Homework 1 of Elec 430
2.5
2001/09/27
2004/01/07 10:01:09.028 US/Central
Behnaam
Aazhang
aaz@ece.rice.edu
Dinesh
Rajan
dinesh@ece.rice.edu
Mohammad
Jaber
Borran
mohammad@ece.rice.edu
Roy
Ha
rha@rice.edu
Shawn
Stewart
mrshawn@alumni.rice.edu
Behnaam
Aazhang
aaz@ece.rice.edu
homework
First homework of Elec 430
Elec 430 homework set 1. Rice University Department of Electrical
and Computer Engineering.
The current I in a
semiconductor diode is related to the voltage
V by the relation
I
V
1
. If V is a random
variable with density function
f
V
x
1
2
x
for
x
, find
f
I
y
; the density function of
I.
Show that if
A
B
{}
then
A
B
c
Show that for any A,
B,
C we have
A
B
C
A
B
C
A
B
A
C
B
C
A
B
C
Show that if A and
B are independent the
A
B
c
A
B
c
which means A and
Bc
are also independent.
Suppose X is a discrete random
variable taking values
0
1
2
…
n
with the following probability mass function
p
X
k
n
k
n
k
θ
k
1
θ
n
k
k
0
1
2
…
n
0
with parameter
θ
0
1
Find the characteristic function of
X.
See problems 3.14 and 3.15 in Proakis and Salehi
Consider outcomes of a fair dice
Ω
ω
1
ω
2
ω
3
ω
4
ω
5
ω
6
.
Define events
A
ω
an even number appears
ω
and
B
ω
a number less than 5 appears
ω
. Are these events disjoint? Are they independent?
(Show your work!)
This is problem 3.5 in Proakis and Salehi.
An information source produces 0 and 1 with probabilities
0.3 and 0.7, respectively. The output of the source is
transmitted via a channel that has a probability of error
(turning a 1 into a 0 or a 0 into a 1) equal to 0.2.
What is the probability that at the output a 1 is observed?
What is the probability that a 1 was the output of the source
if at the output of the channel a 1 is observed?
Suppose X and
Y are each Gaussian random
variables with means
μ
X
and
μ
Y
and variances
σ
X
2
and
σ
Y
2
.
Assume that they are also independent. Show that
Z
X
Y
is also Gaussian. Find the mean and variance of
Z.