Meetings: MWF 1:20-2:10pm Van Vleck B115 |

Instructor: Benedek Valko |

Office: 409 Van Vleck. |

Instructor office hours: Tu 3:30-4:15pm, W 3:45-5pm 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.

** 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**.

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.

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.

Course grades will be based on quizzes, home work and reading assignments (20%), two midterm exams (20% each) and the final exam (40%). Midterm exams will be evening exams on the following dates:

- Exam 1 Wednesday February 26, 2020

- Exam 2 Wednesday April 8, 2020

- Final exam: Friday, May 8, 2020, 7:45AM - 9:45AM, Room TBA

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

Homework
and reading assignments will be posted on the Canvas site of the
course. The assignments will be usually due on Fridays at the
beginning of the class.

Homework assignments will contain exercises related to the covered
material. The reading assignments will include material that is
not covered (or not covered in detail) in class, together with
related exercises. I plan to have a mixture of homework and
reading assignments during the semester.

- Homework is collected in class on the due date. No late
papers will be accepted.

**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.
- Put problems in the correct order and staple your pages together. Do not use paper torn out of a binder.
- Be neat. There should not be text crossed out. Recopy your problems. Do not hand in your rough draft or first attempt. Papers that are messy, disorganized or unreadable cannot be graded.