Teaching

M 362K Probability I (Spring 2017)

I was the TA for Probability I. Some class materials, mostly instructions for the TA project, are preserved here.

• UT email: jamesmurphy AT math DOT [utexas domain], please include "M362K" in the subject line.
• Office: RLM 9.116.
• Official course website: canvas.
• Official office hours: by appointment, email me.
• Unofficial office hours: I will usually be in my office after class 9:30-11:00 Tues/Thurs. You are welcome to stop by unannounced (or follow me from class to my office), but email me and schedule official office hours if you want me to definitely be there.
• I can be very helpful answering questions about probability or math in general, even questions beyond the scope of the class. My Ph.D. research is in probability, specifically random measures and point processes.
• I do not participate in grading anything but the TA project and I cannot change your grades. If you have questions about how you were graded, I can give insight into what the grader may have been thinking or if the grader made an error, but I cannot directly fix any such errors.

The TA project:

• Project goal: Do/learn something interesting related to/using probability beyond what is in the syllabus.
• Don't feel like you have to overdo it, just do one thing well.
• Your project must be approved my me (James Murphy) at least 1 week before your due date.
• Due date: Due dates have been assigned randomly. See the canvas announcement "TA Project due dates, be aware of yours". If you have a due date April 14th or earlier and you have a legitimate reason that this date is too early for you (e.g. you have 3+ exams in the preceding 2 weeks) please contact me IMMEDIATELY and, at my discretion, you may be given a new due date. If you wait until your due date to contact me, you will not be given a new due date and may get a zero on your project.
• You may complete your project on or before your due date, you could be done next week! (But you would need to already know some probability for that to be the case.)
• You may work individually or in groups of up to 5 people. The amount/depth of work produced should be proportional to the number of people in your group. Group members will be graded individually, you will need to keep track of and submit a statement of what you contributed to the project.
• Most students should receive 90+ out of 100 points. Your project does not have to be amazing to get an A. I understand this is not your only class and you should be able to get an A putting 1 week of hard work into your project (2 weeks if group project due to coordination).
• You may need to read ahead in the textbook and learn something ahead of schedule in order to complete your project. If you need help, ask for it.
• You must cite all sources that you use. It is okay if you use internet or other external sources to help your understanding. It is okay if you reference someone else's work or use ideas or explanations that you did not come up with (this is how mathematics progresses, by citing results already known). However, you absolutely may not copy someone else's work as your own.
• You may not use work that you have turned in for another class.

FAQ

• How do I turn in project materials? Via email attachment or canvas messaging attachment. If you have more than 4 files, zip them.
• Do you have any preference on filetypes? For papers a pdf is preferred, for programs turn in the source code files (.java, .c, .h, .m, .py, .nb, etc.), but not binary files.
• Will I be counted off for sloppy language, bad grammar, spelling errors, or otherwise unprofessional work that I turn in? Yes. Please edit your papers for clarity and precision before turning them in. You are trying to prove to me that you understand the material you are writing about.
• What citation format should I use? LaTeX standard, MLA, and APA are all acceptable. Be very clear about what is cited and what is your original work, plagiarism will be taken very seriously.
• Will I be counted off for bad code style or otherwise unintelligible source code? Yes.
• Will I be counted off for inefficiency in my code? If your program produces valid output and halts in less than 3 minutes, then you will not be counted off for inefficiency.
• I'm giving a presentation, when do I present? You must schedule a time to present with me on or before your due date. Most likely, you will present to me during my office hours.
• I picked a very technical topic, is it okay to include things that I don't understand? Generally, no. The best way to handle a technical topic for the purposes of this project is to find a way to cover it that only uses tools we have seen in class. I am always available to help you find a less technical version of your topic. If you really want to do something technical, I still require that you understand what you write. If that means all you can do is give a definition and a simple example, so be it, as long as you understand. I would rather you do one thing well than cover a whole topic broadly.
• My due date is [any date after 04/14], can I get an extension? No.
• What if I have a huge workload the week of my due date? You are in charge of managing your own time, and you are solely responsible for completing your project by your due date. If you are worried about having too much work the week of your due date, then don't wait until the last week to start.
• What happens if I don't turn in my project on time? You receive a zero for your project grade.

Project Ideas

Once you have an idea for your project, you must have me (James Murphy) approve it via email or canvas messaging. I will tell you if what you propose is enough to earn full credit. You can discuss project ideas with me over email or in office hours. Feel free to discuss project ideas with your peers or anyone else as well.

The format of your project is entirely up to you. Some example formats:

• Write a short paper (long enough to do something interesting, but no fluff please)
• Give a short presentation
• Make a video about your topic
• Present a problem/solution (I can help you find a good problem that isn't too hard)
• Take a short written exam on your topic
• Take an oral exam on your topic
• Write a computer program (in a language of your choosing) demonstrating your project
• A combination of these, e.g. if your group had 5 people you could write a paper and a computer simulation to go with it
• Make up your own format (subject to approval)

The subject of your project is also entirely up to you. Your topic can be very applied, very theoretic, or anywhere in between. A great place to find ideas is the Wikipedia page List of probability topics. The following examples are meant to give you ideas. You can use them, but don't feel like you have to use one of the listed topics. Multiple groups/people can choose the same topic if they take the material in a sufficiently different direction. If you choose a topic that is very technical I don't expect you to understand all the details. If you are really interested in a topic but find that you are not able to understand it, ask me if there is a less technical way of handling your topic (there probably is). Some very mathy example topics include:

• Foundations of modern probability (measure theory, \(\sigma\)-algebras)
• \(\pi\)-systems and \(\lambda\)-systems, monotone class theorem
• Existence of random variables with a given distribution (Skorokhod representation)
• Existence of stochastic processes with a given law (Kolmogorov extension theorem)
• Borel-Cantelli lemmas
• Markov processes
• Queueing theory
• Kolmogorov 0-1 law
• Basic convergence theorems (dominated convergence, monotone convergence, Fatou's lemma)
• Types of convergence: a.s., in probability, \(L^p\) and their relationships
• Jensen's Inequality
• Hölder's Inequality
• A version of the Strong Law we don't cover in class
• Weierstrass approximation theorem using binomial random variables
• Measure-theoretic conditional expectation \(\mathbf{E}[X|\Sigma]\)
• Martingales
• Stopping times, optional stopping theorem
• Hitting times for random walks
• Doob's forward/backward convergence theorem
• Lévy's upward/downward convergence theorem
• Kolmogorov's Three-Series theorem
• Doob-Meyer decomposition of a random process
• Quadratic variation, its interpretation as a clock, and its relationship to martingale convergence
• Law of the iterated logarithm
• Large deviation theory, concentration inequalities
• Characteristic functions, Lévy inversion formula
• Weak convergence, Helly-Bray, convergence of characteristic functions
• A multidimensional central limit theorem
• Brownian motion
• Reflection principle
• Itô processes, connections of diffusions
• Random graphs, e.g. Erdős–Rényi
• Stochastic integration, Itô rule
• Probabilistic combinatorics
• Random measures, point processes
• Random closed sets, Fell topology
• Processes of flats
• Stochastic geometry
• Gambler's ruin
• Branching processes
• Entropy, ergodic theorem
• Maximum likelihood estimation
• Applications of probability to prove theorems in other fields

• "Paradoxes" and their resolutions
• Stock market models, arbitrage theory, fundamental theorems of asset pricing
• Risk management
• Monte-Carlo simulations
• Disease outbreaks, epidemic models
• Population dynamics
• Econometrics
• Random algorithms, e.g. randomized sorting algorithms, stochastic gradient descent
• Performance of deterministic algorithms on random inputs
• Analyze a casino game
• Show that something seemingly unlikely is actually not that unlikely
• Signal processing, error correction
• Cryptography, security properties
• Machine learning, neural networks
• Model of DNA replication
• Quantum physics
• Nuclear physics
• Statistical mechanics
• Statistical thermodynamics
• Weather models
• Oceanic flows
• Ask an interesting question and design and implement a statistical experiment