Math 280A

Fall 2020, Lecture B00 (Kemp) TR 12:30-1:50pm

Probability Theory I

Course Information

Instructional Staff

NameRoleOfficeE-mail
Todd Kemp Instructor Zoom tkemp@ucsd.edu
Eva Loeser Teaching Assistant Zoom ehloeser@ucsd.edu

Our office hours, and all relevant scheduled course activities, can be found in the following calendar.

Calendar



Master List of Meetings and Assessments


All dates and times listed below are Pacific Time.

DayTimeLocation
Lectures AsynchronousAsynchronousYouTube
Lecture Q&A Tuesday, Thursday12:30-1:50pmZoom
Quiz 1 Thursday, Oct 81-1:50pm, 7-7:50pmZoom, Gradescope
Quiz 2 Thursday, Oct 221-1:50pm, 7-7:50pmZoom, Gradescope
Quiz 3 Thursday, Nov 51-1:50pm, 7-7:50pmZoom, Gradescope
Quiz 4 Thursday, Nov 191-1:50pm, 7-7:50pmZoom, Gradescope
Quiz 5 Thursday, Dec 31-1:50pm, 7-7:50pmZoom, Gradescope
Homework 1 Monday, Oct 129-9:30pmGradescope
Homework 2 Monday, Oct 199-9:30pmGradescope
Homework 3 Monday, Oct 269-9:30pmGradescope
Homework 4 Monday, Nov 29-9:30pmGradescope
Homework 5 Monday, Nov 99-9:30pmGradescope
Homework 6 Monday, Nov 169-9:30pmGradescope
Homework 7 Monday, Nov 239-9:30pmGradescope
Homework 8 Monday, Nov 309-9:30pmGradescope
Homework 9 Monday, Dec 79-9:30pmGradescope
Take-Home Final Exam Available: Sunday, Dec 13, 2:30pmDue: Friday, Dec 18, 2:30pmGradescope

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Recorded Lectures


The Lectures for this course are pre-recorded, and available on YouTube.

The lectures are not divided into even 80-minute chunks. They are organized by topic, concept, or example.

Below, you will find a list (with links) of the lecture videos you should watch prior to the listed date, along with pdf slides of the tablet output during those lecture videos.

Lecture VideosSlides (Before)Slides (After)
Banach Tarski Banach Tarski 0. Banach Tarski Banach Tarski (Before) 0. Banach Tarski Banach Tarski (After)
Probability Motivation 1.1 Probability Motivation (Before) 1.1 Probability Motivation (After)
Sigma Fields 1.2 Sigma Fields (Before) 1.2 Sigma Fields (After)
Measures: Definition and Examples 2.1 Measures (Before) 2.1 Measures (After)
Finitely Additive Measures 2.2 Finitely Additive Measures (Before) 2.2 Finitely Additive Measures (After)
Stieltjes Premeasures 3.1 Stieltjes Premeasures (Before) 3.1 Stieltjes Premeasures (After)
Outer Measure 3.2 Outer Measure (Before) 3.2 Outer Measure (After)
Outer Pseudo-Metric 4.1 Outer Pseudo-Metric (Before) 4.1 Outer Pseudo-Metric (After)
The Extension Theorem 4.2 Extension Theorem (Before) 4.2 Extension Theorem (After)
Uniqueness, and Sigma-Finite Extension 4.3 Uniqueness, Sigma-Finite (Before) 4.3 Uniqueness, Sigma-Finite (After)
Radon Measures 5.1 Radon Measures (Before) 5.1 Radon Measures (After)
Lebesgue Measure 5.2 Lebesgue Measure (Before) 5.2 Lebesgue Measure (After)
Random Variables Motivation 6.1 Random Variables Motivation (Before) 6.1 Random Variables Motivation (After)
Measurable Functions 6.2 Measurable Functions (Before) 6.2. Measurable Functions (After)
Robustness of Measurability 7.1 Robustness of Measurability (Before) 7.1 Robustness of Measurability (After)
Riemann-Stieltjes Integration 7.2 Riemann-Stieltjes Integration (Before) 7.2 Riemann-Stieltjes Integration (After)
Simple Integration 8.1 Simple Integration (Before) 8.1 Simple Integration (After)
The Lebesgue Integral and the MCTheorem 8.2 MCTheorem (Before) 8.2 MCTheorem (After)
Fatou and Borel-Cantelli 9.1 Fatou BC (Before) 9.1 Fatou BC (After)
$L^1$ and the Dominated Convergence Theorem 9.2 $L^1$ and the DCT (Before) 9.2 $L^1$ and the DCT (After)
Integrals and Derivatives 10.1 Integrals and Derivatives (Before) 10.1 Integrals and Derivatives (After)
Lebesgue vs. Riemann 10.2 Lebesgue vs. Riemann (Before) 10.2 Lebesgue vs. Riemann (After)
Radon-Nikodym 11.1 Radon-Nikodym (Before) 11.1 Radon-Nikodym (After)
Laws Revisited 11.2 Laws Revisited (Before) 11.2 Laws Revisited (After)
$L^2$ and Covariance 12.1 $L^2$ and Covariance (Before) 12.1 $L^2$ and Covariance (After)
The Weak Law of Large Numbers 12.2 WLLN (Before) 12.2 WLLN (After)
Convergence in Measure 13.1 Conv. in Meas. (Before) 13.1 Conv. in Meas. (After)
$L^p$ is Complete 13.2 $L^p$ Complete (Before) 13.2 $L^p$ Complete (After)
Dynkin's Multiplicative Systems Theorem 14.1 Multiplicative Systems (Before) 14.1 Multiplicative Systems (After)
Product Measure 14.2 Product Measure (Before) 14.2 Product Measure (After)
Tonelli's and Fubini's Theorems 14.3 Tonelli-Fubini (Before) 14.3 Tonelli-Fubini (After)
Independence 15.1 Independence (Before) 15.1 Independence (After)
Independent Random Variables 15.2 Independent Random Variables (Before) 15.2 Independent Random Variables (After)
Kolmogorov's Extension Theorem (I) 16.1 Kolmogorov Extension (I) (Before) 16.1 Kolmogorov Extension (I) (After)
Kolmogorov's Extension Theorem (II) 16.2 Kolmogorov Extension (II) (Before) 16.2 Kolmogorov Extension (II) (After)
Supplement: Proof of Tychonoff's Theorem 16.3 Tychonoff Proof
Kolmogorov's 0-1 Law 17.1 Kolmogorov's 0-1 Law (Before) 17.1 Kolmogorov's 0-1 Law (After)
Convolution 17.2 Convolution (Before) 17.2 Convolution (After)
Strong Law of Large Numbers (I) 18.1 SLLN (I) (Before) 18.1 SLLN (I) (After)
Kolmogorov's Convergence Criterion 18.2 Kolmogorov Conv (Before) 18.2 Kolmogorov Conv (After)
Strong Law of Large Numbers (II) 19.1 SLLN (II) (Before) 19.1 SLLN (II) (After)
Renewal Theory 19.2 Renewal (Before) 19.2 Renewal (After)

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 be comfortable with notions such as countable and uncountable sets, limsup and liminf, and open, closed, and compact sets, and should be proficient at writing rigorous epsilon-delta style proofs. Graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter course in 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.

Lectures:  The lectures for this course will be recorded asynchronously, and made available on YouTube. You should engage with the relevant videos before each "Lecture" session. The schedule Lecture times will be devoted to Q&A sessions and quizzes. The Q&A sessions will be recorded, with recordings available on Canvas; the quizzes will not be recorded (but will take place live on Zoom), and will be available in a "second sitting" to accommodate those students in far-flung time-zones.

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 problems during the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you have not 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 problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Quizzes:  There will be 5 quizzes throughout the quarter, to test your fundamental knowledge of the course material. You will write them on Thursdays 1-1:50pm or 7-7:50pm, live on Zoom (so that your instructional team can answer questions if any arise), and turn them in via Gradescope. No collaboration (with other humans or with online resources) is allowed on quizzes.

Lowest scores:   Of the 9 homework assignments, only your highest 7 scores will count towards your final grade. Of the 5 quizzes, only your highest 4 scores will count towards your final grade.

Final Exam:  The final exam will be take-home. It will be available and due during exam week; more details about the exam window will be available later in the term. No collaboration (with other humans or with online resources) is allowed on the final exam. We reserve the right to invite students to follow-up Zoom meetings after the final exam to confirm that the work was completed without collaboration. We reserve the scheduled final exam time-slot for this purpose.

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 the deadline. 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 off some 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 meant as a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the concepts and thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to think and work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you did not receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for example through 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 conversations or complaints after the fact.

Grading: Your cumulative average will be determined by the following weighting:


We reserve the right to add other optional grading schemes at a later date; if so, your final grade will be computed according to whichever scheme gives you the highest score.

Academic Integrity:  UC San Diego's code of academic integrity outlines the expected academic honesty of all students and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers and instructors respectfully in all forms of interaction (Zoom meetings, email, Piazza discussions, etc).

Assessment Versioning: following UCSD (and common) practice, recommended by the Academic Integrity Office, assessments given at non-overlapping times will be comparable, but may not be identical. In particular: the two sittings of each Quiz (and potentially quizzes given within each sitting) may not have exactly the same questions, but will be designed to cover the same material and be of equivalent levels of difficulty. This practice is meant to maintain course integrity, avoiding unpermitted collaboration (either intentional or accidental).

OSD Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter will be issued by the OSD electronically. Please make arrangements to discuss your accommodations with me in advance (by the end of Week 2). We will make every effort to arrange for whatever accommodations are stipulated by your AFA letter. For more information, see here.

Covid-19 Accommodations: Due to the unprecedented Covid-19 pandemic, our lectures and all meetings will be remote, using Zoom, this year. Some of you are residing in different time-zones (and different continents), and these present additional challenges. To accommodate, all lectures are pre-recorded asynchronously, and the Q&A sessions are also recorded. Office hours will not be recorded, but will be distributed throughout different days and times so all students should be able to attend in regular work hours; failing that, individual Zoom meetings can be made. Synchronous quizzes will be held in two "sittings": 1-1:50pm and 7-7:50pm Pacific Time, which should cover everyone enrolled in the class, during daylight hours.

If these accommodations are still insufficient due to a severe Covid-19 pandemic related issue you have, please contact me no later than Friday, October 16, to discuss other arrangements.

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Homework


Weekly homework assignments are posted here. Homework is due by 9:00pm on the posted date, through Gradescope. Late homework will not be accepted.