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
Expected value of a function
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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
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
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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
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Example
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Example
1
( , ) 16f x y xy
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Example
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Example
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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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Correlation Coefficient: Problem
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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Correlation Coefficient: Problem
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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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
elsewhere
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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
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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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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
)( )()(
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Moment Generating Function (MGF)
][],...,[],[ 2 kXEXEXE
m
ii
rr
X xfxtM i1
)(
)()0(
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Moment Generating Function (MGF)
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
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