Prof: | Robert L. Wolpert |
rlw@duke.edu | OH: Mon 1:30–2:30, Zoom | ||

TAs: | Joseph Lawson |
joseph.lawson@duke.edu |
OH: Tue 3:30–5:30, Zoom | ||

& | Pritam Dey |
pritam.dey@duke.edu |
OH: Wed 5:00–7:00, 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) |

Week | Topic | Homework | |
---|---|---|---|

I. Foundations of Probability |
Problems | Due | |

Aug 18-20 | Probability spaces: Sets, Events, and σ-Fields | hw1 | Aug 27 |

Aug 25-27 | Construction & extension of Measures | hw2 | Sep 03 |

Sep 01-03 | Random variables and their Distributions | hw3 | Sep 10 |

Sep 08-10 | Expectation & the Lebesgue Theorems | hw4 | Sep 17 |

Sep 10-11 | Practice in-class Midterm Exam 0 | (Submit via Sakai) | |

Sep 15-17 | Expect'n Ineqs, L_{p} Spaces, & Independence |
hw5 | Sep 22 |

Sep 22-24 | Review & in-class Midterm Exam I ('17, '18, '19) | Hists: | Exam, Course |

Sep 29-01 | Zero-one Laws & Hoeffding's Inequality | hw6 | Oct 08 |

II. Convergence of Random Variables and
Distributions | |||

Oct 06-08 | Convergence: a.s., pr., L_{p},
L_{∞}, UI. |
hw7 | Oct 15 |

Oct 13-15 | Laws of Large Numbers, Strong & Weak | hw8 | Oct 22 |

Oct 20-22 | Fourier Theory and the Central Limit Theorem | hw9 | Oct 27 |

Oct 27-29 | Review & in-class Midterm Exam II ('17, '18, '19) | Hists: | Exam, Course |

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 19 | Review Session for Final Exam | ||

Nov 22 | 2:00-5:00pm Sun: In-class Final Exam ('17, '18, '19) | Hists: | Exam, Course. |

--- TG (Nov 26), Course Ends --- |

- 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*recorded) - Tentative syllabus above may be tailored

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 R^{n},
convergence in metric spaces (especially uniform convergence of functions on
R^{n}), 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, Real
Analysis I (somewhat more advanced
than MTH 431, Introduction to Real Analysis, 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, MTH 228L, STA 240L, STA 231, 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.

- 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 your hand".
- 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 questions.
- 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.

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
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 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:

Fall 2017: | 1st Midterm | 2nd Midterm | Final Exam | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Fall 2018: | 1st Midterm | 2nd Midterm | Final Exam | |||||||

Fall 2019: | 1st Midterm | 2nd Midterm | Final Exam |

- 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 HW assignments),
- 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.