Foundations of the theory of probability

44 IV.MathematicalExpectations

FromIIandIV,weobtaininparticular

V. Inorderthat sequence

(1)

converge

inprobabilitytox,

itissufficientthat

limE(a;

n

-a;)

2

=

.

(4)

If

also

the

totalityofallx

lt

x

2 ,

..

.

,x

n,

..

.

,xisbounded,thenthe

conditionisalsonecessary.

Forproofsof I

• IVseeSlutsky
  [1]

andFrechet

[1].

How-

ever, thesetheorems follow almost

immediatelyfrom formulas

(3)

and

(8)

oftheprecedingsection.

§

5. DifferentiationandIntegrationofMathematicalExpectations

withRespecttoaParameter

Let

us

puteachelementaryevent

$

intocorrespondencewith

a

definite

realfunctionx(t) ofarealvariable

t.

Wesaythat

x(t)

isarandomfunction ifforeveryfixedt,thevariablex(t) isa

randomvariable.Thequestionnowarises,underwhatconditions

canthemathematicalexpectationsignbeinterchangedwiththe

integration

and

differentiationsigns.Thetwofollowingtheorems,

thoughtheydonotexhausttheproblem,canneverthelessgivea

satisfactoryanswertothisquestioninmanysimplecases.

TheoremI://themathematicalexpectationE[x(t)~\isfinite

for

anyt,andx(t) isalways

differ

-entiableforanyt,whilethe

derivativex'(t)

of

x(t) withrespecttotisalwayslessinabso-

lutevaluethansome constantM,

then

^E(x(t))=

E(x'(t)).

Theorem II://x(t) always

remainsless,inabsolutevalue,

thansomeconstantKandisintegrable

intheRiemannsense,then

b r b

JE(x(t))dt=

E jx(t)dt

a

la

providedE[x(t)] isintegrableintheRiemann

sense.

Proof

of

TheoremI.Letusfirstnote thatx'(t)asthelimit

of

therandomvariables

x(t

+

h)-x(t)

1 1

h

n-\, -,...,-,

...

is also a random

variable. Sincex'(t) is bounded, themathe-