STA711: Probability & Measure Theory

Prof:Robert L. Wolpert rlw@duke.edu OH: Mon 1:30–3, Zoom
TAs:Joseph Lawson joseph.lawson@duke.edu   OH: Wed 4–5:30, Zoom
   &Pritam Dey pritam.dey@duke.edu   OH: Tue 12–1:30, Zoom
Class: Tue/Thu 1:45–3:00pm, Zoom
Text:Sidney Resnick, A Probability Path (free pdf) 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, and just $33)
Jacod & Protter, Probability Essentials (2/e) (easier than Resnick, $53)
Rick Durrett, Probability Theory & Examples (5/e) (more complete, free on-line)

Fall 2020 Syllabus

WeekTopicHomework
I. Foundations of Probability ProblemsDue
Aug 18-20 Probability spaces: Sets, Events, and σ-Fields hw1Aug 27
Aug 25-27 Construction & extension of Measures hw2Sep 03
Sep 01-03 Random variables and their Distributions hw3Sep 10
Sep 08-10 Expectation & the Lebesgue Theorems hw4Sep 15
Sep 15-17 Review & in-class Midterm Exam I ('17, '18, '19) Hists: Exam, Course
Sep 22-24 Expect'n Ineqs, Lp Spaces, & Independence hw5Oct 01
Sep 29-01 Zero-one Laws & Hoeffding's Inequality hw6Oct 08
II. Convergence of Random Variables and Distributions
Oct 06-08 Convergence: a.s., pr., Lp, L, UI. hw7 Oct 15
Oct 13-15 Laws of Large Numbers, Strong & Weak hw8 Oct 20
Oct 20-22 Review & in-class Midterm Exam II ('17, '18, '19) Hists: Exam, Course
Oct 27-29 Fourier Theory and the Central Limit Theorem hw9 Nov 05
III. Conditional Probability & Conditional Expectations
Nov 03-05 Cond'l Expectations & the Radon-Nikodym thm hw10 Nov 12
Nov 10-12 Intro to Martingales (Optional: a, b)
Nov 22 2:00pm-5:00pm Sun: In-class Final Exam ('17, '18, '19) Hists: Exam, Course.
--- TG (Nov 26), Course Ends ---


Note re COVID19:

Due to the on-going coronavirus pandemic, many aspects of this class will change, including

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 MTH 631, Real Analysis). Students wishing to continue their study of probability following STA 711 may wish to take any of MTH 641 (Advanced Probability— click here for a summary of the the 2019 edition), 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 MTH 531, Basic Analysis I (somewhat more advanced than MTH 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 = MTH 230, STA 240L, or MTH 340. 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 available (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. Warning: lecture notes and homework problems are also revised from time to time as I try to improve them. SO, it's best not to print out future problem sets or lecture notes until you need them. Lecture notes all have a "last edited" date at the end; homeworks might change any time until the week before they are due.


Note on Lectures:

All lectures will be remote, using Zoom. For easy access, use the Zoom tool within Sakai. Some features of our use of Zoom include:

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 points (usually up to three, but sometimes four or five) based on your success in communicating a correct solution. For full credit 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 (it's good practice anyway). Avoid using proof-by-contradiction wherever possible— it's too easy to create an errant "contradiction" by making some minor mistake.

Each homework assignment must be submitted as a single PDF document (not word, not images like png or jpg, not separate files for different problems). They may be submitted through Sakai or Gradescope. LaTeX typeset homeworks are much better than scanned hand-written ones. If you must scan a hand-written homework, use pen or a very dark pencil; use a scanner, if you have one, or a good cell-phone scanner--- see suggestions from Gradescope (that page also gives instructions for submission via Gradescope) or the New York Times, for instance. Make sure YOU can read the scanned image before you send it. If you can't read it then we can't either. Adjust brightness or other image parameters if necessary.

Most homeworks are due on Thursday before the start of class, so I can answer questions about their solutions. Typically they are returned by the following Tuesday's class. Homeworks assigned the week before exams are due on Tuesday instead of Thursday, so I can answer questions about them before the exam. Remember, homework problems are subject to change up to a week before the due-date.

Note on Exams:

Midterm and Final examinations are open-book and open-notes, but you should not search for answers on-line during the exam. 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 2017: 1st Midterm 2nd Midterm Final Exam
Fall 2018: 1st Midterm 2nd Midterm Final Exam
Fall 2019: 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 share your solutions with the TAs or me during our office hours we'll be glad to give feedback, hints, etc. Note too that the earlier exams were closed book and taken in person, in years before the pandemic led us to off-line solutions.

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. Other features of Sakai we will use include 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% total, lowest HW score is dropped), 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 several 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.