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Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.432
Stochastic
Processes,
Detection
and
Estimation
Problem
Set
1
Spring
2004
Issued:
Tuesday,
February
3,
2004
Due:
Tuesday,
February
10,
2004
Reading:
For
this
problem
set:
Chapter
1
of
course
notes,
through
Section
1.6,
Appendix
1.A
Next:
Chapter
2,
through
Section
2.5.1
Problem
1.1
A
random
variable
x
has
probability
distribution
function
P
x
(
x
)
=
[1
−
exp(
−
2
x
)]
u
(
x
)
where
u
(
·
)
is
the
unitstep
function.
(a)
Calculate
the
following
probabilities:
Pr [
x
1]
,
Pr [
x
±
2]
,
Pr [
x
=
2]
.
(b)
Find
p
x
(
x
),
the
probability
density
function
for
x
.
(c)
Let
y
be
a
random
variable
obtained
from
x
as
follows:
0
x
<
2
y
=
.
1
x
±
2
Find
p
y
(
y
),
the
probability
density
function
for
y
.
Problem
1.2
Let
x
and
y
be
independent
identically
distributed
random
variables
with
common
density
function
p
(
)
=
1
0
0
1
otherwise
.
Let
s
=
x
+
y
.
(a)
Find
and
sketch
p
s
(
s
).
(b)
Find
and
sketch
p
x

s
(
x s
)
vs.
x
with
s
viewed
as
a
known
parameter.

1
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(c)
The
conditional
mean
of
x
given
s
=
s
is
E
[
x
s
=
s
] =
x
p
x

s
(
x s
)
dx.

−

Find
E
[
x
s
=
0
.
5].

(d)
The
conditional
mean
of
x
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 Spring '10
 HyungdongShin
 Electrical Engineering

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