Mathematics 275: Probability Theory
Mathematics 275A - Fall 2007
- Time and place: MWF at 10 in MS 5233
- Instructor: Thomas M. Liggett
- Office hours: MWF 2-3 in MS 7919
- Text: Measure Theory and Probability Theory
by K. B. Athreya and S. N. Lahiri
- There will be a discussion section with a TA (Yen Do) on Thursdays at 10,
beginning October 4. The first lecture is on Friday, September 28.
Yen's office hours will be Tu 11-12 and 1-2 in MS 6142.
- Prerequisites:(a) One (two is even better) quarter of real analysis
(Mathematics 245). As a review of measure theory, you might want to read the following
parts of the text before classes begin: pages 1-25, 39-44, 48-71, 147-164.
exposure to undergraduate probability (e.g., Mathematics 170AB)
is desirable, but not required.
This is formally less important than the real analysis requirement,
but helps to motivate the material. No undergraduate probability
theorems or concepts will be used, however, so students who
have not had undergraduate probability should be able to follow
the course with no difficulty.
- Topics: Introduction to (measure theoretic) probability
theory, including concepts of convergence, independence, weak
and strong laws of large numbers, random series, characteristic
functions, and central limit theorems -- roughly chapters 6-11 of the text.
- The future:This course leads to 275B
(random walks, martingales, ergodic
theorems), followed by 275C (Brownian motion; introduction to
continuous time Markov processes), 275D (stochastic calculus
with applications to mathematical finance),
and 275E (stochastic particle systems).
Courses 275ABC are normally offered each year. Courses 275DE
are generally offered every two or three years.
- Probability theory is a branch of pure mathematics
that has strong ties with many areas of application. In terms
of technique, it is part of analysis. Its motivation comes from
areas such as statistics, economics, business, engineering, biology and
physics. It is an ideal meeting ground for the pure and the
applied. Not only is it used in formulating and analyzing
models in the areas of application listed
above, but it also plays important roles in areas of pure
mathematics such as analysis, partial differential equations,
- Homework will be assigned each Friday and due the
following Thursday in the discussion section (except that the
assignment that would be due on Thanksgiving day will be due
in lecture November 21). It will be returned to you the following
week in discussion section.
- Grades will be based on homework to be assigned from
and the final exam (December 13, 3-6pm). (There will be no midterm.)
You may enroll for a letter grade or on an S/U basis. If you enroll
for a letter grade, you are expected to take the final exam; if
S/U, then the final exam is optional.
- Final exam
- Due Oct. 4. Page 32 #1.4,1.16,1.19 and page 71 #2.1,2.6 (take
T_1 smaller than T_2; what happens if you don't?),2.11,2.15.
- Due Oct. 11. Page 34 #1.23, Page 75 #2.14 and page 212 #6.1,6.3,6.4,6.5 (note
that the phi in part (c) is the one from part (a), not the
one from part (b)),6.6,6.22,6.23.
- Due Oct. 18. Page 228 #7.3,7.4(b,c,e,f),7.6(a),7.8(b),7.20,7.21,7.22.
- Due Oct. 25. Page 231 #7.10,7.12(b) (there is an error in the
statement of this problem; you should prove a correct form, and
find a counterexample to the statement as it is),7.13,7.15(\epsilon_i are iid);
page 279 #8.3,8.7,8.9,8.10.
- Due Nov. 1. Page 229 #7.5,7.9,7.16,7.18; page 279 #8.5,8.8.
- Due Nov. 8. Page 279 #8.2(assume independence;ignore the "nonnegative"
part of the hint),8.11,8.13(a),8.15(note that in the hint, 8.14 should be
- Due Nov. 15. Page 283 #8.28(b,c),8.30; page 309 #9.1,9.2,9.3,9.7,9.8(a).
- Due Nov. 21 (Wednesday!). Page 310 #9.9,9.15,9.17,9.18,9.22,9.24,
- Due Nov. 29 Page 314 #9.32,9.34,9.35(a); page 337 #10.1,10.6,10.10.
- Due Dec. 6 Page 337 #10.3,10.12,10.13,10.14,10.17,10.19.