Stochastic lecture 4

7/30/2019 Stochastic lecture 4

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Copyright Syed Ali Khayam 2009

EE 801: Analysis of StochasticSystems

Inequalities and Limit Theorems

Dr. Muhammad Usman Ilyas

School of Electrical Engineering & Computer Science

National University of Sciences & Technology (NUST)

Pakistan


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Whatwillwecoverinthislecture?

Inthislecture,wewillcoverimportantinequalitiesand

theoremsthatwillleadtoafundamentalresultinprobability

theoryknownastheCentralLimitTheorem

Alistoftopicsthatwillbecoveredisasfollows:

MarkovInequality

ChebyshevInequality BinomialTheorem

TheWeakLawofLargeNumbers

TheCentralLimitTheorem

Forthislecture,Iamborrowingderivations/discussionsfromthe

followingbook:

KishoreS.Trivedi,ProbabilityandStatisticswithReliability,Queuing,and

ComputerScienceApplications,2005.2


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MarkovInequality

ConsiderthatyouaregiventhemeanE{X}= ofanonnegative

randomvariableX

DefineafunctionofXas:

0, if

,

X tY

t X t


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MarkovInequality

ConsiderthatyouaregiventhemeanE{X}= ofanonnegative

randomvariableX

DefineafunctionofXas:

Recallthatafunctionofarandomvariableisalsoarandom

variable Sowhatisthepmf ofY?

0, if

,

X tY

t X t


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MarkovInequality

Yhasadiscretepmf:

TheexpectedvalueofYis:

0, if

,

X tY

t X t


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Relationship BetweenRVsX&Y

X

Y

0 t

t

pX(x)

pY(y)


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MarkovInequality

SinceXY,wehaveE{X}E{Y}

0, if

,

X tY

t X t


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MarkovInequality

ProbabilitythatXtakesvaluesfarfromthemeanislow

Pr{ }X tt

m

9


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ChebyshevInequality

DefineanewrandomvariableW=(X)2

asafunctionofthenonnegativerandomvariableX

Clearly,Wisanonnegativerandomvariable

Also,lets=t2

ThenbyMarkovinequality

Markov

Inequality{ }

Pr{ }X

X tt

E

2

2 22

22 2

2

{ }Pr{ }

{( ) }Pr{( ) }

Pr{( ) }

WW s

s

XX tt

X tt

mm

sm

E

E --

-

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ChebyshevInequality




Also,lets=t2


Markov

Inequality{ }

Pr{ }X

X tt

E

2

2 22

22 2

2

{ }Pr{ }

{( ) }Pr{( ) }

Pr{( ) }

WW s

s

XX tt

X tt

mm

sm

E

E --

-

12


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ChebyshevInequality




Also,lets=t2


Markov

Inequality{ }

Pr{ }X

X tt

E

2

2 22

22 2

2

{ }Pr{ }

{( ) }Pr{( ) }

Pr{( ) }

WW s

s

XX tt

X tt

mm

sm

E

E --

-

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ChebyshevInequality

Nownotethat

Thus:

22 2

2Pr{( ) }X t

t

sm-

2 2Pr{( ) } Pr{ } Pr{ }X t X t X t m m m- = - = -

2

2Pr{ }X t

t

sm-

This is called

the Chebyshev

Inequality

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ChebyshevInequality

Chebyshev inequalitygivesanintuitivesenseforvariance

2 issmall=>Valuesclosetothemeanhavehigherprobabilities

2 islarge=>Valuesfarawayfromthemeanhavehigh

probabilities

2

2Pr{ }X t

t

sm-

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ReadingAssignment

Example3.40inthetextbook



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BernoullisTheorem

ConsiderabinomialdistributionXwithparameters(n,p)

NowapplyingtheChebyshevInequalitygives:

{ } 2

2

1Pr (1 ) 1

(1 ) 1Pr 1

X np k np pk

p pXp k

n n k

- < - -

- - < -

npm = (1 )np ps = -

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BernoullisTheorem

Set

2

(1 ) 1Pr 1

p pXp k

n n k

- - < -

(1 )/k p p n e = -

2

(1 )

Pr 1

p pX

pn ne e

- - < -

lim Pr 1n

Xp

n

e

- < =

This is called

Bernoullis

Theorem

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BernoullisTheorem

BernoullisTheoremstatesthatifwerunalargenumberof

Bernoullitrials(n)thentheprobabilitythattheproportionof

successesinthentrialsdiffersfrompisarbitrarilysmall

BernoullisTheoremisaspecialcaseoftheWeakLawofLargeNumbersdiscussednext

lim Pr 1n

Xp

ne

- < =

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WeakLawofLargeNumbers

LetX1,X2,,Xn,bemutuallyindependentidenticallydistributed

randomvariables

Let representthemeanofthecommondistributionofthese

randomvariables

Thenforlargen,wewouldexpectthatthesamplemean iscloseto

1 2 nx x xxn

m+ + +

=

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Let:

Thenthevarianceofthesamplemeanis:

1 2 nx x xxn

m+ + +

=

1/

n

n i n

i

S X and X S n =

= =

{ } { }{ }

1

2 1

2 2

2

1var var var

1var

nn

i

i

n

ii

SX X

n n

Xn

n

n n

s s

=

=

= =

=

= =

Sample mean

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Thusvarianceofthesamplemeanapproaches0asnapproaches

infinity

Thatis,thedistributionofthesamplemeangetsmoreandmore

concentratedaboutitsmeanasthenumberoftrials(n)approachinfinity

{ }2

var Xn

s=

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Forlargenumberoftrials,thedistributionofthesamplemean

getsmoreandmoreconcentratedaboutitsmean

UsingChebyshevs Inequality,weget

{ }2

var Xn

s=

{ }

2

2 2

var{ }

Pr

X

X n

s

m d d d- = =

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Takingthelimitoftheaboveinequalityyields:

{ }2

2Pr X

n

sm d

d- =

{ }lim Pr 0n

X m d

- = This is called theWeak Law of

Large Number

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Takingthelimitoftheaboveinequalityyields:

{ }2

2Pr X

n

sm d

d- =

{ }lim Pr 0n

X m d

- = This is called theWeak Law of

Large Number

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TheWeakLawofLargeNumbersstatesthatifyourunalarge

numberoftrials(n)thentheprobabilitythatthesamplemean

convergestothetruemeanis1

{ }lim Pr 0n

X m d

- =

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TheCentralLimitTheorem

LetX1,X2,,Xn,bemutuallyindependentrandomvariableswitha

finitemeanE{Xi}=i andafinitevariancevar{Xi}=(i)2.Weforma

normalizedrandomvariable:

sothatE{Zn}=0andvar{Zn}=1.Then,undercertainregularity

conditions,thelimitingdistributionofZn isstandardnormal:

1 1

2

1

n n

i i

i in n

i

i

X

Z

m

s

= =

=

-=

(0,1)nZ N

2 /21lim ( ) lim Pr{ }2n

t

yZ n

n nF t Z t e dy

p-

-

= =

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Example:TheCentralLimitTheorem

ThisisthehistogramoftheweightsofpeopleonRhodeIsland.

Reference: http://www.intuitor.com/statistics/CentralLim.html

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Now,insteadofplottingthehistogramofeachsampleseparately,

weplotaveragesof2samples


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Thenweplothistogramoftheaverageof50samples


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Finally,weplotthehistogramoftheaverageof100samples


The distribution

approaches

normality as we

move keepaveraging over more

and more samples

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ReadingAssignment





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Stochastic lecture 4