STA711: Probability & Measure Theory
|Prof:||Robert L. Wolpert||
||OH: Mon 1:30–2:30, Zoom|
||OH: Tue 3:30–5:30,
| &||Pritam Dey||
||OH: Wed 5:00–7:00,
|| || A Probability Path
Probability and Measure
| ||Jeff Rosenthal,
A First Look at Rigorous Probability Theory (2/e)
||(easier than Resnick, and just $33)|
| ||Jacod &
Probability Essentials (2/e)
||(easier than Resnick, $53)|
| ||Rick Durrett,
Probability Theory & Examples (5/e)||
||(more complete, free on-line)|
I. Foundations of Probability
||Probability spaces: Sets, Events, and σ-Fields
||Construction & extension of Measures
||Random variables and their Distributions
||Expectation & the Lebesgue Theorems
||Practice in-class Midterm Exam
0||(Submit via Sakai)|
||Expect'n Ineqs, Lp Spaces, & Independence
||Review & in-class Midterm Exam
||Zero-one Laws & Hoeffding's Inequality
II. Convergence of Random Variables and
||Convergence: a.s., pr., Lp,
||hw7 ||Oct 15
|| Laws of Large Numbers, Strong & Weak
||hw8 ||Oct 22
||Fourier Theory and the
Central Limit Theorem
||hw9 ||Oct 27|
||Review & in-class Midterm Exam II
III. Conditional Probability &
||Cond'l Expectations & the Radon-Nikodym thm
||Intro to Martingales (Optional: a,
||Review Session for Final Exam
|| 2:00-5:00pm Sun:
In-class Final Exam
||--- TG (Nov 26), Course Ends
Note re COVID19:
Due to the on-going coronavirus pandemic, many aspects of this class will
- Class size will be dramatically reduced to 18 students.
- This means several students formerly enrolled will not be able to register.
- Enrollment will be limited to students from Statistical Science,
Mathematics, and Biostatistics
- Lectures will be given on-line
via Zoom (and recorded)
- All office-hours will be given on-line
via Zoom (and not
- Tentative syllabus above may be tailored
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
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 =
or MTH 340. Students should be (or
become) familiar with the usual commonly occurring probability distributions
(here is a list of many of them).
Weekly lecture notes are 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.
All lectures will be remote, using Zoom, all with the same address:
For easy access, use the Zoom tool within Sakai or
click here. Students trying to
access lectures from some countries may need to use a VPN. Some Zoom
sessions may require a pass code--- if so I will e-mail the code to all
enrolled students, and place it in the "Announcements" section of our Sakai
page. Some features of our use of Zoom include:
- Please enable your camera, and aim it at your face for the entire
class period. Of course you're welcome to take a short break now
and then— no need to ask.
- Please be sure your correct name appears among "Participants". You
can change it if necessary.
- Everyone's audio will be muted initially. Please un-mute yourself
when you are speaking, and then re-mute yourself. This should limit
distracting feedback and background noise.
- Use Zoom's "chat" feature to ask questions during lectures, or "raise
- Feel free to request me to speed up or slow down (click
"participants", then "go faster" or "go slower").
- I'll try to leave 10 or 15 minutes at the end of each class for more
- Lectures are recorded and made available on Warpwire (accessible
through Sakai), for the benefit of those unable to participate in
real-time and for review of items when you wish. Please try to
participate in real-time whenever you can.
- This is my first time teaching remotely, and I'm still learning how
to do it. Expect a few rough edges for a while! I'd welcome your
suggestions as we all cope together with learning during a pandemic.
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
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 (who recommend Scannable for iOS and Genius Scan for
Android) or the New York Times (who recommend Adobe Scan for both iOS and Android), for instance. Others have had good experience
with Cam Scanner with both.
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
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 may be
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.
Midterm and Final examinations are open-book and open-notes this year,
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:
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. Exams are
administered using the "Tests and Quizzes" tool in Sakai, which gives you a
specified length of time to download, solve, scan, and upload your
solutions as a single pdf file, all within a specified time window. It is
your responsibility to obtain and use an adequate scanning app (see
recommendations in the Homework section above) or actual
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.
This term we will be using Piazza for class discussion. The system is
highly catered to getting you help fast and efficiently from classmates,
the TAs, and myself. Rather than emailing questions to the teaching staff,
I encourage you to post your questions on Piazza. If you think you know
the answer to another student's question, go ahead and answer it. Or if
you have a similar or related question, step up! You can edit questions
& answers (your own or others') if you wish. Be nice, of course.
Find our Piazza class page at: https://piazza.com/duke/fall2020/sta71101f20/home
or by clicking on the "Piazza" link from Sakai.
Course grade is based on homework (20% total, lowest HW score is dropped),
midterm exams (20% each), and final exam (40%). Most years grades range
from B- to A, with a median grade just above 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.
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. This year, due to the covid19 coronavirus
pandemic, our TA resources are over-strained and course enrollment is
Unregistered students are welcome to sit in on or (preferably) audit this
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.
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.
- There are enough seats in the room, and
- They are willing to commit to active participation:
- turn in several of the homeworks (or a few problems on each of most
- take one of the midterms or the final exam,
- come regularly to lectures, and ask or answer questions
now and then.
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.