STA711: Probability & Measure Theory

Prof:Robert L. Wolpert wolpert@stat.duke.edu OH: Mon 3:00-4:00pm, 211c Old Chem
TA:T B A tba@stat.duke.edu  OH: Wed 6:00-7:30pm, 211a Old Chem
Class: Tue/Thu 11:45a-1:00p, 025 Old Chem
Text:Sidney Resnick, A Probability Path Additional references
Opt'l:Patrick Billingsley, Probability and Measure (3/e) (a classic)
Jacod & Protter, Probability Essentials (easier than Resnick, $36)
Rick Durrett, Probability Theory & Examples (4/e) (more complete)

Syllabus

WeekTopicHomework
I. Foundations of ProbabilityProblemsDue
Aug 28-30 Probability spaces: Sets, Events, and σ-Fields hw1Sep 06
Sep 04-06 Construction & extension of Measures hw2Sep 13
Sep 09-12 SAMSI Meeting: Massive Datasets
Sep 13 Random variables and their Distributions hw3Sep 20
Sep 18-20 Expectation & the Lebesgue Theorems hw4Sep 27
Sep 25-27 Inequalities, Independence, & Zero-one Laws hw5Oct 04
Oct 02-04 Convergence concepts: a.s., i.p., Lp, Loo hw6Oct 11
Oct 09-11 Uniform Integrability hw7 Oct 18
--- Fall Break (Oct 13-16) ---
Oct 18 Review & in-class Midterm Exam (Thu Oct 20) '10, '11 Results
Oct 23-25 Strong & Weak Laws of Large Numbers hw8 Nov 01
II. Convergence of Distributions
Oct 30-01 Convergence in Dist'n, CLT, & Stable Limits hw9 Nov 08
Nov 06-08 Extremes (notes) hw10Nov 15
III. Conditional Probability & Expectation
Nov 13-15 Radon-Nikodym thm and conditional probability hw11Nov 20
Nov 20 Stein's Method (time permitting)
--- Thanksgiving Break (Nov 21-26) ---
Nov 27-29 Martingales and Markov Chains
Dec 13 Take-home Final Exam due 9am ('07, '10, '12) Hists: exam, course


Description

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 Math 632 (old 241), Real Analysis: F09 vsn) nor to the abstract mathematics of probability theory (see MTH 641 (old 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. Try to answer the questions on this diagnostic analysis quiz to see if you're prepared. Most students who majored in mathematics will be familiar with this material; but students with less background in math should consider taking Duke's Math 531 (203), Basic Analysis I: F09 vsn before taking this course. It is also possible to learn the material by working through one of the standard texts and doing most of the problems, preferably in collaboration with a couple of other students and with a faculty member (maybe me) 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 may be omitted.

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 superseded— reload your browser for the current version.

Course grade is based on homework (20%), in-class midterm exam (40%), and take-home final exam (40%).


Note on Auditing:

Unregistered students are welcome to sit in on or (preferably) audit a course if:

  1. There are enough seats in the room,
  2. S/He is willing to commit to active participation:
    1. turn in about a third or so of the homeworks (or a few problems on each of most HW assignments)
    2. take either the final exam or the midterm
    3. come regularly to lectures, and ask or answer questions now and then.
I expect all students to participate actively. It hurts the class atmosphere and lowers students' expectations when some attenders are just spectators. I try to discourage that by requiring active participation of everyone, including auditors, to make the classes more fun and productive for us all.