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Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.432
Stochastic
Processes,
Detection
and
Estimation
Problem
Set
3
Spring
2004
Issued:
Thursday,
February
19,
2004
Due:
Thursday,
February
26,
2004
Reading:
This
problem
set:
Chapter
2,
Chapter
3
through
Section
3.2.4
Next:
Sections
1.7,
3.2.5,
3.3.1,
3.3.2
Problem
3.1
Suppose
x
and
y
are
the
random
variables
from
Problem
Set
2,
problem
2.4.
Their
joint
density,
depicted
again
below
for
convenience,
is
constant
in
the
shaded
area
x
y
p
(
x
,
y
)
1
2
1
2
and
0
elsewhere.
x,y
1
2
1
2
(a)
In
the
(
P
D
, P
F
)
plane
sketch
the
operating
characeristic
of
the
likelihood
ratio
test
for
this
problem.
Also,
indicate
on
this
plot
the
region
consisting
of
every
(
P
D
, P
F
)
value
that
can
be
achieved
using
some
decision
rule.
2
(b)
Is
the
point
corresponding
to
P
D
=
3
,
P
F
=
5
in
this
region?
If
so,
describe
a
6
test
that
achieves
this
value.
If
not,
explain.
1
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View Full DocumentProblem
3.2
We
observe
a
random
variable
y
and
have
two
hypotheses,
H
0
and
H
1
,
for
its
prob
ability
density.
In
particular,
the
probability
densities
for
y
under
each
of
these
two
hypotheses
are
depicted
below:
p
y

H
0
(
y

H
0
)
=
1
,
0
y
1
p
y

H
1
(
y

H
1
)
=
3
y
2
,
0
y
1
3
1
y
y
1
1
(a)
Find
the
decision
rule
that
maximize
P
D
subject
to
the
constraint
that
P
F
1
.
(b)
Determine
the
value
of
P
D
for
the
decision
rule
speci±ed
in
part
(a).
Problem
3.3
Let
k
denote
the
uptime
of
a
communications
link
in
days.
Given
that
the
link
is
functioning
at
the
beginning
of
a
particular
day
there
is
probability
q
that
it
will
go
down
that
day.
Thus,
the
uptime
of
the
link
k
(in
days)
obeys
a
geometric
distribution.
It
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 Spring '04
 Prof.GregoryWornell
 Electrical Engineering

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