Course: Math 407, Probability Theory I, Fall 2020, Section 39632
Prerequisite: MATH 226 or MATH 227 or MATH 229
Course Content: Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws.
Syllabus: here. Last update: 12 August 2020

Instructor: Steven Heilman, stevenmheilman(@-symbol)
Office Hours: Mondays, 1030AM-1130AM and 430PM-530PM, on zoom [link posted on blackboard]
Lecture Meeting Time/Location: Mondays, Wednesdays, and Fridays, 12PM-1250PM, on zoom [link posted on blackboard]
TA: Andrew Lowy, lowya(@-symbol)
TA Office Hours: Mondays 7PM-9PM Thursdays 8PM-9PM, held online in the Math Center.
Discussion Section Meeting Time/Location:

TA Office Hours: Occur in the Math Center.
You are not required to buy a textbook. Free lecture notes are provided below.
Recommended Textbook: D. P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, 2nd edition. (The book is freely available online)
Another Recommended Textbook: Sheldon Ross, A First Course in Probability, any edition. (The book is freely available online)
Another Recommended Textbook: Elementary Probability for Applications, Durrett.

First Midterm:  Friday, September 18, 12PM-1250PM  [For students in Asia/Pacific time zones, the exam is Friday, September 18, 12AM-1250AM PST.]
Project Proposal due: Thursday, October 8, 2PM
Second Midterm: Friday, October 23, 12PM-1250PM  [For students in Asia/Pacific time zones, the exam is Friday, October 23, 12AM-1250AM PST.]
Progress Report due: Thursday, October 29, 2PM
Final Report due: Friday, November 13, 5PM
Final Exam: Friday, November 20, 11AM-1PM.  [For students in Asia/Pacific time zones, the exam is Thursday, November 19, 11PM PST to Friday, November 20, 1AM PST.]
Other Resources: Supplemental Problems from the textbook. An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments

Zoom Classroom Conduct: Students should attend zoom lectures in a considerate way and abide by the following rules of decorum. Failure to do so could result in a diminished participation grade.  It is preferable (though not required, for equity reasons) that all students have a webcam on during the lecture.
Zoom Security: The zoom links posted on blackboard should not be shared with anyone. You must log into zoom with your USC email address. No one will be admitted to the lecture from the "waiting room" (if you are in the waiting room, you did not log in with your USC email address).
Zoom Technical Support: Technical support for undergraduate students is provided through USC's ITS. Below is the contact information.
Undergraduate Student Technology Support
Phone: 213-740-5555

Lecture Recording: Zoom lectures will be recorded and posted on the blackboard site. It is USC policy to prohibit the sharing of any recording of course lectures with others. Similarly, you should not create your own recording of the lectures.
Time Zone Issues: If the course lectures, office hours, or exam schedules occur outside the range of 7AM-10PM in your current time zone, please alert me to this fact as soon as possible. Late notification of such an issue (e.g. the day before an exam) may result in a denied rescheduling request.  If the course lectures occur outside the range of 7AM-10PM in your current time zone, then you will automatically get a 100% participation grade.

Email Policy:

Exam Procedures: The midterms will be one-hour timed exams, to be submitted on blackboard (a 50 minute exam, with 10 minutes designated for uploading the exam).  In the midterm exams, you are allowed to consult your homeworks, your notes, and your textbook, but these are the only resources you are allowed to use during the exams. So, you are not allowed to use the internet, internet searches, a friend or assistant, etc. Phones must be turned off. If you anticipate issues with a stable internet connection (for obtaining the exam), issues with obtaining a suitable exam environment, etc., please let me know as soon as possible and we can try to come up with a solution to these issues. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the USC Student Conduct Code. (See also here.)
Disability Services: If you are registered with disability services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me as early in the semester as possible. DSP is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.
213-740-0776 (phone)
213-740-6948 (TDD only)
213-740-8216 (fax)

Discrimination, sexual assault, and harassment are not tolerated by the university. You are encouraged to report any incidents to the Office of Equity and Diversity or to the Department of Public Safety This is important for the safety whole USC community. Another member of the university community - such as a friend, classmate, advisor, or faculty member - can help initiate the report, or can initiate the report on behalf of another person. The Center for Women and Men provides 24/7 confidential support, and the sexual assault resource center webpage describes reporting options and other resources.

Exam Resources: Here are the exams I used when I previously taught a similar course: Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Here is a page containing old exams for another similar class. Here is a practice midterm (with solutions). Here are some more exams from a second quarter probability course: Exam 1 Exam 1 Solution Exam 2 Exam 2 Solution Final Final Solution. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions Final Final Solutions. Occasionally these exams will cover slightly different material than this class, or the material will be in a slightly different order, but generally, the concepts should be close if not identical.

Homework Policy:

Grading Policy:

Final Project Guidelines:
The final project is an opportunity to explore your interests and learn something e.g. that we didn't have time to cover in class. A project could begin with an interesting question or a well-known problem, and perhaps lead to a probabilistic model of some phenomena that interests you, investigating or implementing various algorithms, conducting an empirical analysis, etc.

The goal is to say something interesting about a problem in probability, broadly construed to perhaps include statistics, machine learning, etc. You could read some section of a book that we will not cover in the class, and report the most salient facts there, reproducing/streamlining the proofs there. You could perhaps develop new methodology for an existing problem or application that has no fully satisfactory solution. You could alternatively tackle a new problem or application with existing methodology; in this case, you should identify one or more questions without satisfactory answers in your chosen domain and explore how the methodology can help you answer those questions. You could consider some statistical problem and draw inspiration from particular datasets, but your focus should rest not on the data itself but rather on the questions about the world that you can answer with that data.

You may work alone or in a group of two; the standards for a group project will be twice as high. In certain cases I might approve a group of three, but this is unlikely.

We strongly encourage you to come to office hours to discuss your project ideas, progress, and difficulties.

Final Project Milestones: In all cases, ideally you would use LaTeX, but you are not required to use LaTeX.

I: Project Proposal. By this first milestone, you should have selected a question or problem of interest, found some notes or textbooks that discuss your project subject matter, identified relevant data sources (if applicable), begun exploring the literature surrounding the question/topic, and discussed your ideas with the course staff. Your project proposal deliverable is a 1/2 - 1 page report describing the question or problem you intend to tackle, why this question is important or interesting, prior work on this problem, what data you intend to use in your analyses (if applicable), and the principal challenges that you anticipate (if applicable).

If you would like to receive feedback about particular aspects of your proposal, please indicate this in your submission.

I can try to help in problem selection. Ideally, the problem should be something you are very interested in. As such, it might be helpful to first tell me about your interests (maybe after class or in office hours), and we can try to think of something to work on.

II: Progress Report. By this second milestone, you should have some initial results to share; for example, you may have read the section of a book that interests you, you may have implemented and evaluated the performance of some algorithm on a dataset, you may have constructed a probabilistic model for your problem at hand, or you may have conducted an initial study with simulated data to better understand the properties of certain methods, etc.

Your progress report deliverable is a write-up of no more than 2 pages (single-spaced; not including references) describing what you have accomplished so far and, briefly, what you intend to do in the remainder of the term. You should be able to reuse at least part of the text of this milestone in your final report.

III: Final Report. Your final project report (not including acknowledgements and references) should be around 5-8 pages in length (using at most 12 point font, maximum 1 inch margins, and single-spaced) and should follow a typical scientific style (with abstract, introduction, etc.). The write-up should clearly define your problem or question of interest, review relevant past work, and introduce and detail your approach. A comprehensive empirical evaluation could follow, or some proofs of some results, along with an interpretation of your results. Any elucidation of the theoretical properties of an empirical method under consideration is also welcome.

If this work was done in collaboration with someone outside of the class (e.g., a professor), please describe their contributions in an acknowledgements section.

Some Project Ideas:

Machine Learning/ Deep Learning

Proofs of the Central Limit Theorem

Stochastic Processes

Polling/ Elections

Concentration of Measure

Probabilistic Algorithms

Tentative Schedule: (This schedule may change slightly during the course.)

Week Monday Tuesday Wednesday Thursday Friday
1 Aug 17: 1.1, Sets Aug 19: 1.2, Probabilistic Models Aug 20: Homework 0 due (ungraded) Aug 21: 1.2, Probabilistic Models
2 Aug 24: 1.3, Conditional Probability Aug 26: 1.3, Conditional Probability Aug 27: Homework 1 due Aug 28: 1.4, Total Probability Theorem and Bayes' Rule
3 Aug 31: 1.5, Independence Sep 2: 1.5, Independence Sep 3: Homework 2 due Sep 4: 1.6, Counting
4 Sep 7: No class Sep 9: 2.1, Discrete Random Variables Sep 10: Homework 3 due Sep 11: 2.2, Probability Mass Function
5 Sep 14: 2.3, Functions of Random Variables Sep 16: 2.4, Expectation and Variance Sep 17: No homework due Sep 18: Midterm #1
6 Sep 21: 2.5, Joint PMFs, Covariance and Variance Sep 23: 2.6, Conditioning Sep 24: Homework 4 due Sep 25: 2.6, Conditioning
7 Sep 28, 2.7, Independence Sep 30: 2.7, Independence Oct 1: Homework 5 due Oct 2: 3.1, Continuous random variables and PDFs
8 Oct 5: 3.1, Continuous random variables and PDFs Oct 7: 3.2, Cumulative Distribution Functions Oct 8: No homework due. Project Proposal Due. Oct 9: 3.3, Normal Random Variables
9 Oct 12: Joint PDFs of Multiple Random Variables Oct 14: 3.5, Conditioning Oct 15: Homework 6 due Oct 16: 3.5, Conditioning
10 Oct 19: 4.2, Covariance Oct 21: 4.4, Moment Generating Function Oct 22: No homework due. Oct 23: Midterm #2
11 Oct 26: 4.4, Fourier Transform Oct 28: 4.2 Convolution Oct 29: No homework due. Progress Report Due. Oct 30: 7.1, Markov and Chebyshev Inequalities
12 Nov 2: 7.2, Weak Law of Large Numbers Nov 4: 7.3, Convergence in Probability Nov 5: Homework 7 due. Nov 6: 7.4, Central Limit Theorem
13 Nov 9: 7.4, Central Limit Theorem Nov 11: 7.5, Strong Law of Large Numbers Nov 12: No homework due. Nov 13: Final Report Due. 7.5, Strong Law of Large Numbers

Advice on succeeding in a math class:


Homework .tex files


Homework Digest

Supplementary Notes