6)

A

Alltheconceptsjustintroducedarecapableofgeneralization

forconditionalprobabilities.Thesetfunction

9%\A)=?

B

(xc:A)

istheconditionalprobabilityfunctionofx underhypothesisB.

Thenon-decreasingfunction

Ff(a)

=

P

B

(x<a)

is thecorrespondingdistribution function, and,

finally

(in

the

casewhereF^(a) isdifferentiate

)

*?(*)=

j;*VM

istheconditionalprobability

density

of

xat the

point

aunder

hypothesisB.

§

3. Multi-dimensional

DistributionFunctions

Let

now

nrandomvariablesx

lt

x

2

,...,x

n

begiven.The

point

x

=

(x

u

x

2 ,

...

,Xn)

ofthe7i-dimensionalspaceR

n

isa

function

of theelementaryevent £. Therefore,accordingto thegeneral

rules in

§1,

we have a field

«j(*i;

*.••.*>

consisting of