Math 491 - Probability I
Fall 2012

Instructor: Elizabeth Meckes

Office: Yost 208

Phone: 368-5015

Email: ese3 [at] cwru.edu

Office Hours: TBA

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 17. The first draft of the paper is due Friday, November 16 and the final draft is due Friday, December 7 (the last day of classes).

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. 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 August 31:
1.2, 1.4, 1.6 (you don't need to justify the interchange of integral and derivative), 2.15
For September 7:
2.18, 2.19, 3.11
For September 14:
4.2 a,c,d; 4.14, 4.16, class problem, reread proof of LIL
For September 21:
5.1, 5.3, 5.17, extra problem
For September 28:
5.6, 5.11, 5.12, 6.3, 6.6
For October 5:
8.3, 8.8, 8.20, 8.21, 8.29
For October 12:
13.6, 13.12, 14.3, 14.5, 16.8 (the easiest approach uses the dominated convergence theorem)
For October 19:
18.1, 18.4, 18.10, 18.20
For October 26:
20.2, 20.13, 20.16
For November 2:
20.18 20.19, 21.8, 21.10
For November 9:
21.13, 21.14, 21.21, 23.1, 23.3, 23.4
For November 16:
23.5, 23.6
For November 30:
25.4, 25.5, 25.7, Prove Lyapounov's corollary to the Lindeberg CLT (as stated in class)
For December 3:
26.5, 26.10 (this one requires the continuity theorem)
For December 7:
26.20, 27.3, 27.9