STA205: Probability & Measure Theory

Prof:Robert L. Wolpert OH: Thu 2-3pm, 211c Old Chem (684-3275)
TA:Natesh Pillai  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); Additional references


WeekTopicHomework ProblemsDueSol
I. Foundations of Probability
Jan 11,18 Probability spaces: sets, events, and sigma-fields 1/1,3,4,12,24,39Feb 02sol
Jan 23-25 Probability spaces: Constructing & extending measures 2/8,11,14,19Feb 09sol
Jan 30-Feb 1 Random variables and their distributions I 3/2,3,6,15Feb 16sol
Feb 6-8 Random variables and their distributions II 4/1,2,4,7,8,9,10,11Feb 23 sol
Feb 13-15 Independence, sigma-fields & zero-one Laws 4/13,16,18,19,21,24,28Mar 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 10/6,7,8,10,13MGs
May 5 Take-home Final Examination (due 7pm). '02, '03, '04, '05 Results


This is a course about random variables, especially about their convergence and conditional expectations, motivating an introduction to the foundations of modern Bayesian statistical inference. It is a course by and for statisticians, and does not give thorough coverage to abstract measure and integration (for this you should consider MTH241) nor to the abstract mathematics of probability theory (see MTH 287).

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.