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
) =
sin 2
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
p
Y
(
y
), the probability density function for
Y
.
Problem 1.3
A crucial skill in developing simulations for stochastic systems is
random
variable generation
. Most computers (and environments like MATLAB) have software
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 Fall '07
 Lexa
 Signal Processing, Probability theory, probability density function, Cumulative distribution function, Department of Electrical and Computer Engineering Rice University

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