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7/24/2019 10 S241 Moment Generating Functions
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Moment Generating Functions
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0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
( )f x
x
0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
( )f x
x
1
b a
a b
( )f x
x
Continuous Distributions
The Uniform distribution from a to b
( )1
0 otherwise
a x bf x b a
=
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The Norma distribution
!mean, standard de"iation #
( )( ) 2
221
2
x
f x e
=
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0
0.1
0.2
-2 0 2 4 6 8 10
The $%&onentia distribution
( )0
0 0
xe xf x
x
=
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Weibull distributionwith &arameters and.
( )Thus 1x
F x e
=
( ) ( ) 1and 0x
f x F x x e x
= =
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The 'eibu densit()f!x#
0
0.1
0.2
0.3
0.4
0.5
0.*
0.+
0 1 2 3 4 5
!, 0.5) , 2#
!, 0.+) , 2#
!, 0.-) , 2#
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The Gamma distribution
et the continuous random "ariabeX ha"e
densit( function/
( ) ( )1
0
0 0
x
x e xf x
x
=
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$%&ectation of functions of
andom ariabes
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X is discrete
( ) ( ) ( ) ( ) ( )i ix i
E g X g x p x g x p x = =
( ) ( ) ( )E g X g x f x dx
=
X is continuous
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Moments of andom ariabes
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( )k
k E X =
( )
( )
if is discrete
if is continuous
k
x
k
x p x X
x f x dx X
=
The kthmoment ofX.
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the kthcentralmomentof X
( )0k
k E X =
( ) ( )
( ) ( )
if is discrete
if is continuous
k
x
k
x p x X
x f x dx X
=
where=1
= E!X# , the first moment ofX .
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ues for e%&ectation
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Rules/
[ ]1. where is a constantE c c c=[ ] [ ]2. where ) are constantsE aX b aE X b a b+ = +
( ) ( )20
23. "ar X E X = = ( ) ( )
22 2
2 1E X E X = =
( ) ( )24. "ar "ar aX b a X + =
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Moment generating functions
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Moment Generating function of a R.V.X
( )
( )
( )
if is discrete
if is continuous
tx
xtX
Xtx
e p x X
m t E ee f x dx X
= =
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Examples
1. The inomia distribution !&arametersp, n#
( ) ( )tX txXx
m t E e e p x = =
( )0
1n
n xtx x
x
ne p p
x
=
=
( ) ( )0 0
1n n
x n xt x n x
x x
n ne p p a b
x x
= =
= =
( ) ( )1nn ta b e p p= + = +
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( ) 0)1)2)
x
p x e xx
= = K
The moment generating function ofX , mX!t# is/
2. The oisson distribution !&arameter #
( ) ( )tX tx
X
xm t E e e p x = = 0
xn
tx
xe ex
== ( )
0 0
using
t
xt x
e u
x x
e ue e e e
x x
= =
= = =
( )1te
e
=
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( )
0
0 0
xe x
f x x
=
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( )
2
21
2
x
f x e
=
The moment generating function ofX , mX!t# is/
4. The 6tandard Norma distribution !, 0) , 1#
( ) ( )tX tx
Xm t E e e f x dx
= =
2 22
1
2
x tx
e dx
=
2
21
2
xtxe e dx
=
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( )2 2 2 22 2
2 2 21 12 2
x tx t x tx t
Xm t e dx e e dx
+
= =
'e wi now use the fact that( ) 2
221
1 for a 0)
2
x b
ae dx a b
a
= >'e ha"e
com&eted
the s7uare
( ) 22 2
2 2 21
2
x tt t
e e dx e
= =
This is 1
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( ) ( )1
0
0 0
x
x e xf x
x
=
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'e use the fact
( )1
0
1 for a 0) 0
a
a bx
bx e dx a ba
= > >
( ) ( )
( )1
0
t x
Xm t x e dx
=
( )
( )
( )( )1
0
t xtx e dx
tt
= =
$7ua to 1
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ro&erties of
Moment Generating Functions
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1. mX!0# , 1
( ) ( ) ( ) ( ) ( )0
) hence 0 1 1
tX X
X Xm t E e m E e E
= = = =
( )"# Gamma 8ist9n Xm tt
=
( )2
2i"# 6td Norma 8ist9nt
Xm t e=
( )iii# $%&onentia 8ist9n Xm tt
=
( ) ( )1
ii# oisson 8ist9nte
Xm t e
=( ) ( )i# inomia 8ist9n 1
nt
Xm t e p p= +
Note: the moment generating functions of the foowing
distributions satisf( the &ro&ert( mX!0# , 1
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( ) 2 33212. 12 3
kkXm t t t t t
k
= + + + + + +K K
'e use the e%&ansion of the e%&onentia function/
2 3
12 3
ku u u ue u
k= + + + + + +K K
( ) ( )tXXm t E e=2 3
2 312 3
kkt t tE tX X X X
k
= + + + + + +
K K
( ) ( ) ( ) ( )2 32 31
2 3
k kt t ttE X E X E X E X k
= + + + + + +K K
2 3
1 2 31
2 3
k
k
t t tt
k
= + + + + + +K K
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( ) ( ) ( )0
3. 0k
k
X X kk
t
dm m t
dt
=
= =
Now( ) 2 33211
2 3
kkXm t t t t t
k
= + + + + + +K K
( ) 2 1321 2 32 3
kkXm t t t kt
k
= + + + + +K K
( )2 13
1 22 1
kkt t tk
= + + + + +
K K
( ) 1and 0Xm =
( )( )
242 3
2 2
kkXm t t t t
k
= + + + + +
K K
( ) 2and 0Xm =( )
( )continuing we find 0k
X km =
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( ) ( )i# inomia 8ist9n 1n
t
Xm t e p p= +
ro&ert( 3 is "er( usefu in determining the moments of a
random "ariabeX.
Examples
( ) ( ) ( )1
1n
t t
Xm t n e p p pe
= +
( ) ( ) ( )1
0 0 10 1n
Xm n e p p pe np
= + = = =
( ) ( )( ) ( ) ( )2 1
1 1 1n n
t t t t t
Xm t np n e p p e p e e p p e
= + + +
( ) ( )( ) ( )( )
2
2
1 1 1
1 1
nt t t t
nt t t
npe e p p n e p e p p
npe e p p ne p p
= + + + = + +
[ ] [ ] 2 2 21np np p np np q n p npq = + = + = + =
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( ) ( )1
ii# oisson 8ist9nte
Xm t e
=
( ) ( ) ( )1 1t te e tt
Xm t e e e
+ = =
( ) ( ) ( ) ( )1 1 2 121
t t te t e t e tt
Xm t e e e e
+ + + = + = +
( ) ( ) ( )1 2 12 2 1
t te t e t t t
Xm t e e e e
+ + = + + +
( ) ( )1 2 12 3t te t e t te e e
+ + = + +
( ) ( ) ( )1 3 1 2 13 23
t t te t e t e t e e e
+ + += + +
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( )
( )0 1 0
1
0 e
X
m e
+
= = =( )
( ) ( )0 01 0 1 02 22 0
e e
Xm e e
+ += = + = +
( )
3 0 2 0 0 3 2
3 0 3 3
t
Xm e e e = = + + = + +
To find the moments we set t , 0.
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( )iii# $%&onentia 8ist9n Xm tt
=
( ) ( ) 1
X
d tdm t
dt t dt
= =
( ) ( ) ( ) ( )2 2
1 1t t = =
( ) ( ) ( ) ( ) ( )3 3
2 1 2Xm t t t = =
( ) ( ) ( ) ( ) ( ) ( )4 4
2 3 1 2 3Xm t t t = =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )5 54
2 3 4 1 4Xm t t t = =
( )
( ) ( ) ( )
1
kk
Xm t k t
=
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Thus
( ) ( )
2
1
1
0Xm
= = = =
( ) ( )3
2 2
20 2Xm
= = =
( ) ( ) ( ) ( )1
0 kk
k X k
km k
= = =
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( ) 2 332112 3
kkXm t t t t t
k
= + + + + + +K K
The moments for the e%&onentia distribution can be cacuated in
an aternati"e wa(. This is note b( e%&anding mX!t# in &owers of t
and e7uating the coefficients of t
k
to the coefficients in/
( ) 2 31 1 111Xm t u u utt u
= = = = + + + +
L
2 3
2 31
t t t
= + + + +L
$7uating the coefficients of tk we get/
1 or
kkk k
k
k
= =
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( )2
2t
Xm t e=
Te moments for te standard normal distribution
'e use the e%&ansion of eu.2 3
0
1 2 3
k ku
k
u u u ue u
k k
=
= = + + + + + + L L
( ) ( ) ( ) ( ) ( )
2 2 2
22
2
2 3
2 2 2
21
2 3
t
kt t t
tXm t e
k= = + + + + + +L L
2 4 * 212 2 3
1 1 11
2 2 2 3 2
k
kt t t t
k
= + + + + + +L L
'e now e7uate the coefficients tkin/
( )
( )
2 22211
2 2
k kk kXm t t t t t
k k
= + + + + + + +K K K
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:f k is odd/ k, 0.
( )2
1
2 2 k kk k
=For e"en 2k/
( )2
2 or
2 k k
k
k
=
( )1 2 3 4 2
2 4Thus 0) 1) 0) 3
2 2 2 = = = = = =
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!ummar"
Moments
Moment generating functions
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Moments of Random Variables
( )kk E X =
( )
( )
if is discrete
if is continuous
k
x
k
x p x X
x f x dx X
=
Te moment generating function
( )( )
( )
if is discrete
if is continuous
tx
xtX
Xtx
e p x X m t E e
e f x dx X
= =
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Examples
( ) ( )1 0)1) 2) )n xxnp x p p x n
x
= =
K
1. The inomia distribution !&arametersp, n#
( ) ( ) ( )1n n
t t
Xm t e p p e p q= + = +
( ) 0)1)2)
x
p x e x
x
= = K
2. The oisson distribution !&arameter #
( ) ( )1te
Xm t e
=
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( )
0
0 0
xe xf x
x
=
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( ) ( )
1 0
0 0
xx e xf x
x
=
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1. mX!0# , 1
( ) 2 33212. 12 3
kkXm t t t t t
k
= + + + + + +K K
( ) ( ) ( )0
3. 0k
kX X kk
t
dm m tdt
=
= =
#roperties of Moment Generating $unctions
( ) 1i.e. 0Xm =
( ) 20Xm =
( ) 30 ) etcXm =
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et lX !t# , n mX!t# , the og of the moment generating
function
( ) ( )Then 0 n 0 n1 0X Xl m= = =
Te log of Moment Generating $unctions
( ) ( ) ( ) ( )( )1 XX X
X X
m tl t m t
m t m t
= =
( ) ( ) ( ) ( )
( )
2
2
X X X
X
X
m t m t m t l t
m t
=
( ) ( )( )1
0 0
0X
X
X
ml
m
= = =
( ) ( ) ( ) ( )
( )[ ]
2
2 2
2 12
0 0 0 0
0
X X X
X
X
m m ml
m
= = =
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Thus lX !t# , n mX!t# is "er( usefu for cacuating the
mean and "ariance of a random "ariabe
( )1. 0Xl =
( ) 22. 0Xl =
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Examples
1. The inomia distribution !&arametersp, n#
( ) ( ) ( )1n n
t t
Xm t e p p e p q= + = +
( ) ( ) ( )n n tX Xl t m t n e p q= = +
( ) 1 tX tl t n e pe p q = + ( )1
0Xl n p npp q = = =+
( ) ( ) ( )
( )2
t t t t
Xt
e p e p q e p e pl t n
e p q
+ =
+
( ) ( ) ( )
( )
2
20X
p p q p pl n npq
p q
+ = = =
+
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2. The oisson distribution !&arameter #
( ) ( )1te
Xm t e
=
( ) ( ) ( )n 1tX Xl t m t e= =
( ) tXl t e =
( ) tXl t e =
( )0Xl = =
( )2 0Xl = =
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3. The $%&onentia distribution !&arameter #
( )undefined
X tm t t
t
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4. The 6tandard Norma distribution !, 0) , 1#
( )
2
2t
X
m t e
=( ) ( )
2
2n tX Xl t m t = =
( ) ( )) 1
X X
l t t l t = =
( ) ( )2Thus 0 0 and 0 1X Xl l = = = =
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5. The Gamma distribution !&arameters ) #
( )Xm tt
=
( ) ( ) ( )n n nX Xl t m t t = =
( ) 1Xl tt t
= =
( ) ( ) ( ) ( )
( )
2
21 1Xl t t
t
= =
( ) ( )2 2
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*. The ;his7uare distribution !degrees of freedom #
( ) ( )2
1 2Xm t t
=
( ) ( ) ( )n n 1 22
X Xl t m t t
= =
( ) ( )1
22 1 2 1 2Xl t t t
= =
( ) ( ) ( ) ( )( )
2
2
21 1 2 2
1 2X
l t tt
= =
( ) ( )2
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Summary of Discrete Distributions
Name probability function p(x) Mean Variance
Momentgenerating
function MX(t)
DiscreteUniform
p(x) =1N
x=1,2,...,NN+1
2
N2-112
et
NetN-1et-1
Bernoullip(x) =
p x=1
q x=0
p pq q + pet
Binomialp(x) =
N
xpxqN-x
Np Npq (q + pet)N
Geometric p(x) =pqx-1 x=1,2,... 1p q
p2 pe
t
1-qet
NegativeBinomial p(x) =
x-1
k-1pkqx-k
x=k,k+1,...
k
p
kq
p2
pet
1-qetk
Poissonp(x) =
x
x!e- x=1,2,...
e(et-1)
Hypergeometric
p(x) =
A
x
N-A
n-x
N
n
n
A
N n
A
N
1-A
N
N-n
N-1
not useful
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Summary of Continuous Distributions
Nameprobability
density function f(x) Mean arianceMoment generating
function MX(t)
ContinuousUniform
=otherwise
bxaabxf
0
1#!
a+b
2 (b-a)2
12 ebt-eat
[b-a]t
Exponential