Math 491 - Probability I
Fall 2009
Instructor: Elizabeth Meckes
Office: Yost 208
Phone: 368-5015
Email: ese3 [at] cwru.edu
Office Hours: MTF 11:00 - 12:00, or by appointment
Textbook: Probability and Measure, 3e by Patrick
Billingsley
Topics: We will cover most of chapters 1 -- 5 of the textbook.
For some of the foundational measure theory (mostly in chapters 2 and 3), I will
state results without proof in class. You are welcome (in fact, encouraged)
to read the proofs in the book, but you will not be responsible for them on
homework or the final. The topics covered in the first five chapters of
the book are: an introduction to rigorous probability theory, general theory of
measures, Lebesgue integration, random variables and expectations, and
convergence of distributions.
Grading: I expect you to attend the lectures, take notes (I
may cover some material not in the text), and read the text book, (it is
certain that not all
material will be discussed in class). This book is very thorough and
well-written, and
will be a valuable resource for the course. There will be student
projects (worth 20% of the course grade),
one final (worth 40%) and weekly homework assignments (worth 40%). Selected
homework problems will be graded.
Projects: For your project, you should pick an expository
article on some probability topic, read it, and write your own exposition
of what's discussed in the article, filling in details and adding
your own perspective. Once you've picked an article, come talk to me about
it so we can come up with a detailed plan (e.g., you may need to pick just
a section or two if it's longer). You are welcome to choose your own topic;
the following is a list of some reasonable possibilities (in reverse
chronological order); in general,
probability articles from the monthly are mostly reasonable choices.
You must talk to me about which paper you're going to read by Wednesday,
October 14. The first draft of the paper is due Friday, November 20
and the final draft is due Friday, December 4 (the last day of
classes).
- Larry Goldstein. "A probabilistic proof of the Lindeberg-Feller
central limit theorem". Amer. Math. Monthly 116 (2009),
no. 1, 45--60.
- Byron Schmuland. "Random harmonic series". American Mathematical
Monthly 110 (2003), no. 5, 407--416.
-
Lou Billera, Ken Brown and Persi Diaconis. "Random walks and plane
arrangements in three dimensions". Amer. Math. Monthly 106
(1999), no. 6, 502--524.
- Jack Cassidy. "The last round of betting in poker". Amer. Math.
Monthly 105 (1998), no. 9, 825--831.
- Bennet Eisenberg and Rosemary Sullivan. "Random triangles in n
dimensions". Amer. Math. Monthly 103 (1996), no. 4, 308--318.
- Tom Liggett and Peter Petersen. "The law of large numbers and
$\sqrt{2}$". Amer. Math. Monthly 102 (1995), no. 1, 31--35.
- Michael Steele. "Le Cam's inequality and Poisson approximations".
Amer. Math. Monthly 101 (1994), no. 1, 48--54.
Homework Problems: Doing the homework problems is probably the
single biggest factor in determining how much you get out of the course.
If you are having trouble with the problems, please come ask for help; you will
learn much more (and probably get a rather better grade) if you figure out
all of the homework problems, possibly with help in office hours, than if you
do them alone when you can and skip the ones you can't.
Homework is weekly, due at the beginning of class on Wednesdays.
I strongly suggest starting early so that there is time to
ask for help if you need it. In particular, office hours are MTF so that
you can get started in time to ask initial questions on Friday, work hard
over the weekend, and then have the opportunity to ask more questions Monday
and Tuesday before turning in the assignment on Wednesday.
Problems will be posted below; selected problems will be graded.
You may discuss the homework with other students, however, you must
write up solutions on your own.
Assignments: Problems are numbered as n.m, with
n being the section number and m the problem number.
When a problem has an up arrow indicating that previous results are needed,
you may use them without proof.
For September 2:
- 1.2, 1.4, 1.6 (you don't need to justify the interchange of
integral and derivative), 2.15, 2.18, 2.19.
For September 9:
- 3.11, 4.2 (a,c,d), 4.13, 4.14, 4.15.
Extra problem: Show that if {A1,...,
An} are independent, then so are {B1,...,
Bn}, where for each i, Bi
is either Ai or Aic.
For September 23:
- 5.1, 5.3, 5.6, 5.11, 5.12, 5.17.
For September 30:
- 6.3, 6.6, 6.7, 6.11 9.4, 9.5 (this claims to need 9.4, but I don't think it does).
For October 7:
- 10.2, 10.3, 12.1, 12.12, 13.6, 13.17.
For October 14:
- 14.3, 14.5, 14.8.
For October 21:
- 16.2, 16.8, 18.4, 18.10, 18.20.
For October 28:
- 20.2, 20.8, 20.13, 20.16, 20.18, 20.19.
For November 4:
- 21.8, 21.9, 21.10, 21.13, 21.14, 21.21.
For November 11:
- 23.1, 23.2, 23.3, 23.4, 23.5, 23.6.
For November 20:
- 25.2, 25.4, 25.5, 25.7.