| Prof: | Sayan Mukherjee | sayan@stat.duke.edu | OH: Mon 9:15-10am, 112 Old Chem (684-4608) | ||
| TA: | Jianyu Wang | jw163@stat.duke.edu | OH: tba | ||
| Class: | Tu/Thu 11:40-12:55pm | 025 Old Chem | |||
| Text: | Sidney Resnick, | A Probability Path | |||
| Opt'l: | Patrick Billingsley, |
Probability and Measure (3rd edn); | |||
| Week | Topic | Homework | |
|---|---|---|---|
| I. Foundations of probability | Problems | Due | |
| Aug 25 | Introduction: Motivation, set theory, and set operations | ||
| Aug 27 | Probability spaces: fields and sigma-fields | hw1 | Sep 8 |
| Sep 01-03 | Probability spaces: constructing & extending measures | hw2 | Sep 15 |
| Sep 08-10 | Random variables and their distributions | hw3 | Sep 22 |
| Sep 15-17 extra notes | Markov, Chebychev, Hoeffding, Azuma | hw4 | Sep 29 |
| Sep 22-24 | Integration & expectation: Lebesgue's MCT & DCT | hw5 | Oct 13 |
| Sep 29, Oct 01 | Independence & zero-one laws | hw6 | Oct 20 |
| --- Fall break (Oct 02-07) --- | |||
| II. Convergence of random variables & probability | |||
| Oct 13,15 | Convergence concepts: a.s., i.p., Lp, Loo | hw7 | Oct 22 |
| Oct 20-23 | Law of large numbers revisited | hw8 | Oct 29 |
| Oct 27-29 | Review and in-class Midterm (Thu Oct 29) '05, '06, '07, '08, '09, | Results | |
| Nov 03-05 | Strong & weak laws of large numbers | hw9 | Nov 10 |
| Nov 10-12 | Convergence in distribution & C.L.T. | hw10 | Nov 17 |
| Nov 17 | Exchangeability & de Finetti representation | hw11 | Dec 01 |
| III. Conditional Prob & Expectation | |||
| Nov 19 | Radon-Nikodym thm and conditional probability | hw12 | Dec 01 |
| --- Thanksgiving (Nov 24-30) --- | |||
| Dec 01 | Martingales | ||
| Dec 04 | Take-home Final Exam (due 2pm) '04, '05, '06, '07, '08 | Results | |
| May 1 | Histogram of Course Averages | ||
Students are expected to be well-versed in real analysis at the level of W. Rudin's Principles of Mathematical Analysis or M. Reed's Fundamental Ideas of Analysis--- the topology of Rn, convergence in metric spaces (especially uniform convergence of functions on Rn), infinite series, countable and uncountable sets, compactness and convexity, and so forth. Most students who majored in mathematics will be familiar with this material; students with less background in math should consider taking Duke's Math 203, Basic Analysis I. It is also possible to learn the material by working through standard text, doing most of the problems, preferably in collaboration with a couple of other students and with a faculty member to help out now and then. More advanced mathematical topics from real analysis, including parts of measure theory, Fourier and functional analysis, are introduced as needed to support a deep understanding of probability and its applications. Topics of later interest in statistics (e.g., regular conditional density functions) are given special attention, while those of lesser statistical interest (e.g., extreme value theorems) may be omitted.
Some problems and projects may require computation; you are free to use whatever environmnent you're most comfortable with. Most people find R (lots of on-line some documentation is available) or Matlab (a primer is available) easier to use than compiled languages like FORTRAN or C. Homework problems are of the form chapter/problem from the text. Not all of them will be graded, but they should be turned in for comment; This syllabus is tentative, last revised , and will almost surely be superceded- RELOAD your browser for the current version.
Course grade is based on homework (20%), in-class midterm exam(30%), and take-home final exam (50%), the top 2 students on the midterm exam have an option of a challenging final project instead of the final exam.
My rules about auditors are that a student can sit in on or (preferably) audit a course if: