# Probability Theory Homework ^NEW^

A mathematically rigorous course in probability theory which uses measure theory but begins with the basic definitions of independence and expected value in that context. Law of large numbers, Poisson and central limit theorems, and random walks.

## Probability Theory Homework

Other textbooks that you can look at for a different perspective: Measure Theory by Cohn or Real Analysis: Modern Techniques and Their Applications by Folland for measure theory background. Probability and Measure by Billingsley, A Course in Probability Theory by Chung, A First Look at Rigorous Probability Theory by Rosenthal for the main subject material of the course.

Homework will usually be due on Fridays. Each student is granted two free passes to turn in homework up to a week after the posted due date. Beyond this, late work will not be accepted without a compelling reason. You may not use a late pass on the final homework assignment.

You are encouraged to work with each other on the homework. Everything you write should be in your own individual words; direct copying is forbidden! Also, though it is perfectly acceptable to consult outside resources for help on occasion, you should spend a reasonable amount of time thinking about the problem and attempting your own solution before doing so, and you are required to explicitly cite all sources other than the official text.

Homework, Quizzes, etc.: There will be 12 weekly quizzes, 12 homeworks, and a computer project. Check the schedule for the exact dates. You should know how to solve every homework problem and turn in each homework on the corresponding due date, but do not expect homework problems to be thoroughly graded. You are welcome to use any help with all the work other than quizzes and exams. During quizzes and exams, you are on your own, with only a writing/erasing instrument, and a calculator (without internet connection or any other communication capabilities). Devices such as tablets, smart phones, and laptop computers are not allowed during exams. If in doubt, please talk to me in advance about the particular calculator you plan to use.

Please keep in mind that homework assignments are minimal requirements. To succeed in the class, you need to solve more problems, from the book and/or from other sources. Keep all your notes, including scratch paper, until after you are completely done with this class.

Quizzes will take place during discussion sections, either on Tuesday or on Thursday. The exact dates are in the class schedule. The teaching assistant is responsible for preparing, administering, and grading quizzes and for grading the homeworks; I will grade the computer project and (most probably) the exams.

Missed work. The general rule: no make-up exams or quizzes, and no late submissions of homeworks or projects (but early submissions, especially in electronic format, are welcome). Emergencies will be handled on a case-by-case basis. If you miss the final exam, with a valid excuse, you get an incomplete in the class; an incomplete is a major inconvenience for a number of people, including yourself, so, please, do not miss the final exam.

Textbook: The main source we will follow are Bruce Driver's excellent Probability notes:Probability Tools with Examples, by Bruce DriverHere are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel. A Probability Path by Sidney Resnick Probability Theory: A Comprehensive Course by Achim Klenke If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here.

Coursework: There will be weekly homework assignments due on Mondays (starting in Week 2); they are posted below.There will be 5 quizzes, in weeks 1, 3, 5, 7, and 9 of the quarter; they will take place during the scheduled Thursday lecture time, withan alternate sitting available in the late evening to accommodate those in distant time-zones. And there will be a take-home final examduring exam week. Timing and due dates for all courses assessments can be found below.

Piazza is an online discussion forum. It will allow you to post messages (openly or anonymously) and answer posts made by yourfellow students, about course content, homework, quizzes, etc. The instructor and TA will also monitor and post to Piazza regularly.You can sign up here. Note: Piazzahas an opt-in "Piazza Careers" section which, if you give permission, will share statistics about your Piazza use withpotential future employers. It also has a "social network" component, based on other students who've shared a Piazza-basedclass with you, that comes with the usual warnings about privacy concerns. Piazza is fully FERPA compliant, and is an allowed resourceat UCSD. Nevertheless, you are not required to use Piazza if you do not wish.

Gradescope is an online tool for uploading and grading assignments and exams (it is now under the umbrella of Turnitin). You willturn in your homework, quizzes, and final exam through Gradescope, and you will access your graded assessments there as well. Access the class Gradescope sitehere.

We will be communicating with you and making announcements through an online question and answer platform called Piazza (sign up link: piazza.com/ucsd/winter2016/math180a).We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response.Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework throughPiazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors,in office hours. You can also post private messages to instructors on Piazza, which we prefer to email.

Below are the tablet slides from synchronous class meetings (Q&A sessions). DateActivityTablet Slides Oct 6Q&A 280A-Zoom-Notes-10-6.pdf Oct 8Q&A 280A-Zoom-Notes-10-8.pdf Oct 13Q&A 280A-Zoom-Notes-10-13.pdf Oct 15Q&A 280A-Zoom-Notes-10-15.pdf Oct 20Q&A 280A-Zoom-Notes-10-20.pdf Oct 22Q&A 280A-Zoom-Notes-10-22.pdf Oct 27Q&A 280A-Zoom-Notes-10-27.pdf Oct 29Q&A 280A-Zoom-Notes-10-29.pdf Nov 3Q&A 280A-Zoom-Notes-11-3.pdf Nov 10Q&A 280A-Zoom-Notes-11-10.pdf Nov 17Q&A 280A-Zoom-Notes-11-17.pdf Nov 19Q&A 280A-Zoom-Notes-11-19.pdf Nov 24Q&A 280A-Zoom-Notes-11-24.pdf Dec 1Q&A 280A-Zoom-Notes-12-1.pdf Dec 8Q&A 280A-Zoom-Notes-12-8.pdf Dec 10Q&A 280A-Zoom-Notes-12-10.pdf Syllabus Math 280A is the first quarter of a three-quarter graduate level sequence in the theory of probability. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability.A strong background in undergraduate real analysis at the level of Math 140AB is essential for success in Math 280A. In particular, students should becomfortable with notions such as countable and uncountable sets, limsup and liminf, and open, closed, and compact sets, and should be proficient at writingrigorous epsilon-delta style proofs. Graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter coursein stochastic processes which will be offered in Winter 2021. See also this page,maintained by Ruth Williams, for more information on graduate courses in probability at UCSD.

According to the UC San Diego Course Catalog, the topics covered in the full-year sequence 280ABC include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, convergence in distribution, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. Given the current pandemic crisis and emergency remote teaching modality, it is more difficult than usual to predict what pace we will work through this material, and where the dividing line between 280A and 280B will occur.

Prerequisite: Students should have mastered the fundamentals of real analysis in metric spaces, as covered in MATH 140AB, before taking this course. An undergraduate course in probability, comparable to MATH 180A, and further courses in stochastic processes, comparable to MATH 180BC, would also be an asset, but are not absolutely necessary.

Homework: Homework assignments are posted below, and will be due by 9pm (with a 30-minutes"late" grace period in case of technical glitches)on Mondays throughout thee quarter. You must turn in your homework through Gradescope; if you have produced it on paper,you can scan it or simply take clear photos of it to upload. You must select pages corresponding to your solutions of problemsduring the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you havenot selected pages when the TA begins grading, the TA will not grade your assignment and you will receive a grade of 0 on it.No appeals of this policy will be considered. It is allowed and even encouraged to discuss homework problemswith your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Regrade Policy: Your quizzes, homeworks, and final exam will be graded using Gradescope.For quizzes and the final exam, you will be able to request regrades through Gradescope for a specified window of time.Be sure to make your request within the specified window of time; no regrade requests will be accepted after thedeadline. For homework, any clerical erros (such as a problem or page that the TA accidentally missed when grading)should be discussed with the TA during office hours. Grading rubrics are not negotiable; if the TA has taken offsome number of points from your solution, there is a sound pegagogical reason for this. This is a PhD class in mathematics.We are not focused on numerical grades here; we are focused on learning deep and challenging material. The grading is meantas a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the conceptsand thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to thinkand work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you didnot receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for examplethrough Piazza), should you consider approaching your TA or instructor for further explanation of grading choices. However,please understand that these conversations will not result in a change in your grade unless there has been some clear clerical error,such as the TA accidentally missing part of your solution. The TA will not change their assessment of a students work due to conversationsor complaints after the fact.