University Problem Set #11 ECE 313
of Illinois Page 1 of 2 Spring 2003
Assigned: Wednesday, April 9
Due: Wednesday, April 16
Reading: Yates and Goodman: Chapter 5
Reminder: Hour Exam II is scheduled for Monday April 14, in class.
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 5-10 is included on the exam. It will also help
you if you make an early start on this Problem Set since it will provide you with
valuable practice in problem-solving.
Lecture material covered on the Exam is Lectures 12-14 and 18-28 of the Powerpoint
slides available on the class web page. Also, discrete random variables treated earlier
in the course re-appeared in Poisson random processes, DeMoivre-LaPlace theorem
etc., so don’t neglect them entirely.
You should have the pdfs/pmfs, means, variances etc. of binomial, Poisson,
geometric, negative binomial; uniform, exponential, gamma, and Gaussian random
variables on your sheet of notes
Problems:
1. [“A chip off the old block”] We return to Problem 4 of Problem Set #9 where you were asked
to show that Y, the lifetime of a TMR system (with infallible majority gate and VLSI chips
with lifetimes that are exponential random variables with parameter λ = –ln 0.999/week),
has complementary CDF 1 – FY(t) = P{Y > t} = 3exp(–2λt) – 2exp(–3λt), t > 0.
(a) Sketch fY(t), the pdf of Y and also fX(t), the pdf of X, the lifetime of a chip, on the same
set of axes. Use Mathematica/Matlab etc. for this. Don’t just wing it!
(b) Find all values of t for which fY(t) = fX(t). An analytical answer is desired. Don’t just
guesstimate the locations of the crossing-points of the two pdf curves from your graph in
part (a). [Hint: set x = exp(–λt)]. For small values of t, which is larger: fY(t) or fX(t)?
(c) Sketch the hazard rate hY(t) of the TMR system and the hazard rate hX(t) of a VLSI chip on
the same set of axes. For what values of t does the TMR system have a larger hazard rate
than the VLSI chip?
(d) Which is more likely to fail in the next minute: a one-week old TMR system or a one-week
old VLSI chip?
(e) Which is more likely to fail in the next minute: a twenty-year old TMR system or a twenty-
year old VLSI chip? For the purposes of this problem, twenty years = 1040 weeks, (which
reminds me: income tax returns must be filed by April 15…)
2. The random variable X has probability density function fX(u) =
2(1 – u), 0 ≤ u ≤ 1,
0, elsewhere.
Let Y = (1 – X)2.
(a) What is the CDF FY(v) of the random variable Y? Be sure to specify the value of FY(v)
for all v, –∞ < v < ∞.
(b) What is the pdf fY(v) of the random variable Y? Be sure to specify the value of fY(v) for
all v, –∞ < v < ∞.
3. [“Give me an A! Give me a D! Give me a converter! What have we got? An A/D converter! Go Team!”]
A signal X is modeled as a unit Gaussian random variable. For some applications,
however, only the quantized value Y (where Y = α if X > 0 and Y = –α if X ≤ 0) is used.
Note that Y is a discrete random variable.
(a) What is the pmf of Y?
