603 ECE - Probability and Random Processes, Fall 2010
Midterm Exam #1
October 20th, 6:30-8:30pm
Overview
The exam consists of ve problems for 130 points. The points for each part of each problem are given
in brackets - you should spend your two hours accordingly.
The exam is closed book, but you are allowed one page-side of notes. Calculators are not allowed. I
will provide all necessary blank paper.
Testmanship
Full credit will be given only to fully justied answers.
Giving the steps along the way to the answer will not only earn full credit but also maximize the
partial credit should you stumble or get stuck. If you get stuck, attempt to neatly dene your approach
to the problem and why you are stuck.
If part of a problem depends on a previous part that you are unable to solve, explain the method for
doing the current part, and, if possible, give the answer in terms of the quantities of the previous part
that you are unable to obtain.
Start each problem on a new page. Not only will this facilitate grading but also make it easier for you
to jump back and forth between problems.
If you get to the end of the problem and realize that your answer must be wrong, be sure to write this
must be wrong because . . . so that I will know you recognized such a fact.
Academic dishonesty will be dealt with harshly - the minimum penalty will be an F for the course.
Hint: You may nd the following fact useful as you solve this exam:
$
7$CB!A @987 54321)'&%#"
6 0 ( $ !
1. Consider the network:
F F TF G F
S
U
F F QF G F
P
E
E
F
F FG F
E
E
F F IF G F
R
D
F F IF G F
H
WG
and let
denote the probability that switch is closed (i.e. that packets can ow
through that switch). Now, suppose that the events
have the following properties:
RG
S
fG
.
.
RV
V V TH
ys
V
ts
u V TH
(
V g$
q ($
H v7$ P
xbqf wv7$ P
i ($
rdfph$ P
qi (
form a partition.
and
.
$
$ H V
.
in terms of unions and intersections of the
and
$P V
$
7$ P V
and
can be re-designed, but
$
$
V
$P V
$
7$ P V
t V TH V
s
t V TH V
s
$
7$ P V t V s H V
How would you specify
y V TH V
s
y V TH V
s
$
7$ P V y V s H
How would you specify
and
is closed.
such that
such that
$ H V
[10] (d) Suppose that
remain the same as in (a).
H
G
occurs, nd the probability that
s (and their comple-
W
V
SV
V
V
V
[8] (b) Write an expression for
ments), and nd
.
[5] (c) Given that
D
V
be the event that there is a connection between
[7] (a) Find
and
.
U
PV
and
P
QG G
are independent.
are independent of the status of switches
HG
d b a Y ( X W
`c7Q`Q` C7 CV
Let
and
and
W
eV
The status of switches
must
is maximized?
is minimized?
, of
(
fi (
D
$ ( 7$ G 7
$
$ PG P
by:
(restricted to
,
fi
[10] Dene (a) the random variable
is dened by
2. Suppose that the probability space
.
course), and
i
Y
!
U
$
PH $
D
?
%
&
'
Apple, Banana, Cardinal, Dove
b
4
5
Y
Y
b
!4
65
!
i
!
!
!
Qa Y
#$"
(
g$
D
D
U
(
g$
1
)11
0
11
13
&
Apple, Cardinal . Find
(
(
8
&
Apple, Banana and
8
(
(
7
Dene the events
7
U
2
Find the induced probability space for (feel free to use the rst letters A, B, C, and D
of Apple, Banana, Cardinal, and Dove, respectively, to save you space)
$s
[10] (b) Consider the mapping (not a random variable!)
given by
(
.
falls in the interval
Apple
Banana
Cardinal
Dove
What is the probability
for
Y
Find the probability density function
.
3. Suppose that I conduct the following experiment. I ip a coin continually, and I write down the result
of the ips to get an outcome . For example, a possible might look like:
q$ q q
of all possible outcomes . Is
9
$
9
(
[10] (a) Consider the sample space
Tail, Tail, Tail, Head, Head,
countable or uncountable?
[10] (b) Construct a rich (i.e. non-trivial) probability space
for this experiment. You should
end up with an
with an uncountable number of elements that should allow one to answer any
question of interest.
9
D
be a random variable with probability density function:
i
(
g$
f i
#"
be a random variable with probability density function
$ Y Ya ! $ a g$
(
and let
else
4. Let
"
Answer the following six parts independently. In each case, the answer and a single line of justication is sufcient.
.
Y
fi
Qd
(
v$
$
.
.
"
. Find
$
4
P
!
and
!
P
b
)(
"
[7] (d) Find
U
[8] (e) Let
Find
else
"
, where
.
Y
$
$ (
D U
(
D
U
Y! (
DU
[5] (c) Let
. Find
"
[5] (b) Let
. Find
$
[5] (a) Let
.
[5] (f) Suppose that I run 10 trials, each resulting in an independent observation of a random variable
with probability density function
. What is the probability that the sum of the outcomes exceeds
15?
$
"
D
has probability density function:
h$
(
#"
which is shown below (Be sure to look at the plot!).
5. The random variable
[10] (a) Find .
D
If you cannot get part (a), proceed now do do parts (b) through (d) as if
is Gaussian with
mean and variance . Make sure to note on your paper that you are doing such.
PD
a
. (Hint: You do not have to do any integrals - look at your equation for nding the
[5] (b) Find
variance of a random variable.)
.
D
Y! b! d
PD
D
.
0.25
0.2
f(x)
0.15
0.1
0.05
0
10
8
6
4
2
0
x
2
4
Figure 1: Probability density function of
6
D
[5] (d) Find
$
[5] (c) Find
8
10
for Problem 5.