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MATH 6040/6080: Mathematical Probability (Fall 2023)

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.

Students in 6080 attend the SAME LECTURES and complete IDENTICAL HOMEWORK as 6040 students.

However, while

Achieving an A in 6040 earns a qual high pass, a B+/A- earns a qual pass. On the other hand,

Students taking 6080 can still take a Math 6040 qual exam afterwards, in January or August.

They also may have more detailed discussions of some of the topics.

A copy is available on CANVAS.

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.

you need to really learn how things work and not just blindly follow someone else's solution.

resources such as Stack Exchange. Soliciting help from such sites will be considered as

Once the due date passes you can find solutions to the homework problems here.

The password is the same as the Piazza access code.

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

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 $e^{2\pi i\alpha}$.)

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

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

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

5.5, 5.8, 5.14, 5.18

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

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

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)

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

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

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

The exercises in this file.