| Prof: | Robert L. Wolpert | wolpert@stat.duke.edu | OH: Thu 2-3pm, 211c Old Chem (684-3275) | ||
| TA: | Natesh Pillai | natesh@stat.duke.edu | OH: Tue 3-5pm, 222 Old Chem (684-8840) | ||
| Class: | Mon/Wed 2:50-4:05pm | 025 Old Chem | |||
| Text: | Sidney Resnick, | A Probability Path | |||
| Opt'l: | Patrick Billingsley, |
Probability and Measure (3rd edn); | |||
| Week | Topic | Homework Problems | Due | Sol |
|---|---|---|---|---|
| I. Foundations of Probability | ||||
| Jan 11,18 | Probability spaces: sets, events, and sigma-fields | 1/1,3,4,12,24,39 | Feb 02 | sol |
| Jan 23-25 | Probability spaces: Constructing & extending measures | 2/8,11,14,19 | Feb 09 | sol |
| Jan 30-Feb 1 | Random variables and their distributions I | 3/2,3,6,15 | Feb 16 | sol |
| Feb 6-8 | Random variables and their distributions II | 4/1,2,4,7,8,9,10,11 | Feb 23 | sol |
| Feb 13-15 | Independence, sigma-fields & zero-one Laws | 4/13,16,18,19,21,24,28 | Mar 02 | sol |
| Feb 20-22 | Integration & expectation I (Lebesgue's theorems) | 5/1,4,7, 16, 17 | Mar 09 | |
| Feb 27-Mar 1 | Integration & expectation II (Fubini's theorem) | 5/19,21,24,29,32 | Mar 09 | |
| Mar 6-8 | Review and in-class Midterm Exam (Wed Mar 9) | '02, '03, '04a,b, '05 | Results | |
| --- Spring Break (Mar 11-19) --- | ||||
| II. Convergence of Random Variables & Distributions | ||||
| Mar 20-22 | Convergence concepts: a.s., i.p., Lp, Loo | 6/5,7,8,10,13,14,17,30 | ||
| Mar 27-29 | Strong & weak laws of large numbers | Martingale Notes | ||
| Apr 3-5 | Convergence in distribution & C.L.T. | 7/1,4,8,43,44,46 | ||
| Apr 10-12 | Stable limit theorem & ID limits (notes: ps, pdf) | 8/2,3,12; 9/5,6,9,10 | ||
| III. Conditional Prob & Expectation | ||||
| Apr 17-19 | Radon-Nikodym thm and conditional probability | 10/6,7,8,10,13 | MGs | |
| May 5 | Take-home Final Examination (due 7pm). | '02, '03, '04, '05 | Results | |
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 (some notes are 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; Tuesday classes will begin with a class solution of one or two of the preceeding week's problems. Some weeks will have lecture notes added (click on the "Week" column if it's blue or green). This is syllabus is tentative, last revised , and will almost surely be superceded- RELOAD your browser for the current version.