603 ECE - Probability and Random Processes, Fall 2006
Midterm Exam #1
October 16th, 6:00-8:00pm, Marston 132
Overview
The exam consists of six problems for 120 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:
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1. [15] I ip a fair coin twice (assume the ips are independent) and record the outcome of these ips
in order. For example, for head followed by head, the outcome is HH. Your job is to dene a
probability model that will used by one of your co-workers to analyze the experiment. You are not
aware of what questions he/she might want to ask, so you want to generate as complete a model as
for this
possible (e.g. you do not want to use a trivial ). Provide a probability space
experiment. Since the size of the sets involved here is not that large, be explicit about how each of
these three things are dened. In particular, write out all of the sets in and give the probability of
each.
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2. Your buddy who works in the modeling department provides you with the following probability space:
,
(where the Borel eld is restricted to
, of course) and, for any interval,
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ac b
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[5] (a) Find the probability of the outcome .
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[10] (b) Find the probability of the of set irrational numbers in
.
3. The word algebra contains four consonants (l, g, b, and r) and three vowels (a, e, a).
Suppose I place these seven letters in a bag along with an eighth symbol that is a blank space.
I draw the eight symbols out of the bag at random one at a time and place them left to right in the
order drawn. The blank space will cause there to always be two words in the resulting expression (if
the blank space is drawn rst and thus appears at the left end, assume the rst word is empty; if the
blank space is drawn last and thus appears at the right end, assume the second word is empty).
Answer each part of this problem independently.
[8] (a) Suppose that the blank space is the third symbol drawn (and thus I have a two-letter word
and a ve letter word). What is the probability that both words have at least one vowel?
[12] (b) This part has nothing to do with part (a). Read the question again, skip over part (a),
and answer this part. What is the probability that each word will contain at least one consonant?
4. Alex, Brian, and Chris take turns rolling a fair six-sided die (in that order: Alex, Brian, Chris, Alex,
Brian, Chris, Alex, ). The game stops when somebody rolls a 6, and the person rolling the 6 is
declared the winner. Assume that rolls of the die are independent.
[10] (a) Find the probability that Alex wins the game.
be the number of times Alex rolls the die in a given game. Find the probability that
.
G (
[8] (b) Let
for
(
[7] (c) Suppose that Alex wins on his sixth roll. Let be the total number of 5s that have appeared
on the die before that point. What is the probability that
?
, where:
f
g g G
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f
otherwise
.
r
.
be dened by:
G
G
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.
Gg
Find the probability density function of
[13] (f) Let the random variable
[6] (e) Find the probability that
is given by
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(
[5] (a) Find the value of the constant .
[6] (d) Find the probability that
5. The probability density function of a random variable
6. Suppose that I am observing a network connection that is good (G) with probability 0.9 and bad
(B) with probability 0.1. Let (in seconds) be the time until the rst packet arrives.
has probability density function:
else
G
G
f b
!
( $
has probability density function:
else
G
b
G
If the connection is bad,
If the connection is good,
( $
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[7] (a) What is the probability that the rst packet arrives during the rst second?
[8] (b) Given that the rst packet arrives during the rst second, what is the probability that the link is
good?