Foundations of the theory of probability

46 IV. MathematicalExpectations

Therefore,S*convergestoE(J),fromwhichresultstheequation

b

Ex(t)dt

=

limS*

n

=

E(/).

/'

TheoremII caneasilybe generalized for double andtriple

andhigherordermultipleintegrals.Weshallgiveanapplication

ofthistheoremtooneexampleingeometricprobability.LetGbea

measurableregionoftheplanewhoseshapedependsonchance

;

inotherwords,letusassigntoeveryelementaryevent

£

ofafield

of probability a definite measurable plane region G. We shall

denoteby/theareaoftheregion

G,

andby ?(x,

y)

theprob-

abilitythatthepoint (x,
y)

belongstotheregion

G.

Then

E{J)=jj?{x,y)dxdy.

Toprovethis

it

is

sufficienttonotethat

/

=s

fif(x,y)dxdyl

P(x;y)=

Ef(x,y),

where f(x,y)

is the characteristic function of the region G

(fix,y)

—

1 onGand

f(x,y)

=

outsideofG)

6

.

A-

6

Cf.A.Kolmogorov andM.Leontovich,ZurBerechnungdermittleren

BrownschenFldche,Physik.Zeitschr.d.

Sovietunion,v.

4,

1933.