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### Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

### Probability theory (Spring 2015)

Classes: Tue, Thu, 5:40-7:10, LH-1, Mathematics department

Description: This is a first course in measure theoretical probability theory. It is optional for everyone, but undergraduates, int-phd and phd students are all welcome to credit or audit. The prerequisites are basic real analysis, at least a first course in discrete probability. Measure theory is required for the course, but it is possible to learn it concurrently. If you are planning to do that, please know that it requires you to work hard and make extra effort. In the first five weeks, I shall cover various aspects of measure theory, but that is not a substitute for a full course.

Grading: There will be a final exam (50%) and two mid-term exams. Yet to decide on homeworks.

Texts and other resources: I shall put my lecture notes up on the course page (but not always in a timely fashion). There are many good books on the subject, some of which I mention below.
1. Rick Durrett Probability: theory and examples. Click for a copy!
2. John Walsh Knowing the odds
3. Patrick Billingsley Probability and measure, 3rd ed. Wiley India. Available in Tata book house
4. Richard Dudley Real analysis and probability, Cambridge university press
5. David Williams Probability with martingales, Cambridge University Press (available in Indian edition)
6. J V Uspensky Introduction to mathematical probability, McGraw-Hill (1937)
7. William Feller An Introduction to Probability Theory and its Applications, Vol 2, 2ed, Wiley (Indian ed.)(1971)
Some comments: Durrett's book has both theory and interesting examples, but it is a graduate level text and some of the students may find it a little difficult. Billingsley's book is well-written, very detailed and has elaborate explanations. Dudley's book appears to be extremely good and reasonably concise. It has both measure theory and probability theory and the relevant functional analysis but perhaps lacks interesting examples of interest in modern probability theory. David Williams book is exceedingly well-written (and easy compared to the others, except perhaps Walsh's book). The last two books are much older. Uspensky's book contains certain things that are not usually found in modern books. So does Feller's book, but it is not well-suited for the beginning of the course. For specialized topics like characteristics functions, there is no better resource that I know of.
Notes and homeworks/links: See earlier versions of this course for many problem sets, exams etc. If I make new ones, they will appear below.