20091107 Stochastics 3.9 TransformMethods
Transcript of 20091107 Stochastics 3.9 TransformMethods
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Transform Methods Chapter 3: Random Variables
College of Electrical & Mechanical EngineeringNational University of ciences & Technology !NUT"
EE#$%3 tochastic ystems
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Random Variables ' Transform Methods
Transform Methods
[ ]1 2 1 2( ) ( ) ( ) ( ) f x f x ω ω ∗ =F F F
(ogarithms are )sef)l comp)tational aids for performingm)ltiplications* )sing additions only
Transform methods held red)ce comp)tational effort e+g+
convol)tion may be performed in transform domain
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Random Variables ' Transform Methods
Characteristic Function
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,
Random Variables ' Transform Methods
Characteristic Function: Definition
( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫
E-pected val)e of af)nction of .
pdf and its characteristic f)nction form a )ni/)e 0o)rier
transform pair
pdf and its characteristic f)nction form a )ni/)e 0o)rier
transform pair
0o)rier Transform ofthe pdf
1nverse 0o)rier Transform of
Characteristic 0)nction gives pdf
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Random Variables ' Transform Methods
Example 3.47 (Exponential Random
aria!le"( )
( )
j X X
j x
X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
=
∫
pdf of e-ponential
random variable
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%
Random Variables ' Transform Methods
Characteristic Function for Discrete
Random aria!le( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫ iscrete random variable X
0o)rier Transform of the se/)ence0o)rier Transform of the se/)ence
1nteger#val)ed random variable X
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Random Variables ' Transform Methods
#eriodicit$ of Characteristic Function
( 2 ) 2 .1 j k j k j k j k j k e e e e eω π ω π ω ω += = =
5eriodicity of discrete variable5eriodicity of discrete variable
0o)rier Transform of the se/)ence
periodic 6ith period pi
0o)rier Transform of the se/)ence
periodic 6ith period pi
1nteger#val)ed random variable X
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Random Variables ' Transform Methods
Reco%erin& #ro!a!ilities from
Characteristic Function
0o)rier Transform of the se/)ence
periodic 6ith period pi
0o)rier Transform of the se/)ence
periodic 6ith period pi
1nverse 0o)rier transform
5robabilities5robabilities
Coefficients of 0o)rier eries of
periodic f)nction
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Random Variables ' Transform Methods
Moment Theorem
( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫ Moments can be comp)tedfrom Characteristic 0)nction
5roof:5roof: E-pand into po6er series j xe
ω
If power series converges, pdf and characteristic f)nction are completely
determined by moments of .
If power series converges, pdf and characteristic f)nction are completely
determined by moments of .
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Random Variables ' Transform Methods
Example 3.4' (Computin& Mean and
ariance usin& Moment Theorem"( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫ Characteristic f)nction
of e-ponential random
variableifferentiating once
Mean
ifferentiating once again
Mean#s/)are
Variance
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Random Variables ' Transform Methods
#ro!a!ilit$ eneratin& Function
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Random Variables ' Transform Methods
#ro!a!ilit$ eneratin& Function:
Definition
0
( )
( )
N N
k N
k
G z E z
p k z ∞
=
=
= ∑
E-pected val)e of af)nction of N
#Transform of the pmf
Relation bet6een Characteristic0)nction and 5robability
;enerating 0)nction
( ) ( ) j N N G e ω ω Φ =
j z e
ω =
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Random Variables ' Transform Methods
Moment Theorem (#F"( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫ Moments can be comp)tedfrom 5robability ;enerating
0)nction !5;0"
0irst derivative of 5;0
econd derivative of 5;0
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8,
Random Variables ' Transform Methods
Moment Theorem (#F" ) continued( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫ Mean
Variance'' ' ' 2
[ ] (1) (1) ( (1)) N N N VAR N G G G= + −
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Random Variables ' Transform Methods
Example 3.*+ (Computin& Mean and
ariance usin& Moment Theorem"
( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫
5robability ;enerating
0)nction of 5oisson
Random Variable
ifferentiating once
Mean
ifferentiating once again
Variance
pmf of poisson random variable
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8%
Random Variables ' Transform Methods
,aplace Transform of pdf
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84
Random Variables ' Transform Methods
,aplace Transform of pdf
( )
( )
j X X
j x X
E e
f x e dx
ω
ω
ω
∞
−∞
Φ =
= ∫
E-pected val)e of a
f)nction of .
Moment TheoremMoment Theorem
(aplace Transform of
the pdf