EE 511 Problem Set 3
1. The receiver of a communication system receives random variable
Y
which is defined as
Y
=
X
+
N
in terms of the input random variable
X
and the channel noise
N
.
X
takes on
the values

1
/
4 and 1
/
4 with
P
[
X
= 1
/
4] = 0
.
6. Let
f
N
(
n
) denote the pdf of the channel
noise and let
X
and
N
be independent. The receiver must decide for each received
Y
=
y
whether the transmitted
X
was

1
/
4 or 1
/
4. If
N
is uniform in (

1
/
2
,
1
/
2), (a) determine
f
Y
(
y

X
= 1
/
4),
f
Y
(
y

X
=

1
/
4) and
f
Y
(
y
) (b) determine the optimal rule such that the
probability of correct decision is maximised.
2. If random variables
X
and
Y
are related via
Y
=
g
(
X
) where
g
(
.
) is a monotonically increasing
function, show that their CDF’s satisfy
F
Y
[
g
(
α
)] =
F
X
(
α
).
3. Let
X
be a random variable with pdf
f
X
(
x
). Define
Y
=
g
(
X
) where
g
(
x
) =
x

x
 ≤
2

2
x <

2
2
x >
2
(i) Determine
f
Y
(
y
) in terms of
f
X
(
.
), and (ii) Sketch
f
Y
(
y
) when
X
is a uniform random
variable over the interval [3,3].
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 Spring '11
 TU
 Probability theory, 2k, FN, Rayleigh

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