8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

1/30

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

1 of 45

EXPECTED VALUE OF A FUNCTION &

JOINT MGF

EXPECTED VALUE OF A FUNCTION &

JOINT MGF

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

2/30

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

2 of 45

Sessions target

Nilai harapan bersama (Joint expected value)

Kovarian (Covariance)

Korelasi (Correlation)

Nilai Harapan Bersyarat (Conditional Expected Value)

MGF Bersama (Joint MGF)

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

3/30

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

3 of 45

Expected value (Review)

Expected value of discrete R.V.

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

1

1 0 1 1 0

0

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

x

E X x p p p p p p p

12 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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

4/30

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

4 of 45

Expected value (Review)

21

21( ) ;2

xX f x e x

2 21 1

2 2

01 1( ) 2 02 2

x xE X x e dx x e dx

2 21 1

2 2 22 2

0

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 ka

x e dx k a

Expected value of continuous R.V.

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

5/30

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

5 of 45

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

6/30

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

6 of 45

Contoh

A joint pdf of two random variablesX, Yis given by

otherwise

yxforyxf

xy 51,40

0),( 96

4 5

0 1

4 5 2

0 1

54 2 2

0 1

4 2

0

43

0

[ ] ( , )

96

96

192

8

824 3

x y

E X x f x y d xd y

xy

x d yd x

x yd y d x

x yd x

xd x

x

4 5

0 1

4 5 2

0 1

54 3

0 1

4

0

42

0

[ ] ( , )

96

96

288

31

72

31 31144 9

x y

E Y y f x y dxdy

xy

y dydx

xydydx

xydx

xdx

x

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

7/30

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

7 of 45

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-rataXu 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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

8/30

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

8 of 45

IfXand Yhas a jointpdf f(x,y) and if is afunction ofX and Y, maka

Discrete random variables:

Continuous random variables:

( ) ( , ) ( , )x y

E H u x y f x y

dxdyyxfyxuHE

),(),(...)(

),( YXuH

Expected value of a function

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

9/30

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

9 of 45

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

Expected value of a function

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

10/30

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

10 of 45

Example

Joint pdf of two random variablesX, Yis given by

Let

The expected value of His

960 4, 1 5

( , )0

xy x yf x y for

otherwise

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

11/30

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

11 of 45

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 Yx 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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

12/30

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

12 of 45

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

13/30

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

13 of 45

Some properties of Covariance

If Xand Yare random variables and a andb are constant, then

IfXand Yare independent, then

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

),(),( YXCovYbXaCov )(),(),( XaVarXXaCovbaXXCov

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

14/30

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

14 of 45

Covariance: Example

A joint pdf of two random variablesX, Yis given by

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

otherwise

yxforyxf

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

xyxy dxdy x y dydx

0

9

31

3

8

27

248][].[][

YEXEXYEXY

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

15/30

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

15 of 45

Variance of a function

Some properties of variance

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

IfXand Yare independent random variables, thenVar[X Y] = Var[X] + Var[Y]

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

where a, b are constants

222222 ])[(][][])[( XEXEXEXEX

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

16/30

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

16 of 45

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 = 0 does

not imply independence

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

17/30

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

17 of 45

Correlation Coefficient: Problem

Assume the lengthXin minutes of a particular type oftelephone conversation is a random variable with

probability 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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

18/30

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

18 of 45

Correlation Coefficient: Problem

5

255

1

255

5

1

5155

5

1

55

5

1

51][

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:

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

19/30

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

19 of 45

Correlation Coefficient: Problem

505

250

2505055

1

2551055

1

55105

5

1

1055

1

5

1][

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

20/30

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

21/30

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

21 of 45

Correlation Coefficient: problem

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

el sewh ere

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

22/30

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

22 of 45

Correlation Coefficient: problem

In order to calculate the covariance, we need the values of E[XY],E[X], and E[Y]. First compute the marginal pdf ofXand Y

20 < x < 40 20 < y < 40

Then from the marginal pdf calculate E[X], and E[Y]. The E[XY] iscalculated from the joint pdf

x

dyxgx

005.02.0200

1)(40

1.0005.0

2001)(

20

y

dxyhy

67.26

)005.02.0(][40

20

dxxxXE33.33

)1.0005.0(][40

20

dyyyYE

9002

800

200

1

200

1][

40

20

3

4040

20

dxx

x

dydxxyXYEx

Jadi,XY

= Cov[XY]

Cov[XY] = E[XY] E[X]*E[Y]

= 900 26.67(33.33)= 11.09

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

23/30

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

23 of 45

Correlation Coefficient: problem

In order to calculate the correlation coefficient, we need the values of

E[X2], E[Y2], Var [X] and Var[Y].

X2 = Var[X] = E[X2] E[X]2 = 733.33 (26.67)2 = 22.204

Y2 = Var[Y] = E[Y2] E[Y]2 = 1133.33 (33.33)2 = 22.244

Thus the correlation coefficient is

33.733

)005.02.0(][ 4020

22

dxxxXE33.1133

)1.0005.0(][40

20

22

dyyyYE

[ ] 11.090.499

[ ] [ ] 22.204 22.244

XY

X Y

C o v E X

V ar X Var Y

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

24/30

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

24 of 45

Correlation Coefficient: problem

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

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

25/30

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

25 of 45

Moment

The kth moment about the origin of a random variableXis

The kth moment about the mean is

continuousisXifxfx

discreteisXifxfxXE

k

x

k

k

k

)(

)(]['

kk

k XEXEXE )()]([

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

26/30

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

26 of 45

Correlation Coefficient: problem

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

2122 )'(' '1

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

27/30

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

27 of 45

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.

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

28/30

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

28 of 45

Moment Generating Function (MGF)

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

)( )()(

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

29/30

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

29 of 45

Moment Generating Function (MGF)

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

m

i

i

rr

X xfxtM i1

)( )()0(

8/13/2019 Pertemuan 3-4 - Nilai Harapanxxxx Dan MGF

30/30