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.

** 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 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

the grading.