# Math 531 Probability Theory

## Spring 2023

 Meetings: MWF 1:20-2:10pm Van Vleck B115 Instructor: Benedek Valko Office: 409 Van Vleck. Instructor office hours: Tu 3:45-4:45pm, Fr 2:20-3:20pm or by appointment

This is the course homepage. Part of this information is repeated in the course syllabus that you find on Canvas. Here you will find our weekly schedule and updates on scheduling matters. The Mathematics Department has also 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 natural science, social science 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, 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

MATH 376, 421, or 521 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program.

### 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. Here are two textbooks that could be used as additional resources. These books are not required for the course, but they can be recommended for students who wish to have a book to rely on. These two books present the material on different levels of sophistication and hence are somewhat complementary to each other.

Students who would benefit from reading a gentle introduction to probability on the side can consider acquiring the textbook for Math 431:
Anderson-Seppäläinen-Valkó: Introduction to Probability,  Cambridge University Press, 2017.

The following is an example of a textbook that is pitched more or less at the right level for 531:
Grimmett-Stirzaker: Probability and Random Processes,  Oxford University Press, 3rd edition.
Grimmett-Stirzaker is a more comprehensive book. It covers also part of the material of Math 632.

### Canvas

All relevant course materials will be posted on Canvas.

### 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.

### Evaluation

Course grades will be based on quizzes, home work assignments (15%), two midterm exams (25% each) and the final exam (35%).

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.

Final letter grades are not curved but the grade lines above may be lowered at the end. Class attendance is not part of