MATH 6040/6080: Mathematical Probability (Fall 2023)
Time & Place: MWF 10:45-11:35 AM, JWB 208
Instructor: Firas Rassoul-Agha
E-Mail:, Office: LCB 209
Office Hour: Wednesdays 1-2 PM or by appointment.

Forum: We will be using Piazza for class discussion and answering questions. The system is highly catered to getting you help fast and efficiently from classmates and myself.
Rather than emailing questions, I highly encourage you to post your questions on Piazza. I also highly encourage you to answer your classmate's questions, if you think you know the answer.
One advantage over CANVAS Discussion Board is that you can post anonymously, if you prefer.
You can sign up here but you will need an access code that you can find in the version of the syllabus on CANVAS.

6040 vs 6080: This fall, we are offering Math 6080, a course linked to Math 6040 but introduced for those interested in 6040 WITHOUT IT BEING A QUAL COURSE.
Students in 6080 attend the SAME LECTURES and complete IDENTICAL HOMEWORK as 6040 students.
However, while 6040's grade is based on homework and two exams, 6080's grade is based solely on homework.
Achieving an A in 6040 earns a qual high pass, a B+/A- earns a qual pass. On the other hand, Math 6080 CANNOT be used to get a qual pass, neither low nor high.
Students taking 6080 can still take a Math 6040 qual exam afterwards, in January or August.

Videos: Here, you can find recorded videos from a previous semester. We may not follow these lectures word for word, but you can take advantage of these videos to clarify things you did not get during class.
They also may have more detailed discussions of some of the topics.

Textbook: We will follow Probability, by Davar Khoshnevisan. (AMS Graduate Studies in Mathematics, 2007)
A copy is available on CANVAS.

Homework will be mostly from the textbook. Make sure you do homework in a timely fashion, otherwise you WILL fall behind.
You should submit your homework on CANVAS. Make sure you include your name (at least the last name) as part of the file name.
I highly prefer that you type the homework up, with LaTex being the most favored.

If you are taking Math 6080, Homework will make 100% of your grade.

If you are taking Math 6040, then:

  • Homework will make 20% of your grade.
  • One two-hour midterm makes 40% of your grade. It is on Friday October 20 at 9:30-11:30 AM in JWB 240, and will cover chapters 3 to 5 of the textbook.
  • One two-hour final makes 40% of your grade. It will be on Friday December 15 at 10:30AM-12:30PM in JWB 308, and will cover and chapters 6 to 8 of the textbook.


    On homework: feel free to collaborate with classmates and friends, with the understanding that homework is preparing you for the tests and hence
    you need to really learn how things work and not just blindly follow someone else's solution.

    On exams: you are allowed to use calculators (although really won't need them). You cannot use notes nor textbooks. Also, you are NOT allowed to seek help from others, including online
    resources such as Stack Exchange. Soliciting help from such sites will be considered as academic misconduct and will be dealt with accordingly.

    Homework Problems:

    Once the due date passes you can find solutions to the homework problems here.
    The password is the same as the Piazza access code.

    Review (not due, but very recommended):

    1.1, 1.3, 1.9, 1.11, 1.12, 1.13, 1.16, 1.22, 1.24, 1.28
    2.1, 2.3, 2.4, 2.5, 2.7

    Due Monday Sep 4:

    Prove Lemmas 3.4, 3.9, and 3.11 (one of the two claims needs finiteness, which one?!)
    Prove that the Borel sigma-algebra on (0,1] is the same as the sigma-algebra generated by intervals of the form (a,b].

    3.2, 3.9, 3.10, 3.11(you can assume that there are $a<b$ such that $\mu((a,b])<\infty$), 3.14, 3.17(this one shows that the existence of nonmeasurable sets relies on the axiom of choice! There is a typo though in this exercise: the circle should be parameterized as $S^1=\{e^{2\pi i\theta}:\theta\in(0,1]\}$ and, therefore, intervals $(0,2\pi]$ should be $(0,1]$ and quantities like $e^{i\alpha}$ should be $e^{2\pi i\alpha}$.)

    Due Monday Sep 25:

    4.1, 4.6, Construct a counter example to Fatou's lemma (that shows that if we remove the non-negativity assumption, then the conclusion may fail), 4.16, 4.18

    Not due, but very recommended:

    4.9-4.11, 4.17, 4.21, 4.22, 4.23, 4.24, 4.27

    Due Monday Oct 16:

    4.28, 4.29, 5.2, 5.3, 5.4, 5.7, 5.13, 5.17

    Not due, but very recommended:

    5.5, 5.8, 5.14, 5.18

    Due Monday Nov 6:

    6.2, 6.4, 6.5, 6.7, 6.9, 6.10, 6.14, 6.19, 6.27, 6.28, 6.29, 6.30, 6.37

    Not due, but very recommended:

    6.3, 6.6, 6.16, 6.17, 6.20, 6.23, 6.25, 6.26, 6.32

    Due Wednesday Nov 22:

    8.3, 8.4, 8.6, 8.7, 8.8, 8.14, 8.15, 8.25, 8.31, 8.34, 8.35, 8.39, 8.40, 8.51
    8.14 continued: prove that $S_n$ is recurrent when $p=1/2$ and transient when $p\not=1/2$, i.e. $P(S_n\not=0\ \forall n\ge1)=0$ for $p=1/2$ and $>0$ for $p\not=1/2$.
    Prove also that when $p=1/2$, $S_n$ is in fact null recurrent, i.e. that $E[\text{return time to }0]=\infty$. (When this is finite, the process is called positive recurrent)

    Not due, but very recommended:

    8.16, 8.20, 8.26, 8.27, 8.30, 8.36, 8.38, 8.43, 8.47, 8.48, 8.56

    Due Friday Dec 8:

    7.2, 7.7, 7.10, 7.16, 7.18 (can use 7.14(3)), 7.19, 7.20, 7.25, 7.26, 7.30, 7.32, 7.40

    Not due, but very recommended:

    7.3, 7.6, 7.8, 7.14(3), 7.15, 7.21, 7.42, 7.44

    Not Due:

    The exercises in this file.