# Math 531 - Probability Theory

## Spring 2021

 Meetings: MWF 1:20PM - 2:10 PM.  Virtual classroom. Instructor: David Anderson Office: Van Vleck 617 Instructor office hours: Mondays and Thursdays, 9am - 10am

This is the course homepage. Part of this information is repeated in the course syllabus that you will find on Canvas. Here you will find our weekly schedule and updates on scheduling matters. The Mathematics Department  also has a general information page on this course. Deadlines from the Registrar's page.

### Course description

Probability theory is the part of mathematics that studies random phenomena. From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic, and computation. Probability theory is ubiquitous in the natural sciences, social sciences, and engineering, so a course in probability can be valuable in conjunction with many different majors.

Math 531 is a mathematically rigorous introduction to probability theory at the undergraduate level. This means that some rigorous analysis is required as background, but no measure theory. Math 531 is not a course in statistics. Statistics is the discipline mainly concerned with drawing inferences from data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.

Math 531 gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and goes over some classical parts of probability theory with proofs, such as the weak and strong laws of large numbers, the DeMoivre-Laplace central limit theorem, the study of simple random walk, and applications of generating functions. Math 531 serves both as a stand-alone undergraduate introduction to probability theory and as a sequel to Math/Stat 431 for students who wish to learn the 431 material at a deeper level and tackle some additional topics.

After 531 the path forward in probability theory goes as follows. At the undergraduate level there are two courses on stochastic processes: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus. Another alternative is to take 629 Measure Theory or 721 Real Analysis I as preparation for graduate probability Math/Stat 733-734.

### Prerequisites

A proof-based analysis course (such as Math 421, Math 521, or Math 375-376) or consent of the instructor. Let me emphasize that comfort with proofs and rigorous reasoning is a necessary prerequisite.

### Math 431 or 531 or both?

With the goal of avoiding regrets, here is some guidance for two types of students: (i) those who have had proof-based analysis but no probability and are choosing between 431 and 531, and (ii) those who have had both analysis and an introduction to probability such as 431 and may be wondering whether to take 531.

The great majority of the probability topics covered by 431 and 531 are the same. In 531 we gain a deeper understanding of the limit theorems (law of large numbers and central limit theorem) of probability. Math 431 is an intermediate course. It is more challenging than the recipe-oriented standard calculus and linear algebra courses, but it is not as demanding as rigorous 500 level math courses. Math 431 concentrates on calculations with examples. Examples are important in 531 also, but much class time is spent on developing theory and many examples are left to the students. In 531 homework and exams are a mixture of examples and proofs.

Recommendations. (i) If you enjoy proofs and are eager to work harder for a deeper introduction to probability, then 531 is your course. Otherwise take 431 for your introduction to probability. (ii) If you have already had analysis and 431 and wish to move ahead to new topics in probability, look at 632 and 635 for stochastic processes, and possibly at 629 as preparation for graduate probability. On the other hand, if you are looking to repeat an undergraduate introduction to probability, this time with more mathematical depth, then 531 is right for you.

### Course material

The course will follow lecture notes provided by the instructor. The notes have been written by Professors Benedek Valko and Timo Seppalainen for this course.

### Canvas

Homework and reading assignments, solutions to homework, and lecture notes will be posted on Canvas.  However, you will turn in your assignments to Gradescope.

### Piazza

Piazza is an online platform for class discussion. Post your math questions on Piazza and answer other students' questions. Our class Piazza page can be accessed from the Canvas page of the course and from this link:  piazza.com/wisc/spring2021/sp21math531001/home.

### Evaluation

Course grades will be based on quizzes and home work (20%), two midterm exams (25% each) and the cumulative final exam (30%).

• Exam 1 Tuesday, March 2nd, evening exam.  7:15PM - 9:15PM (Madison time)
• Exam 2 Tuesday, April 13th, evening exam.  7:15PM - 9:15PM (Madison time)
• Final exam: Tuesday, May 4, 2021, 7:45AM - 9:45AM.  (I did not pick this time, and there is nothing to be done about it.)

Here are the grade lines that can be guaranteed in advance. A percentage score in the indicated range guarantees at least the letter grade next to it.

[100,89) A,   [89,87) AB,  [87,76) B,  [76,74) BC,  [74,62) C,  [62,50) D,  [50,0] F.

Homework and reading assignments will be posted on the Canvas site of the course. The assignments will be usually due on Fridays at 4pm. You will submit your assignment to Gradescope.
Homework assignments will contain exercises related to the covered material, and material from sections of the textbook that have been assigned, but not explicitly covered in class.

### Quizzes

We will have a few quizzes throughout the semester.  These will be done online using Honorlock.  Each quiz is worth (in terms of grading) the same as a homework assignment.

### Instructions for homework

• Homework will be due at 4pm on Fridays.
• No late assignments will be collected.
• Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions. You are certainly encouraged to discuss the problems with your fellow students, but in the end you must write up and hand in your own solutions.
• Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answers from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.
• As always in mathematics, numerical answers alone carry no credit. It's all in the reasoning you write down.
• Be neat.
• The use of LaTeX or Overleaf is strongly suggested.  Please ask me for help with LaTeX if you have interest.

### Spring 2021 Weekly Schedule

Here we record the topics covered.  Section numbers refer to the lecture notes on Canvas.  The specific dates for the topics and readings are tentative and subject to slight changes.

 Week Topics Covered Sections in text (which you should read beforehand) 1 Jan. 25 - 29 Monday: Warm-up: random experiments with equally likely outcomes. The birthday problem, Buffon's needle problem. Wednesday and Friday: The probability space.  Probability spaces with equally likely outcomes.  Sigma-algebras and Borel sets. Warm Up, Sections 1.1 and 1.5. 2 Feb. 1 - 5 Quiz #1.  Will be 4:15pm on Tuesday or 9am on Wednesday.  Whichever time you can make. Monday: equally likely outcomes (examples), and infinite sequences of rolls of a die (i.e., Examples 1.8, 1.12, 1.13). Wednesday and Friday: Properties of probability measures, inclusion-exclusion, bound on probabilities, continuity of probability measures. Sections 1.1, 1.2. 3 Feb. 8 - 12 Finish continuity of probability measures Conditional probability independence of 2 events Sections 1.2, 1.3, beginning of 1.4. 4 Feb. 15 - 19 independent events, random variables probability mass function for discrete random variables, cumulative distribution function Sections 1.4, and 2.1 5 Feb. 22 - 26 Probability density function for absolutely continuous random variables, RVs that are neither discrete nor continuous. Random vectors Sections 2.1 and 2.2. 6 March 1 - 5 Notions of equality. Functions of random variables and vectors Independence of random variables.  MIDTERM 1 Sections 2.3 and 2.4.  Section 3.1 7 March 8 - 12 Independent random variables and independent trials. Sections 3.1 and 3.2. 8 March 15 - 19 Independent trials and sums of random variables--convolutions (and lots of named distributions) Sums of absolutely continuous random variables Sections 3.2 and 3.3. 9 March 22 - 26 exchangeable random variables (read) random walks (Monday and Wednesday) Expectations (Friday) Sections 3.4 (read on own), and 3.5. Section 4.1 10 March 29 - April 2 Expectations Sections 4.1 - 4.3 11 April 5 - 9 More expectations Sections 4.1 - 4.3 12 April 12 - 16 Laws of large numbers, Markov and Chebyshev inequalities Borel Cantelli MIDTERM 2 Sections 5.1 - 5.4 13 April 19 - 23 Convergence in distribution Central limit theorem Chapter 6 14 April 26 - 30 Confidence intervals Conditional expectations Chapters 7 and 8.