Meetings: MWF 8:509:40 Van Vleck B115 
Instructor: Benedek Valko 
Office: 409 Van Vleck. 
Instructor office hours: M 1112 or by appointment 
TA: Xiao Shen

TA office hours: Tu 46, Sat 35, Sun 35 in VV 101. 
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, DeMoivreLaplace central limit theorem, the study of simple random walk, and applications of generating functions. Math 531 serves both as a standalone 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 733734.
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 recipeoriented 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:
AndersonSeppäläinenValkó:
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:
GrimmettStirzaker: Probability and Random Processes,
Oxford
University Press, 3rd edition.
GrimmettStirzaker 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:
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 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.
Week  Covered Topics 
Suggested reading for next week 

1 
Warmup: random experiments with equally likely outcomes.
The birthday problem, Buffon's needle problem The Kolmogorov axioms of probability. 
Ch 1 
2 
Infinite probability spaces, the probability space of
infinitely many die rolls, 
Ch 1 
3 
simple consequences of the additivity property, the continuity of measure, conditional probability  Sections 1.4, 2.1, 2.2 
4 
independent events, random variables probability mass function for discrete random variables, cumulative distribution function 
Sections 2.12.4 
5 
Probability density function for absolutely continuous
random variables, random vectors 
Sections 3.1, 3.2 
6 
functions of random variables, independent random
variables, independent trials MIDTERM 1 
Sections 3.13.2 
7 
Named distributions from independent trials (Bernoulli,
binomial, geometric, negative binomial, Poisson,
exponential, multinomial), convolution 
Sections 3.33.5 
8 
convolution, exchangeability, simple random walk 
Section 4.14.3 
SPRING BREAK 

9 
expectation of random variables 

10 

11 

12 

13 

14 