Math 3770, Probability and Statistics, Fall 2008.
For a copy of the course syllabus, click
Homework 1 (To be turned in Monday, September 8).
Page 20-22, #12, 18, 20.
Page 30, #34.
Page 39-40, #44, 56.
Page 57-59, #12, 18, 24.
Page 65-66, #30, 34, 36, 38.
Click for a note on uses of
Bayes's theorem to testing (medical, drug, etc.).
Here is an article appearing in the Telegraph
newspaper about ``false positives''. It will make you think twice about
testing results! Click here for a link to a Nature article on this.
Homework 2 (to be turned in Monday, September 22).
Page 74-75, #46, 50, 52, 56, 60.
Page 80, #70, 72, 74, 76, 78.
Page 89-90, #1, 4, 8.
Page 98-99, #12, 14, 18, 20.
Page 106-107, #30, 32, 34, 36, 38.
Click for a copy of notes
on Bayesian spam filtering (in the note the set Sigma is the set of
It will cover all material up to and including
section 3.2 (which we have yet to get to).
Click for an old Math 3225 exam, a few of the questions from which
would be appropriate for your first exam.
Click for a copy of another 3225 exam (don't worry if the questions
look too hard -- I will post a study sheet at the level of Math 3770
soon.). And, here is another.
Click for an old note I wrote (for
an honors version of our class) on the ``birthday paradox''. In this
note, Sigma denotes the set of events -- it is the set of subsets of the
sample space S.
for a fascinating article on the misuse of math in criminal cases
(unjustified independence assumptions and the prosecutor fallacy), written
by Keith Devlin, mathematician and regular on NPR's Weedkend Edition.
Click for a link to Simpson's paradox and
baseball scores (the Derek Jeter and David Justice example from class today).
Remember the random triangle example from the beginning of class
on Monday, Sept. 8? Click for a webpage about it.
Click here for a copy of some
selected solutions to the homework you just turned in.
for an article by AICPA (American Institute of Certified Public
Accountants) on how accountants can use Benford's Law (the leading digit
law from class today) to detect fraud.
There will be 5 questions.
One will be a definition question (you will be asked to define 5 terms).
One will be a proof (you will be asked to verify a simple probability
identity). One will be a tricky combinatorial problem about drawing
marbles from a bag. One will be a tricky Bayes's theorem question. And
one will be a problem on statistics. I will let this info, and the info
presented in class today, stand in for a ``study sheet'' for your exam.
for a note on Zipf's Law from class today -- the empirical law about the
ranking of English words.
Homework 3 (due Monday, Oct. 6).
Page 113-114, #46, 48, 50, 52, 54.
Page 120-121, #68, 70, 72, 74.
Page 125-126, #80, 82, 86, 92.
Page 135-136, #1, 2, 3, 4, 10.
Page 143-144, #12, 14, 22.
Click for a note on the ``Feigenbaum constant'' from class today.
another webpage about ``bifurcation diagrams'' that come up in
determining the Feigenbaum constant.
Click for a link to the Google Tech talk
given by our own Vijay Vazirani.
here for an article about ``the wisdom of crowds'' from class today.
is another webpage about it.
is a Youtube video of fireflies in Sync.
Here is a link to a webpage about Steven Strogatz's work.
Here is an
article about Kirkwood gaps.
Click for an
article about Marian Rejewski from class today.
Click for a link about Jonathan Farley's work on catching terrorists
using mathematics (i.e. probability theory).
Click for a note on how
to solve the ``two slips of paper'' problem from the beginning of class
lecture note on Poisson random variables. Basically, it contains material
I presented in class that is not covered in your book.
Click for the
``two portfolios'' problem from the beginning of class today.
Homework 4 (Monday, October 20).
Page 154-155, #28, 36.
Page 162-163, #60, 62, 64, 68.
Page 194-196, #2, 4, 15.
Page 201-202, #24, 26, 28.
For a link about the Google Pagerank Algorithm, and a discussion
of the probability theory it uses, click
Click for a copy of some
notes I typed up on the chi-squared random variable. Much of the material
is not in your book.
for a survey article on using Hidden Markov Models (a particularly
common type of statistical model) to do speech recognition. Although the
article is quite old (written in 1989), it is a classic and gives a good
introduction to the subject. Click here for an article on uses of HMM's in
Click for an article on statistics, prime numbers, and
energy levels of atoms. Click here
for another note on this.
for an article on John Scott Russell and solitions.
for an amazing video of a Falaco Soliton (a type of topological soliton)
in a swimming pool. Go here for an article
about a solition found in the upper atmosphere.
Click for a nice Wikipedia article on the counterintuitive
Banach-Tarski Paradox (one small correction to the Wikipedia article:
The five sets in the sphere decomposition can be chosen so that they are
each ``connected sets'', not mere ``infinite scatterings of points''.
This was shown by deGroot and Dekker way back in 1956.).
for a rather technical Wikipedia article on the Black-Scholes model, a
very useful statistical model to price options. Click
here for an article titled ``Don't blame the quants'',
which concerns recent economic troubles and quants' role in them.
for an MIT Technology Review article on the subprime crisis and quants
(registration required). Here is another link to this
article which may work without registration.
Homework 5 (due Wednesday, November 5).
Page 218, #46, 48.
Page 240, #2, 4.
Page 251-252, #20, 22, 29.
for an article about a statistical model that predicts why some traffic
jams form for no apparent reason. Click
here for the article
I mentioned in class about the Biham-Middleton-Levine Traffic Model.
It concerns how interesting patterns emerge as traffic exceeds a certain
critical threshold of around 32 to 34 percent full.
Click for an article on the clouds of Saturn distributed as
a hexagon. Click here for an article about strange Earth hurricane
eyewall shapes, in particular hurricane Isabel. Click
here for a fascinating article on the
Great Red Spot of Jupiter by James Gleick, author of the book Chaos: The
Making of a New Science. Click
here for a
link about the upcoming Nova episode about fractal geometry.
I have finished writing up the lecture note on
moment generating functions, which is a topic not covered in your book.
Click for a copy of
I have also written up a lecture note on applications of
the central limit theorem, including the hypothesis testing example from
class about smokers (taking into account Benjamin Davis's comments).
This note also has some material on Maximum Liklihood Estimates, which
we haven't yet gotten to. Click for a copy of this lecture note.
I will be speaking this weekend (Saturday Oct. 25) at a math
conference on somewhat recent research of mine (and A. Granville,
R. Pemantle, and P. Tetali) on rigorous analysis of
``integer factoring algorithms'' (which are used to break certain
cryptosystems). Since this is probability-related, I will mention
it in class today. Here is a copy of our paper on this. It is a bit technical,
but perhaps you will be able to read the introduction.
I have scheduled your second exam for
. I have decided to allow you to bring
a non-scientific calculator to the exam. Also, I have decided to make
at least one of your exam questions the same as one of your homeworks,
verbatim; and maybe I will take two of the questions from HWs -- I haven't
decided yet. One of your questions will be a definition, and one will be
an easy proof. The remaining question(s) will be calculations.
I hope that the median score will be much higher for this exam than it was
for the first one.
Click for a copy of the Geman and Geman article
on using a ``Gibbs Sampler'' to do image reconstruction.
here for a link to Gary Miller's work on image segmentation.
Click here for a copy of the paper by Miller and Tolliver.
here for an interesting Youtube video about the a mathematics
child prodigy, who happens to also be the son of the former Georgia Tech
math grad student Mark Ladue (who studied under Bill Green).
Michael Barnsley was once a professor of math at Georgia Tech,
and was a pioneer in the use of randomness and fractals to do image
compression (he founded the company Iterated Systems Incorporated).
here for a Wikipedia entry related to his work. Click
Barnsley's homepage (at Australian National University).
copy of the study sheet for the exam.
Click for a link about the Rhind Papyrus that I spoke about
on Halloween -- basically, one of the oldest math ``textbooks'' at the
time of its discovery, copied by the Scribe Ahmes. The scribes studied
language and mathematics, and worhiped the god Thoth, click
While looking through the TED foundation website the other day,
I came across an interesting online video about the use of statistics in
genetics and in court cases (basically, the things we discussed early in
the course). Click here for this fascinating video.
(And here is one of my favorite non-math TED talks.)
Here are three articles on Daniel Bernoulli's work on the
epidemiology of smallpox:
Homework 6 (due Friday, Nov. 21).
Page 262, #2, 3.
Page 269, #14.
Page 276-277, #28, 32, 34, 36.
Page 293, #2, 4.
Page 304, #16, 18.
Page 310, #36.
Page 317, #46.
Page 465, #12.
Click for a note on one of the early ``branching processes'',
called the Galton-Watson process, which concerns the loss of noble
surnames through time (it was worked out in the late 19th century). Click
here for some applications to nuclear physics.
Click for a paper on
some applications of probability theory to a certain inventory problem
in systems engineering. Here is a paper
with a few algorithms (not as mathematical as the first one).
Most people I have talked with have said that they think the
exam today was quite a bit more difficult than the first one. I certainly
tried to make it easier than the first exam (the questions on this one
are more standard and basic), though maybe because there was so much
material to study, and because you didn't have enough time, you didn't
perform as well as you would have otherwise. If it is
the case that the median score is lower than 80, then I will curve the
exam by adding enough points to everyone's score to bring the median up to 80.
Homework 7 (turn it in with the final exam).
Page 475, #30.
Page 492, #58.
Page 483, #44.
I have decided to make the final exam optional for everyone; HOWEVER,
you will not be allowed to keep the 27.5 curve on the second exam.
If you wish to avoid taking the final, you must use a curve of 18 points;
and, your grade if you skip the final will be computed as follows:
Final grade = (2/7)(Exam1 + Exam2 + 18) + (3/7)(Homework grade). So,
if you got a 60 on the first exam, and 65 on the second (pre-curve),
and a grade of 90 on the HW, the grade you would get if you didn't
take the final is (2/7)(60 + 65 + 18) + (3/7)(90) = 79.429, which is a C.
Of course, if you DO take the final, then you get to use the curve of
I came across an interesting article by the CS titan Edsger Dykstra,
which you can view here .
Click here for a copy of
the final exam.