EE 178
Handout #14
Probabilistic Systems Analysis
Thursday, February 19, 2009
Homework #6
Due Thursday, February 26
1.
Uncorrelation vs. Independence.
Let
X
and
Y
be random variables with joint pdf
f
X,Y
(
x, y
) =
braceleftbigg
c
if 0
≤ 
x
 ≤ 
y

,
0
≤ 
y
 ≤
1
0
otherwise,
where
c
is a constant.
a. Find
c
.
b. Are
X
and
Y
independent? Justify your answer.
c. Are
X
and
Y
uncorrelated? Justify your answer.
2.
Correlation Coefficient.
Let
X
and
Y
have correlation coefficient
ρ
X,Y
. What is the correlation coefficient between
X
and
aY
+
b
, where
a
and
b
are constants and
a
negationslash
= 0 ?
3.
Modified additive noise channel.
Consider the additive noise channel shown in the figure below.
a
b
X
Z
Y
=
b
(
aX
+
Z
)
The signal
X
and the noise
Z
have zero mean, average powers
P
and
N,
respectively,
and are uncorrelated, and
a
and
b
are constants. Find the best linear MSE estimate of
X
given
Y
and its MSE in terms only of
P
,
N
,
a
, and
b
.
4.
Repetition code.
We wish to transmit 10
3
data bits
U
1
, U
2
, . . . , U
10
3
, where
U
i
∼
Bern(0
.
5), for 1
≤
i
≤
10
3
,
over a binary symmetric channel.
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 Spring '09
 Eggers
 Systems Analysis, Correlation, Probability theory, Probabilistic Systems Analysis, i.i.d. Bern

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