4. nilai harapan dan mgf bersamaa

7/26/2019 4. Nilai Harapan Dan MGF Bersamaa

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EXPECTED VALUE OF A FUNCTION &JOINT MGF


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Sessions target

Nilai harapan bersama (Joint expected value)

Kovarian (Covariance)

Korelasi (Correlation)

Nilai Harapan Bersyarat (Conditional Expected Value)

MGF Bersama (Joint MGF)


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Expected value (Review)

Expected value of discrete R.V.

1( ) (1 ) ; 0,1x xX f x p p x

11 0 1 1 0

0( ) (1 ) 0 (1 ) 1 (1 )

x x

xE X x p p p p p p p

1

2 2 1 2 0 1 2 1 0

0

( ) (1 ) 0 (1 ) 1 (1 )x x

x

E X x p p p p p p p

22 2( ) [ ] [ ] (1 )V X E X E X p p p p


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Expected value (Review)

21

21

( ) ;2

x

X f x e x

2 21 1

2 2

0

1 1( ) 2 0

2 2

x x

E X x e dx x e dx

2 21 1

2 2 22 20

1 1( ) 2 12 2

x x

E X x e dx x e dx

22 2( ) [ ] [ ] 1 0 1V X E X E X

0( 1) xx e dx

12

1

0!

xk kax e dx k a

Expected value of continuous R.V.


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Joint pdf and expected value

LetXand Ybe random variables with joint probability f(x, y).Their expected values (means) are written as

Discrete random variables:

or

Continuous random variables:

or

( , )Xx y

x f x y ( , )Yy x

y f x y

( , )X x f x y dydx

( , )Y y f x y dxdy

( )Xx

x f x ( )Yy

y f y

( )X x f x dx

( )

Y y f y dy


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Contoh

A joint pdf of two random variablesX, Yis given by

otherwise

yxforyxf

xy 51,40

0),( 96

3

8

24

8

192

96

96

),(][

4

0

3

4

0

2

4

0

5

1

22

5

1

24

0

4

0

5

1

x

dxx

dxyx

dydxyx

dydxxy

x

dxdyyxxfXE

x y

931

14431

72

31

288

96

96

),(][

4

0

2

4

0

4

0

5

1

3

5

1

24

0

4

0

5

1

x

dxx

dxxy

dydxxy

dydxxy

y

dxdyyxyfYE

x y


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Expected value of a function

E H

( ). ( )x

u x f x

( ) ( )x

u x f x dx

; X diskrit

; X kontinu

Misalkan ( ) adalah sembarang fungsi dari X, makaH u X

Jika ( ) [ ( )] [ ] Rata-rata Xu X X E u X E X X

22

22 2

Jika ( ) ( [ ]) ( ) ( )

( ) [ ]X

u X X E X E u X E X E X

E X E X V X

R.V. ( )X f x



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IfXand Yhas a jointpdff(x,y) and if is afunction ofX andY, then

Discrete random variables:

Continuous random variables:

( ) ( , ) ( , )x y

E H u x y f x y

dxdyyxfyxuHE

),(),(...)(

),( YXuH




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If adalah joint fungsi darimaka untuk

1

1 1( ) ... ( ,..., ) ( ,..., )k

k k

x x

E H u x x f x x

1 2( , ,..., )kH u X X X


1 2, ,..., kX X X1 2( , ,..., )kf X X X

Jika diskrit1 2, ,..., kX X X

Jika kontinu1 2, ,..., kX X X

1

1 1 1( ) ... ( ,..., ) ( ,..., ) ...

k

k k k

x x

E H u x x f x x dx dx



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Example

Joint pdf of two random variablesX, Yis given by

Let

The expected value ofH

is

otherwise

yxforyxf

xy 51,40

0),( 96

4 5

0 1

4 5 2 2

0 1

[2 3 ] (2 3 ) ( , )

(2 3 )

96

48 32

47

3

x y

E X Y x y f x y dxdy

xyx y dxdy

x y xydydx

( , ) 2 3H u X Y X Y



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Covariance: Definition

X,Y discrete

X,Y continuous

Ukuran keeratanhubungan linear

antara 2 R.V.

Misalkan ( , ) ( [ ])( [ ])H u X Y X E X Y E Y

Covar ,

XY

X Y

( , )

( , )

X Y

X Y

x y

X Y

y x

E X Y

X Y f X Y

X Y f X Y dxdy

[ ] [ ]E H E X E X Y E Y



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Variance vs. Covariance

Case of 1 R.V.

2

2

22

Var

( )

( ) [ ]

XX

E X E X

E X E X

Case of 2 R.V.

Covar ,

[ ] [ ]

( ) [ ] [ ]

XYX Y

E X E X Y E Y

E XY E X E Y

Rumus hitungVariance

Rumus hitungCovariance

Var( ) Covar( , )X X X



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Some properties of Covariance

If Xand Yare random variables and a andbare constant, then

IfXand Yare independent, then

( , ) ( , )Cov aX bY abCov X Y ),(),( YXCovYbXaCov

)(),(),( XaVarXXaCovbaXXCov

0)().()(),( YEXEXYEYXCov


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Example



Example



Example

1

( , ) 16f x y xy



Example



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Covariance: Example

A joint pdf of two random variablesX, Yis given by

Thus, Cov(X,Y) = E[XY] E[X]E[Y] :

otherwise

yxf oryxf

xy 51,40

0),( 96

3

8

96][

4

0

5

1

x y

dxdyxy

xXE9

31

96][

4

0

5

1

x y

dxdyxy

yYE

4 5 4 5

2 2

0 1 0 1

[ ] ( , )

1 248

96 96 27x y

E XY xyf x y dxdy

xy

xy dxdy x y dydx

0

9

31

3

8

27

248][].[][

YEXEXYEXY



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Variance of a function

Some properties of variance

If cis a constant, Var[cX] = c2Var[X]

IfXand Yare independent random variables, then

Var[X Y] = Var[X] + Var[Y]

Var[aX+ bX]= a2Var[X] + b2Var[Y],

where a, bare constants



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Correlation Coefficient

Correlation is another measure of the strength ofdependence between two random variables.

It scales the covariance by the standard deviation ofeach variable.

IfXand Yare independent, then = 0, but = 0does

not imply independence



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Correlation Coefficient: Problem

Assume the lengthXin minutes of a particular type oftelephone conversation is a random variable withprobability density function

Determine

The mean length E(X) of this telephoneconversation.

Find the variance and standard deviation ofX

Find

5/

5

1)( xeXf x0

])5[( 2XE


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Prepared by : M. Dokhi, Ph.D.9:01

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5

255

12555

1

5155

5

1

555

1

5

1][

0

55

0

55

0

55

0

5

xx

xx

xx

x

exe

exe

dxexe

vduuv

dxxeXE5

5

5 x

x

evdxdu

dxedvxu

Use integration by part:


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505

250

250505

5

1

2551055

1

551055

1

1055

1

51][

5552

0

5552

0

5552

0

552

0

522

xxx

xxx

xxx

xx

x

exeex

exeex

dxexeex

dxxeex

vduuv

dxexXE

5

5

5 x

x

evdxdu

dxedvxu

25550))(()()var(222 XEXEX

5)var()( XXstd

5

52

52 x

x

evxdxdu

dxedvxu


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Correlation Coefficient: problem

E[(X + 5)2] = E[(X2+ 10X + 25)]

= E[X2] + 10E[X] + E[25]

= 50 + 10[5] + 25

= 125


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The joint density function ofXand Yis given by

Find the covariance and correlation coefficient ofXand Y

120 40

200( , )

0

x y

f x y

elsewhere


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Consider the joint density function

x>2; 0 < y< 1;

elsewhere;

Compute f(x), f(y), E[X], E[Y], E[XY], XY, XY.

3

16),(

x

yyxf

0),( yxf


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Moment

The kth

moment about the origin of a random variableXis

The kthmoment about the mean is

continuousisXifxfx

discreteisXifxfx

XEk

x

k

k

k

)(

)(

]['

kkk XEXEXE )()]([


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Moment

Moments are useful in characterizing some features of the

distribution

The first and the second moment about the origin are given by

We can write the mean and variance of a random variable as

The second moment about the mean is the variance.

The third moment about the mean is a measure of skewness of

a distribution.

22

2 ][ XE

]['1 XE ][' 2

2 XE

2

12

2 )'(' '1


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Moment Generating Function (MGF)

Moment-generating function is used to determine the moments of

distribution It will exist only if the sum or integral converges.

If a moment-generating function of X does exist, it can be used to

generate all the moments of that variable.


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m

i

i

tx

X xfetM i

1

)()(

m

i

i

tx

iX xfextM i

1

' )()(

m

i

i

txrr

X xfextM i

i

1

)( )()(


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][],...,[],[ 2 kXEXEXE

m

ii

rr

X xfxtM i1

)(

)()0(


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Find the moment-generating function of the binomial random

variableXand then use it to verify that and

First derivation, E[X]

Second derivation, E[X2]

Setting t = 0 we get

Therefore,

np npq2

nt

n

x

xnxt

n

x

xnxtx

X

qpe

qpe

x

n

qpx

netM

0

0

)(

The last sum isthe binomialexpansion of(pet+q)n

tntX peqpendt

tdM 1)(

tnttnttX eqpepeqpenenp

dttMd 12

2

2

1)(

npqpnpnp )1(2'22'

1

11][][ '22'

1

1 pnnpXEnpXE


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Conditional Expectation

If X and Y are jointly distributed R.V. then the conditionalexpectation of Y given X=x is given by

y xyfyxYE )|()|(

y

dyxyfyxYE )|()|(

X dan Y diskrit

X dan Y kontinu


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Conditional Expectation: latihan

101,x0),( yyxyxfCarilah )|( xyE

1y05.0)(

),(

)|(

x

yx

xf

yxf

xyf


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Conditional Expectation: latihan

Jika X dan Y memiliki joint distribusi maka

)(| YEXYEE YYX

| ( | ). ( )

( | ) ( )

( )

X Y

x

x y

Y

E E Y X E Y x f x dx

y f y x f x dydx

E Y

M


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Joint MGF

Definisi : Joint MGF dari vector random adalah),...,,( 21 kXXXX

k

i

iiXtEtM1

exp)(X

Dimana dan for h > 0),...,( 1 kttt hth i

4. nilai harapan dan mgf bersamaa

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