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
%LQRL53¶
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
Enter the password to open this PDF file:
File name:
-
File size:
-
Title:
-
Author:
-
Subject:
-
Keywords:
-
Creation Date:
-
Modification Date:
-
Creator:
-
PDF Producer:
-
PDF Version:
-
Page Count:
-
Preparing document for printing…
0%
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο