Mathematics 495
Spring, 2011
Instructor: Edward N. Wilson
Cupples I, Room 18
E-mail: enwilson@math.wustl.edu
Office Tel: 935-6729 (O.K. to leave messages, better to use e-mail)
Office Hours: MWF
12-1, MWF 2-3
and by appointment
Class Meeting Times
and Place: MWF 1-2 Psych 251
Textbooks: 1. Introduction To Stochastic Processes, (paperback
edition), by Paul G. Hoel, Sidney C.
2. Introduction to Stochastic Processes, 2nd
Edition, by Gregory F. Lawler, published in 2006 by Chapman&Hall/CRC.
Overview: We’ll use both of the textbooks, trying to get through well over half of each one over the course of the semester. Homework assignments will involve problems from each book. We’ll refer to these book as HPS (Hoel, Port, & Stone) and L (Lawler). Although there’s a considerable amount of overlap between HPS and L, HPS goes deeper into the subject and is less “user-friendly”. Especially in the early stages, students will probably find L easier to read. On the other hand, HPS is regarded as a classic book in stochastic processes which is often referred to by practioners and by instructors of more advanced courses. Having HPS in one’s personal library is therefore beneficial to everyone who uses stochastic processes and, when the HPS exposition is murky, being able to do a side-by-side comparison with L is also beneficial. This is the reason we are following the recommendation of Prof. Baernstein, the instructor of 495 for the past 3 or 4 times it’s been offered, to use both books.
We’ll begin with the treatment of finite state space Markov chains. These are covered in Chapter 1 of L with Chapter 2 of L moving on to infinite state space Markov chains. On the other hand, HPS do the finite and infinite cases simultaneously in their Chapters 1 and 2. At a minimum, we’ll also try to cover Chapters 3,5, and 8 in L; these compare roughly with Chapters 3 and 4 in HPS. As time permits at the end of the semester, we’ll dip into Chapter 5 of HPS and/or Chaper 4 of L .
In brief, stochastic processes are the probabilistic version of dynamical systems.
Both of these areas are enjoying great popularity with those who want to develop a model for how some complicated phenomenon (economic development, business cycles, the weather, ecological systems, …) evolves over time. In dynamical systems, deterministic models are used with a fixed transition function assumed to exist and, if we only knew it, would predict the next stage of evolution based on the current stage (or state) . Stochastic processes also want to predict futures based on the current state but do so only in probabilistic fashion, giving probabilities for entering various states tomorrow based on the current state. Linear algebra plays a very large role in both subjects. For the finite state space Markov chains with discrete time increments (days, weeks,…) , this comes down to studying certain types of matrices called Markov matrices. For the infinite state space Markov chains with discrete time increments, we have to go on to Markov matrices with infinitely many rows and columns. Linear algebra is still important but we have to rely more on probability to prove the basic theorems. Not surprisingly, the results aren’t as “clear cut” as in the finite state space case. When we move to the more realistic setting of continuous time Markov processes, we’re essentially obliged to limit attention to random variables having nice probability density functions and to use calculus tools.
Homework: There will be weekly homework assignments to be written up and handed in during a designated class. During the week of an exam, we’ll suspend homework.
The homework problems will be drawn from both HPS and
L. Each has a wide range of problems,
some considerably more difficult than others.
When a number of students have trouble with a particular problem, we’ll
try to go over it in class. Homework
will be an important part of the course for the usual reasons: it’s virtually impossible to learn a
mathematical subject without working through problems.
Examinations: There will be two in-class exams during the semester plus a final exam.
The date of the final exam is determined by the
for the missed exam. Otherwise, to the extent possible, make-up exams will be avoided by simply excusing the student from the missed exam and basing the course grade on remaining exam scores and homework.
Academic Integrity: When there is evidence strongly suggesting that cheating took place on an exam, the evidence will be forwarded to the Arts and Sciences Integrity
Committtee. If, after a hearing with the instructor and affected student(s), a majority of the Committee members are convinced that cheating did occur, the Committee will assess a penalty and inform the instructor and sudent(s) of its decision. For exam cheating, the most common penalties are either a failing grade on the exam or a failing grade in the course. In brief, please don’t cheat on exams since cheating cases, however they turn out, are very painful for all concerned.
In sharp
contrast to the above paragraph, students are encouraged to discuss homework problems with other students in the
course. But each student must write up
his/her own solutions and, when parts of solutions were obtained in
collaboration with others, should write the names of the collaborators at the
top of the first page. There will be no
penalty for acknowledged collaboration.
On the other hand, when collaboration clearly took place and wasn’t
acknowledged, the instructor will assess a penalty (e.g, a reduction in the grade for the assignment). This is simply to encourage students to
openly acknowledge help from others.
Later on in life, not doing so is a form of plagiarism which can end a
career.
Grading Scale: Each of the two in-class exams will account for 20% of the course grade average, homework for 30%, and the final exam will for the remaining 40%. Letter grades for exams and homework/course averages will be determined as follows:
A 90-100%
B 80-89%
C 65-79%
F below 65%
In no case will final grades be lower than indicated by this scaling. However, the instructor reserves the right to switch to a more lenient scaling at the end of the semester if he decides that it is warranted.