Random Processes - Robert Marks.org

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copyright Robert J. Marks II

EE 505

Random Processes
-

Example Random Processes

copyright Robert J. Marks II

Example RP’s

Example Random Processes


Gaussian

Recall Gaussian pdf






Let
X
k
=X(t
k
) , 1


欠


渮

Then if, for all
n
, the
corresponding pdf’s are Gaussian, then the
RP is Gaussian.

The Gaussian RP is a useful model in signal
processing.



)
(
K
)
(
2
1
2
/
1
2
/
1
|
K
|
2
1
)
(
m
x
m
x
n
X
T
e
x
f












copyright Robert J. Marks II

Flip Theorem

Let A take on values of
+1

and
-
1

with equal
probability

Let X(t) have mean m(t) and
autocorrelation R
X

Let Y(t)=AX(t)

Then Y(t) has mean zero and
autocorrelation R
X

What about the autocovariances?




copyright Robert J. Marks II

Multiple RP’s





X(t) & Y(t)


Independence

(
X(t
1
), X(t
2
), …, X(t
k
)
)


is independent to




(
Y(

1
), Y(


2
), …, Y(


j
)
)


…for All choices of
k

and
j

and

all sample locations



copyright Robert J. Marks II

Multiple RP’s





X(t) & Y(t)


Cross Correlation

R
XY
(t,


⤽䕛堨)⥙(

)
崠


Cross
-
Covariance

C
XY
(t,


⤽)R
XY
(t,


⤠
-

䕛E⡴(崠䕛](

)
崠


Orthogonal:

R
XY
(t,


⤠㴠0


Uncorrelated:

C
XY
(t,


⤠㴠=


Note: Independent

啮捯U牥污l敤Ⱐ扵琠湯琠
the converse.


copyright Robert J. Marks II

Example RP’s

Multiple

Random Process
Examples


Example

X(t) = cos(

琫

)
Ⱐ†⡴(㴠獩s(

琫

)
Ⱐ

䉯瑨⁡牥⁺敲漠敡渮

䍲潳猠䍯牲敬慴e潮㴿




p.338

copyright Robert J. Marks II

Example RP’s

Multiple

Random Process Examples


Signal + Noise

X(t) = signal, N(t) = noise

Y(t) = X(t) + N(t)

If X & N are independent,R
XY
=?


p.338

Note: also, var Y = var X + var N




N
var
X
var
SNR

copyright Robert J. Marks II

Example RP’s

Multiple

Random Process Examples (cont)


Discrete time RP’s

X[n]

Mean

Variance

Autocorrelation

Autocovariance


Discrete time i.i.d. RP’s


Bernoulli RP’s

%LQRL53¶
p.340


Binary vs. Bipolar


Random Walk

p.341
-
2




copyright Robert J. Marks II

Autocovariance of Sum Processes

X[k]
’s are iid.





Autocovariance=?







n
k
n
]
k
[
X
S
1
X
n
]
S
[
E
n

)
X
var(
n
]
S
var[
n

copyright Robert J. Marks II

Autocovariance of Sum Processes







When
i=j
, the answer is var(
X
). Otherwise, zero.

How many cases are there where
i = j?

















































k
j
j
n
i
i
k
n
k
k
n
n
S
X
X
X
X
E
X
k
S
X
n
S
E
S
S
S
S
E
k
n
C
1
1
)
(
)
(
)
)(
(
)
)(
(
)
,
(
)
X
var(
)
k
,
n
min(
)
k
,
n
(
C
S


)
k
,
n
min(
copyright Robert J. Marks II

Autocovariance of Sum Processes

For Bernoulli sum process,




For Bipolar case




pq
)
k
,
n
min(
)
k
,
n
(
C
S

pq
)
X
var(

pq
)
X
var(
4

pq
)
k
,
n
min(
)
k
,
n
(
C
S
4

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process


Place n points randomly on line of length T




T
t
p
;
q
p
k
n
]
s
int
po
k
Pr[
k
n
k












T

t



Choose any subinterval of length t.



The probability of finding k points on the
subinterval is

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)


The Poisson approximation: For k big and p small…

!
)
/
(
!
)
(
]
points

Pr[
/
k
T
nt
e
k
np
e
q
p
k
n
k
k
T
nt
k
np
k
n
k














copyright Robert J. Marks II

Continuous Random Processes

The Poisson Approximation…


For n big and p small (implies k << n since p


欯渼㰱k

!
)
(
k
np
e
q
p
k
n
k
np
k
n
k











!
!
)
1
)...(
2
)(
1
(
)!
(
!
!
k
n
k
k
n
n
n
n
k
n
k
n
k
n
k
















n
p
n
k
n
k
n
e
p
p
q
)
(
)
1
(
)
1
(








Here’s why…

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)







Let n




獵捨⁴桡琠

㵮⽔=㴠晲敱略湣礠潦⁰潩湴猠
牥r慩湳a捯湳c慮琮




!
)
(
]

interval
on

points

Pr[
k
t
e
t
k
k
t




!
)
/
(
!
)
(
]
points

Pr[
/
k
T
nt
e
k
np
e
q
p
k
n
k
k
T
nt
k
np
k
n
k














copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)




This is a Poisson process with parameter



潣捵牲敮捥猠灥爠畮楴⁴業e


Examples: Modeling


Popcorn


Rain (Both in space and time)


Passing cars


Shot noise


Packet arrival times


!
k
)
t
(
e
]
t
k
Pr[
k
t





interval
on

points

t

copyright Robert J. Marks II

Continuous Random Processes

Poisson Counting Process


Poisson Points


!
k
)
t
(
e
]
k
)
t
(
X
Pr[
k
t





)
t
(
X
copyright Robert J. Marks II

Continuous Random Processes

Recall for Poisson RV with parameter a




Poisson Counting Process Expected Value is
thus


t
)]
t
(
X
[
E


!
k
)
a
(
e
]
k
X
Pr[
k
a



a
)
X
var(
X


copyright Robert J. Marks II

Continuous Random Processes

The Poisson Counting Process is independent
increment process. Thus, for




琠慮


椬






)!
(
)
(
!
]
)
(
)
(
Pr[
]
)
(
Pr[
]
)
(
)
(
,
)
(
Pr[
]
)
(
,
)
(
Pr[
)
(
i
j
e
t
i
e
t
i
j
t
X
X
i
t
X
i
j
t
X
X
i
t
X
j
X
i
t
X
t
i
j
t
i





























copyright Robert J. Marks II

Continuous Random Processes

Autocorrelation: If


㸠
























t
t
t
t
)
t
(
t
)
t
(
X
E
)
t
(
X
)
(
X
E
)
t
(
X
E
)
t
(
X
E
)
t
(
X
)
(
X
)
t
(
X
E
)
t
(
X
)
t
(
X
)
(
X
)
t
(
X
E
)
(
X
)
t
(
X
E
)
,
t
(
R
X






























2
2
2
2
2
)
,
t
min(
t
)
,
t
(
R
X







2
t




copyright Robert J. Marks II

Continuous Random Processes

Autocovariance of a Poisson sum process










)
,
t
min(
t
)
,
t
min(
t
)
(
X
E
)
t
(
X
E
)
,
t
(
R
)
,
t
(
C
X
X

















2
copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process


Random telegraph signal


)
t
(
X
Poisson Points

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal













|
t
|
X
e
)
,
t
(
C






2
|
|
2
)]
(
[
t
e
t
X
E



PROOF…

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t>0,




















odd

is

0
on

points

of
number
Pr


even

is

0
on

points

of
number
Pr
]
1
)
(
Pr[
)
1
(
1
)
(
Pr
1
)]
(
[
,t)
(
,t)
(
t
X
t
X
t
X
E











copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t>0,


















t
e
t
t
e
,t)
(
t
t





cosh
...
!
4
)
(
!
2
)
(
1
even

is

0
on

points

of
number
Pr
4
2





















copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t>0.

Similarly…















)
sinh(
...
!
5
)
(
!
3
)
(
odd

is

0
on

points

of
number
Pr
5
3
t
e
t
t
t
e
,t)
(
t
t





















copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t>0.

Thus





















0
;
)
sinh(
)
cosh(
odd

is

0
on

points

of
number
Pr


even

is

0
on

points

of
number
Pr
]
1
)
(
Pr[
)
1
(
1
)
(
Pr
1
)]
(
[
2

















t
e
t
t
e
,t)
(
,t)
(
t
X
t
X
t
X
E
t
t




|
|
2
)]
(
[
t
e
t
X
E



For all t…

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

,









X(t)

X(

)

-
1

1

1

-
1













1
)
(
Pr
1
)
(
|
1
)
(
Pr

1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr

1
)
(
,
1
)
(
Pr
]
1
)
(
)
(
Pr[



























X
X
t
X
X
X
t
X
X
t
X
X
t
X
X
t
X
copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

,


















)
(
cosh
even

is

)
,
(
on

points

of
number
Pr
1
)
(
|
1
)
(
Pr
1
)
(
|
1
)
(
Pr
)
(


















t
e
t
X
t
X
X
t
X
t




























e
e
t
X
X
t
X
X
t
X
t
)
cosh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr
)
(
Thus…

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

,

















)
cosh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X









And…









)
sinh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X














X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

.

Onward…


















)
(
sinh
odd

is

)
,
(
on

points

of
number
Pr
1
)
(
|
1
)
(
Pr
1
)
(
|
1
)
(
Pr
)
(



















t
e
t
X
t
X
X
t
X
t
copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

.

















)
cosh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X











And…









)
sinh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X












X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

.

















)
cosh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X











And…









)
sinh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr







t
e
t
X
X
t
X
X
t
X












X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes


Random telegraph signal. For t >

.



























)
sinh(
)
(
cosh
)
cosh(
)
(
sinh
)
sinh(
)
(
sinh
)
cosh(
)
(
cosh
1
)
(
)
(
Pr
1
1
)
(
)
(
Pr
1
)
(
)
(
)
,
(




















t
t
t
t
X
e
t
e
t
e
t
e
t
X
t
X
X
t
X
X
t
X
E
t
R




















X(t)

X(

)

-
1

1

1

-
1

In general…

|
|
2
)
(
)
(
)
,
(
)
,
(










t
X
X
e
X
t
X
t
R
t
C
copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process


Poisson point process, Z(t)

Let X(t) be a Poisson sum process. Then









pp.352


)
S
t
(
)
t
(
X
dt
d
)
t
(
Z
n
n





)
t
(
Z
Poisson Points

copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process


Shot Noise, V(t)

Z(t)



V(t)












pp.352


)
S
t
(
h
)
t
(
V
n
n



Poisson Points

h(t)

)
t
(
V
copyright Robert J. Marks II

Continuous Random Processes

Wiener Process


Assume bipolar Bernoulli sum process with jump
bilateral height h and time interval



E[X(t)]=0; Var X(n

⤠㴠㑮灱栠
2

= nh
2



Take limit as h


〠慮搠




〠步数0湧n


㴠栠
2

/


constant and t = n

.


Then
Var X(t)

†





By the central limit theorem, X(t) is Gaussian with zero
mean and
Var X(t)
=







We could use any zero mean process to generate the
Wiener process.

p.355

copyright Robert J. Marks II

Continuous Random Processes

Wiener Processes:


=
1


copyright Robert J. Marks II

Continuous Random Processes

Wiener processes in finance

S= Price of a Security.


㴠楮晬慴楯湡特a景牣攮†

If there is no risk…interest earned is proportional to investment.



Solution is

With “volatility”

, we have the most commonly used model in
finance for a security:




V(t) is a Wiener process.

S
dt
dS
dt
)
t
(
S
)
t
(
dS





t
e
S
)
t
(
S

0

)
t
(
dV
)
t
(
S
dt
)
t
(
S
)
t
(
dS