EE 278
September 30, 2009
Statistical Signal Processing
Handout #2
Homework #2
Due: Wednesday October 7
1.
Probabilities from cdf.
The cdf of random variable
X
is given by
F
X
(
x
) =
braceleftBigg
1
3
+
2
3
(
x
+ 1)
2

1
≤
x
≤
0
0
x <

1
a. Find the probabilities of the following events.
A
=
{
X >
1
3
}
,
B
=
{
X
 ≥
1
}
,
C
=
{
X

1
3

<
1
}
,
D
=
{
X <
0
}
.
b. Does
X
have a pdf? Explain your answer.
2.
Distance to nearest star.
(Bonus)
Let the random variable
N
be the number of stars in a
region of space of volume
V
. Assume that
N
is a Poisson random variable with pmf
p
N
(
n
) =
e

ρV
(
ρV
)
n
n
!
,
n
= 0
,
1
,
2
, . . . ,
where
ρ
is the “density” of stars in space. We choose an arbitrary point in space and define
the random variable
X
to be the distance from the chosen point to the nearest star. Find the
pdf of
X
in terms
ρ
.
3.
Additive Gaussian noise channel.
A communication channel has a realvalued input signal
X
and an output
Y
=
X
+ 2
Z
, where
Z
∼ N
(0
,
0
.
09). Suppose that
X
=

1 is sent. Use the
attached table of the
Q
(
·
) function to find the probability of the event
{
Y >
0
}
.
4.
Lognormal pdf.
Let
X
∼ N
(0
, σ
2
). Find the pdf of
Y
=
e
X
.
5.
Random phase signal.
(Bonus)
Let
Y
(
t
) = sin(
ωt
+ Θ) be a sinusoidal signal with random
phase Θ
∼
U[

π, π
] . Find the pdf of the random variable
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 BalajiPrabhakar
 Normal Distribution, Signal Processing, Probability theory, Alice, Cumulative distribution function

Click to edit the document details