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Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.432
Stochastic
Processes,
Detection
and
Estimation
Problem
Set
7
Spring
2004
Issued:
Thursday,
April
1,
2004
Due:
Thursday,
April
8,
2004
Reading:
For
this
problem
set:
Sections
4.34.6,
4.A,
4.B,
Chapter
5
Next:
Chapter
5,
Sections
6.1
and
6.3
Problem
7.1
Let
N
(
t
)
be
a
Poisson
counting
process
on
t
∀
0
with
rate
�
.
Let
{
y
i
}
be
a
collection
of
statistically
independent,
identicallydistributed
random
variables
with
mean
and
variance
E
[
y
i
]
=
m
y
var
y
i
=
ω
y
2
,
respectively.
Assume
that
the
{
y
i
}
are
statistically
independent
of
the
counting
pro
cess
N
(
t
)
and
define
a
new
random
process
y
(
t
)
on
t
∀
0
via
⎧
⎧
0
N
(
t
)
=
0
N
(
t
)
y
(
t
)
=
⎨
.
⎧
⎧
y
i
N
(
t
)
>
0
i
=1
(a)
Sketch a
typical
sample
function
of
N
(
t
)
and
the
associated
typical
sample
function
of
y
(
t
).
(b)
Use
iterated
expectation
(condition
on
N
(
t
)
in
the
inner
average)
to
find
E
[
y
(
t
)]
and
E
[
y
2
(
t
)]
for
t
∀
0.
(c)
Prove
that
y
(
t
)
is
an
independentincrements
process
on
t
∀
0
and
use
this
fact
to
find
the
covariance
function
K
yy
(
t,
s
)
for
t,
s
∀
0.
Problem
7.2
(practice)
(a)
Let
x
(
t
)
be
an
independent
increments
process
on
t
∀
0
whose
covariance
func
tion
is
K
xx
(
t,
s
),
for
t,
s
∀
0.
Show
that
K
xx
(
t,
s
)
=
var[
x
(min(
t,
s
))]
for
t,
s
∀
0
.
1
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(b)
Suppose
x
(
t
)
in
part
(a)
has
stationary
increments.
Show
that
m
x
(
t
)
=
at
+
b,
for
t
∀
0
,
and
K
xx
(
t,
s
)
=
c
min(
t,
s
)
+
d
for
t,
x
∀
0
where
a,
b
are
constants
and
c,
d
are
nonnegative
constants.
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 Spring '04
 Prof.GregoryWornell
 Electrical Engineering, Stochastic process, Stationary process, covariance function

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