University Problem Set #7 ECE 313
of Illinois Page 1 of 3 Spring 2002
Assigned: Wednesday, February 27, 2002
Due: Wednesday, March 6, 2002
Reading: Ross, Chapter 3
Noncredit Exercises: Ross pp. 104-117: 53, 58, 59, 62, 63, 70-74, 78, 81
Reminder: Hour Exam I is scheduled for Monday March 4, 7:00 pm to 8:00 pm,
in Room 119 Materials Science and Engineering Building
One 8.5" by 11" sheet of notes is allowed.
Calculators, laptop computers, Palm Pilots, pagers, SMS, etc are not allowed.
The material covered on Problem Sets 1-6 is included on the exam. This Problem Set
also has material on decision theory, and thus, working these problems will help you
prepare for the exam. Coverage of material from Ross is as follows:
Chapter 1 (except Section 1.6)
Chapter 2 (except Section 2.6)
Chapter 3 (except for Section 3.4 on independent events. However, note that we
considered Example 4f on pp. 86-87 in the context of random variables)
Chapter 4.1–4.8.2. Material on hypergeometric and Zipf random variables is not
included, nor is the material on the CDF (Section 4.9)
However, you must know (or have on your sheet of notes) the pmfs,
means, variances etc. of the binomial, Poisson, geometric, and negative
binomial random variables.
Additional material (not always in Ross but covered in class and on homework) can
be found in the Powerpoint slides and my Lecture Notes.
Problems:
1. [“…From the town of Bedrock, They’re a page right out of history…”] Fred suggests that he and Wilma
play a game in which they will take turns tossing a fair coin; the first one to toss a Head wins. Fred
proposes that he will toss first. Wilma agrees to this, but, having taken ECE 313, she knows that she is at
an disadvantage. So, she demands that in succeeding games, the loser of the previous game gets to toss
first. For n ≥ 1, let pn and qn = 1 – pn respectively denote the probabilities that Fred and Wilma win the
n-th game. We saw in class that p1 = 2/3 > q1 = 1/3,
(a) Use the theorem of total probability to show that p2 = 4/9 < q2 = 5/9. More generally, show that for n ≥
2, pn = (2/3) – (1/3)pn-1 and qn = (2/3) – (1/3)qn-1 and use these difference equations to find pn and qn in
terms of p1and q1.
(If you never learned in Math 285/286 or Math 315 (or ECE 310) how to solve difference equations, assume
that for all values of n, pn can be expressed as a + bαn. Substitute into the above difference equation
and solve for α and a; the value of b is obtained from the “initial condition” p1 = 2/3. Repeat for qn — it
is the same difference equation but the “initial condition” q1 = 1/3 is different.)
(b) What is the limit as n → ∞ of pn and qn ? Is this game asymptotically fair?
(c) Fred now proposes that instead of the first one to toss a head winning the game, the first one who matches
the previous toss (whether the previous toss is part of the current game or the last toss of the previous
game) wins. Wilma accepts but generously insists that, as before, Fred still toss first (so that the poor
schmuck has no previous toss to match on his first toss!). What are p1 and q1 now? p2 and q2? p3 and
q3? Is this game asymptotically fair? Assume as before that the loser of each game tosses first in the next
game.
2. [“Reach out and touch someone…”] The probability that you can hear the sound of a pin dropping onto a
table during a long-distance telephone call from Urbana to Hollywood is p0 if the call is being carried over
the AT&T network and p1 if the call is being carried over the Sprint network. Assume that both p0 and p1
are quite small and that p1 > p0. You call Miss Candice Bergen from a payphone owned by Sleazo
Telecom Corporation which happens to lease its long-distance lines either from AT&T or Sprint (but you
don’t know which!), and she very graciously agrees to drop pins one by one onto a table until you hear the
sound of one dropping. Thus, if the first pin that you hear is the Xth, then X is a geometric random
