Mathematics 171 - Spring 2005
Stochastic Processes
- Time and place: MWF at 10 in MS 5117
- Instructor: Thomas M. Liggett (tml@math.ucla.edu)
- Office hours: MWF 1-2 in MS 7919 (Students should feel free to
send questions by email as well, though we can usually accomplish
more by speaking in person.)
- Teaching Assistant: Tim Prescott. His office hours are
Monday (4-5), Tuesday (2-3) and Thursday (3-4) in MS 6154.
- Text: "Essentials of Stochastic Processes",
by Rick Durrett. There is a review chapter at the
beginning of the text, which is a good summary of
the probability that is assumed in the course. You
should read this to make sure you are up to speed.
The main subject of the course is Markov chains,
so we will begin by covering Chapter 1, Chapter 3
(except Section 3.3) and the first four or five
sections of Chapter 4. If time permits, we will
then cover a couple of topics from Chapters 2 or 5.
- A stochastic process is simply a collection
of random variables indexed by time. These come up
in many contexts: position of a particle that is
influenced by some random phenomenon, price of an
option or stock, winnings of a gambler, size of
a queue (a queue is a waiting line, such as at a
checkout counter, in the skies above LAX, or in
a buffer at a node in a communications network),
and many others.
The main issue we will discuss is the behavior of
the random system after it has evolved for a long
time -- is it close to being in equilibrium, does
it grow without bound,...?
- Prerequisites: Mathematics 170A or equivalent.
- Discussion sections: (Tuesdays at 10 in
MS 5117) These provide an opportunity
for you to ask questions about and discuss more fully
issues that are covered in the lectures. The TA will discuss
the homework that is due that day, but all questions related
to the course are fair game. In the first discussion section
(April 5), the TA will do some reviewing of probability
theory, based on the material in "Review of Probability" at
the beginning of the text. In preparation for this session,
you should look at (and try to do) problems 1.15, 1.17,
1.21, 1.25, 1.28, 2.9, 2.17, 2.19, 3.11, 3.15, 3.17,
3.20, 3.22 of this
pre-chapter. They are not to be turned in, however.
- Homework will be assigned each Wednesday (and
listed on this web page), and
will be due at the beginning of the discussion section the following Tuesday.
It is essentially impossible to learn mathematics without
doing a lot of problems. I urge you to do the homework
with care and on time. You should also look through and
attempt as many of the unassigned problems as you have
time for. The basic rule is: The more problems you do,
the more thoroughly you will learn the material.
Note that there are answers to many of the problems at the
end of the book, so you can check your work that way.
- Midterms: There will be two midterms, tentatively on Monday, April 25 and Wednesday, May 18.
- The first midterm will cover through page 54 + computation
of simple absorption probabilities. (The harder computations
of absorption probabilities is part of Section 1.6, and will be covered
on the second midterm.)
- The second midterm will cover all of Chapter 1
and the first two sections of Chapter 3.
- Final exam: The final exam will be Tuesday, June
14, 3-6. It will cover all of Chapter 1, Chapter 3 except for
Section 3, and Chapter 4, Sections 1-4.
- Finals week OH: Tu 1pm
- Grades will be based on homework (20%), midterms (20% each)
and final exam (40%).
Homework
- Due April 12: Page 88, # 1,2,3,4,5 + handout
- Due April 19: Page 89, # 6,7,8,9,11,13,14,15
- Due April 26: Page 90, #12,16,18,23,24,26,31,32
- Due May 3: Page 90, #10,33,34,35,36,37,41,45
- Due May 10: Page 96, #39,42,47,48,49,50,51,53
- Due May 17: Page 97, #44 and page 152, #1,2,3,4,5,6,7
- Due May 24: Page 137, #2.2,2.3; page 153, #11,17,19,22,23,25
- Due May 31: Page 155, #33,34,35,36,39,43; page 200, #2
- Due June 7: Page 200, #3 (do it two different ways: (a)
solve the Kolmogorov backward or forward equations and (b) use
properties of the Poisson process), 4,5,6,8,11,20,21,27,28