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



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



3


Characteristic Function



,


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



2


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



%


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




#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



1nteger#val)ed random variable X


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Reco%erin& #ro!a!ilities from

Characteristic Function





1nverse 0o)rier transform

5robabilities5robabilities

Coefficients of 0o)rier eries of

periodic f)nction




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 .




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




#ro!a!ilit$ eneratin& Function




#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 =



83


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



8,


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= + −



82


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



8%


,aplace Transform of pdf



84


,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

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