STA711: Probability & Measure Theory

Prof:Robert L. Wolpert wolpert@stat.duke.edu OH: Mon 3:00-4:00pm, 211c Old Chem
TA:Anurag Ghosh ag246@stat.duke.edu  OH: Wed 5:30-7:00pm, 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 (2/e) (easier than Resnick, $36)
Rick Durrett, Probability Theory & Examples (4/e) (more complete)

Syllabus

WeekTopicHomework
I. Foundations of Probability ProblemsDue
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 Workshop: Massive Datasets
Sep 11-13 Random variables and their Distributions hw3Sep 20
Sep 19-21 SAMSI Workshop: AstroStats
Sep 18-20 Expectation & the Lebesgue Theorems hw4Sep 27
Sep 25-27 Inequalities, Independence, & Zero-one Laws hw5Oct 04
II. Convergence of Random Variables and Distributions
Oct 02-04 Convergence concepts: a.s., i.p., Lp, Loo hw6Oct 18
Oct 09-11 Review & in-class Midterm Exam I '10, '11 Exam, Course
--- Fall Break (Oct 13-16) ---
Oct     -18 Uniform Integrability
Oct 23-25 Strong & Weak Laws of Large Numbers hw7 Nov 01
--- Halloween Break (Oct 30) ---
Nov    -01 Fourier Theory and Convergence in Distribution hw8 Nov 08
III. Conditional Probability & Conditional Expectations
Nov 06-08 Central Limit Theorem & Stable Limit Theorem hw9 Nov 20
Nov 13-15 Review & in-class Midterm Exam II Exam, Course
Nov 20 Radon-Nikodym thm hw10Dec 04
--- Thanksgiving Break (Nov 21-26) ---
Nov 27-29 Conditional probability & Martingales hw10Dec 04
Dec 04 Extremes & Review for Final Exam
Dec 12 9am Wed: In-class Final Exam ('07, '10, '11) 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 as undergraduates will be familiar with this material, but students with less background in math should consider taking Duke's Math 531 (old 203), Basic Analysis I: F09 vsn (somewhat more advanced than Math 431 (old 139), Advanced Calculus I) before taking this course. It is also possible to learn the material by working through one of the standard texts (like the two listed above) 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.

Most students in the class will be familiar with undergraduate-level probability at the level of STA 230 (old 104). While this isn't required, students should be or become familiar with the usual commonly occuring probability distributions (here is a list of many of them).

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.

Note on Evaluation:

Course grade is based on homework (10%), in-class midterm exams (25% each), and final exam (40%). Most years grades range from B- to A, with a median grade near the B+/A- boundary. Grades of C+ or lower are possible (best strategy: skip several homeworks, skip several classes, tank on the exams), as is A+ (given about once every two or three years for exceptional performance).

Note on Enrollment:

Space in this course is reserved for Statistical Science PhD students. While other well-prepared students are welcome, space in the course is limited and in some years it is over-subscribed. Early applicants and participants in the concurrent Statistical Science MS program have the best chance of enrolling. Occasionally one or two exceptionally well-prepared undergraduate students wishes to take the course; there is a surprisingly cumbersome process for obtaining permission for that described on the Trinity College website.

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.

Note on Absence:

No excuse is needed for missing class. Class attendance is entirely optional. You remain responsible for turning in homework on time. Try not to get sick for scheduled examination times.

Academic Integrity

Cheating on exams, copying or plagiarising homeworks or projects, lying about an illness or absence and other forms of academic dishonesty are a breach of trust with classmates and faculty, and will not be tolerated. They also violate Duke's Community Standard and will be referred to the University Judicial Board.