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FALL 2011
Instructor: Sujay Sanghavi
sanghavi@mail.utexas.edu
Homework 10 Solution
Problem 1
A radar works by transmitting a pulse, and seeing if there is an echo. Ideally, an echo means object is
present, and no echo means no object. However, some echoes might get lost, and others may be generated
due to other surfaces. To improve accuracy, a radar transmits
n
pulses, where
n
is a ﬁxed number, and sees
how many echoes it gets. It then makes a decision based on this number.
Let
p
1
be the probability of an echo for a single pulse when there is no object, and
p
2
be the probability
when there is an object. Assume
p
1
< p
2
. What is the maxlikelihood estimation rule for whether the object
is present or absent?
Sol
:
Let
X
be the random variable denoting the number of pluses where echoed and detected on the
n
. Let
H
0
be the hypothesis that there is no object,
H
1
be that there is an object. Then
p
X
(
x
;
H
0
) =
(
n
x
)
p
x
1
(1

p
1
)
n

x
,
p
X
(
x
;
H
1
) =
(
n
x
)
p
x
2
(1

p
2
)
n

x
.
So if using the ML estimation rule, we need
L
(
x
) =
p
X
(
x
;
H
0
)
p
X
(
x
;
H
1
)
= 1
,
which further implies
x
=
ln(1

p
2
)

ln(1

p
1
)
ln(
p
1
)

ln(
p
2
) + ln(1

p
2
)

ln(1

p
1
)
n.
So that we determine that an object is present if
x
exceeds the expression on the right hand side of the
inequality above. If
x
is smaller than this expression, then we decide that there is no object present.
Problem 2
A source emits a random number of photons
K
each time that it is triggered. We assume that the PMF of
K
is
p
K
(
k
;
θ
) =
c
(
θ
)
e

θk
,
k
= 0
,
1
,
2
,....
where
θ
is the inverse of the temperature of the source and
c
(
θ
)
is a normalization factor. We also assume
that the photon emissions each time that the source is triggered are independent. We want to estimate the
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