STA711: Probability & Measure Theory

Prof:Robert L. Wolpert wolpert@stat.duke.edu OH: Mon 3:00-5:30pm, 211c Old Chem
TA:Sheng Jiang sheng.jiang@duke.edu   OH: Tue 7:00-9:00pm, 211a Old Chem
Class: Mon/Wed 1:25p-2:40p, 111 Soc Sci
Text:Sidney Resnick, A Probability Path Additional references
Opt'l:Patrick Billingsley, Probability and Measure (a classic)
Jeff Rosenthal, A First Look at Rigorous Probability Theory (2/e) (easier than Resnick, $33, free on-line)
Jacod & Protter, Probability Essentials (2/e) (easier than Resnick, $53)
Rick Durrett, Probability Theory & Examples (4.1/e) (more complete, free on-line)

Syllabus

WeekTopicHomework
I. Foundations of Probability ProblemsDue
Aug 29-31 Probability spaces: Sets, Events, and σ-Fields hw1Sep 07
Sep 05-07 Construction & extension of Measures hw2Sep 14
Sep 12-14 Random variables and their Distributions hw3Sep 21
Sep 19-21 Expectation & the Lebesgue Theorems hw4Sep 28
Sep 26-28 Expect'n Ineqs, Lp Spaces, & Independence hw5Oct 03
Oct 03-05 Review & in-class Midterm Exam I ('13, '14, '15) Hists: Exam, Course
--- Fall Break (Oct 08-11) ---
Oct     -12 Zero-one Laws & Hoeffding's Inequality hw6Oct 19
II. Convergence of Random Variables and Distributions
Oct 17-19 Convergence: a.s., pr., Lp, Loo, UI. hw7 Oct 26
Oct 24-26 Laws of Large Numbers, Strong & Weak hw8 Nov 02
Oct 31-02 Fourier Theory and the Central Limit Theorem hw9 Nov 09
III. Conditional Probability & Conditional Expectations
Nov 07-09 Cond'l Expectations & the Radon-Nikodym thm hw10 Nov 14
Nov 14-16 Review & in-class Midterm Exam II ('13, '14, '15) Hists: Exam, Course
Nov 21- Introduction to Martingales (a, b) hw11 Nov 30
--- Thanksgiving Recess (Nov 23-27) ---
Nov 28-30 Heavy tails and Extreme Values & Final Review
Dec 13 Tuesday office hours, 2-5pm (none Monday)
Dec 18 2-5pm Sun: In-class Final Exam ('13, '14, '15) 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 631, Real Analysis). Students wishing to continue their study of probability following Sta 711 may wish to take any of MTH 641 (Advanced Probability), MTH 545 (Stochastic Calculus), or STA 961 (Stochastic Processes - 2016 version).

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 analysis diagnostic 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, Basic Analysis I (somewhat more advanced than Math 431, Advanced Calculus I, but that's a good second choice) before taking this course. It is also possible to learn the background material by working through one of the standard texts (like Rudin's or Reed's books 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. While this isn't required, students should be or become familiar with the usual commonly occurring probability distributions (here is a list of many of them).

Most weeks will have lecture notes added (click on the "Week" column above 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 Homework:

This is a demanding course. The homework exercises are difficult, and the problem sets are long. The only way to learn this material is to solve problems, and for most students this will take a substantial amount of time outside class— six to ten hours per week is common. Be prepared to commit the time it will take to succeed, and don't expect the material to come easily. Working with one or more classmates is fine; but write up your own solutions in your own way, don't copy someone else's solutions (that's plagiarism).

Homework problems are awarded zero to three points each, based on your success in communicating a correct solution. For the full three points the solution must be clear, concise, and correct; even a correct solution will lose points or be returned ungraded if it is not clear and concise. Neatness counts. Consider using LaTeX and submitting your work in pdf form if necessary (it's good practice anyway). Avoid of proof-by-contradiction wherever possible. Homeworks may be submitted in paper form (just bring them to class on Wednesday) or electronically in the form of pdf files (submit as Sakai "Assignments" or send by e-mail to the course TA, sheng.jiang@duke.edu). Homeworks before midterm exams are collected on Monday instead of Wednesday, so you can ask questions about them before the exam.

Note on Exams:

In-class Midterm and Final examinations are closed-book and closed-notes with one 8½"×11" sheet of your own notes permitted. Tests from recent STA711 offerings are available to help you know what to expect and to help you prepare for this year's tests:
Fall 2013: 1st Midterm 2nd Midterm Final Exam
Fall 2014: 1st Midterm 2nd Midterm Final Exam
Fall 2015: 1st Midterm 2nd Midterm Final Exam
Solutions are not made available for these, because many students can't resist looking up the answer when they get stuck and then the exams lose their value. If you bring your solutions to the TA or to me during our office hours we'll be glad to give feedback, hints, etc.

Note on Sakai:

Use your Duke University uid and password to log into the Sakai learning management system to see scores for your homework assignments and examinations. You may also submit homework as "Assignments" in Sakai. At present those are the only features of Sakai we are using, but if interest warrants we may use some others such as "Forums". It's a good idea to check your homework and exam scores in Sakai every now and then, to make sure your scores were recorded correctly.

Note on Evaluation:

Course grade is based on homework (20%), in-class midterm exams (20% 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 an exam or two), as is A+ (given about once every two or three years for exceptional performance). Your current course average and class rank are available at any time on request.

Note on Enrollment:

Some spaces in this course are reserved for PhD students from the Statistical Science, Mathematics, and Biostatistics & Bioinformatics Departments. 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 Statistical Science MS programs 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 this course if:
  1. There are enough seats in the room, and
  2. They are 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 one of the midterms or the final exam,
    3. come regularly to lectures, and ask or answer questions now and then.
All students are expected to participate actively. It hurts the class atmosphere and lowers students' expectations when some attenders are just spectators. I try to discourage that by asking active participation of everyone, including auditors, to make the classes more fun and productive for us all. Past experience suggests that most auditors stop attending midway through the semester, when they have to balance competing demands on their time; if this course material is important to you, it is better to take the class for credit.

Note on Absence:

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

Academic Integrity

You may discuss and collaborate in solving homework problems, but you may not copy— each student should write up his or her solution. Cheating on exams, copying or plagiarizing 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 Graduate School Judicial Board or the Dean of the Graduate School. Additionally, there may be penalties to your final course grade. Please review Duke's Standards of Conduct.