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ELEC 531: STATISTICAL SIGNAL PROCESSING
Department of Electrical and Computer Engineering
Rice University
Fall 2007
Problem Set 1
Due: September 4, 2007
Problem 1.1
Which of the following are probability density functions? Indicate your
reasoning. For those that are valid, what is the mean and variance of the random
variable?
(a)
p
X
(
x
) =
e

x

2
(b)
p
X
(
x
) =
sin2
πx
πx
(c)
p
X
(
x
) =
(
1
 
x
 
x
 ≤
1
0
otherwise
(d)
p
X
(
x
) =
(
1

x
 ≤
1
0 otherwise
(e)
p
X
(
x
) =
1
4
δ
(
x
+ 1) +
1
2
δ
(
x
) +
1
4
δ
(
x

1)
(f)
p
X
(
x
) =
(
e

(
x

1)
x
≥
1
0
otherwise
Problem 1.2
A random variable
X
has cumulative distribution function
P
X
(
x
) = [1

exp(

2
x
)]
u
(
x
)
,
where
u
(
x
) 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 of
X
.
(c) Let
Y
be a random variable obtained from
X
as follows:
Y
=
(
0
X <
2
1
X
≥
2
.
Find
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This homework help was uploaded on 02/11/2008 for the course ELEC 531 taught by Professor Lexa during the Fall '07 term at Rice.
 Fall '07
 Lexa
 Signal Processing

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