Math 531 - Probability Theory

Spring 2016

Meetings: TuTh 9:30AM - 10:45AM, B139 Van Vleck
Instructor: Benedek Valkó
Hans Chaumont
409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: Tu 1pm-2pm or by appointment

This is the course homepage that also serves as the syllabus for the course. Here you will find homework assignments, our weekly schedule, and updates on scheduling matters. The Mathematics Department also has a general information page on this course.

I will use the class email list to send out corrections, announcements, please check your email from time to time.

Important deadlines.

Course description

Math 531 is a rigorous introduction to probability theory on an advanced undergraduate level with an emphasis on problem solving. Only a minimal amount of measure theory is used (in particular, Lebesgue integrals will not be needed). The course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some of the classical results of probability theory with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on Z, applications of generating functions).

Possible subsequent probability courses are Math 632 - Introduction to Stochastic Processes and Math 635 - Introduction to Brownian Motion and Stochastic Calculus.


Grimmett - Stirzaker: Probability and Random Processes,  Oxford University Press; 3 edition

We will also use additional notes that will be posted on the Learn@UW site of the course.


A proof based analysis course (421 or 521 or 375-376) or consent of instructor.

Course content

We will cover selected sections from the first 5 chapters of the textbook. Any additional material will be uploaded to the Learn@UW site of the course.

We will cover the following topics:


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

The midterm exams will be evening exams scheduled on Wednesdays. Please check the dates at the beginning of the semester. There will not be any makeup exams for the midterms.

Exam dates:

The final grades will be determined according to the following scale:
A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D: [62,50), F: [50,0].
There will be no curving in the class, but the instructor reserves the right to modify the final grade lines.


I will use the Learn@UW website of the course to post homework assignments, solutions and additional material.

How to succeed in this course

We will cover a lot of material during the course and some of you will find the pace of the lectures a bit fast. Try to read ahead in the textbook (I will post the relevant section numbers below), this way it will be a lot easier to follow the lectures.
We might not cover the sections of the textbook in order and I will cover extra material as well. Because of this, it makes sense to attend the lectures regularly and it helps to take notes.
The midterm and final exams will contain problems which will be similar to the homework problems in difficulty. The best way to prepare for these is to do as many practice problems from the book as you can. This will also help you understand the theory a lot better!
If you have trouble solving the homework (or practice) problems then come see me in my office hours (or set up an appointment).

Homework and reading assignments

Homework and reading assignments will be posted on the Learn@UW site of the course. The assignments will be usually due on Thursdays 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 the class, together with related exercises. I plan to have a mixture of homework and reading assignments during the semester.

Instructions for the assignments


The following schedule is based on last year's course. I will update it with the actual material during the semester. I will also post a more detailed list of covered topics at the Learn@UW page.

Covered topics
Suggested reading for next week
Week 1
(1/19, 1/21)
Warm-up: random experiments with equally likely outcomes
The birthday problem, Buffon's needle problem
The Kolmogorov axioms of probability, the probability space of infinitely many coin flips
simple consequences of the additivity property
Sections 1.2-1.4
Week 2
(1/26, 1/28)
Random variables, consequences of the additivity property, inclusion-exclusion, the continuity of probability measure,

Sections 1.4-1.6
Week 3

Borel-Cantelli lemma, conditional probability, independence of events
Sections 2.1, 2.3, 2.4
Week 4

Independence of events, constructing independent experiments,
Distribution of random variables, the probability mass function, the cumulative distribution function
Sections 2.5, 3.1-3.3, 4.1-4.3
Week 5

Probability density function, joint distribution of several random variables,
Joint CDF, PMF and PDF,
Sections 3.3, 4.3
Week 6

Independence of random variables, marginal distribution from joint distribution
Expectation of random variables, general definition, discrete and continuous cases, properties of expectation
Sections 3.3, 3.4, 4.3, 7.3
Week 7
Further properties of expectation, the indicator method, variance and moments, Section 3.10
Week 8.
Coupon collector's problem, Markov and Chebyshev's inequality, Weak Law of Large numbers

Week 9
Strong Law of Large Numbers with finite fourth moment, Random walks

Week 10
The normal distribution, Stirling's formula, De Moivre-Laplace Central Limit Theorem, scaling limits of the symmetric random walk
The Poisson limit of binomial distribution.

Week 11
Applications of the CLT, sum of independent random variables
Sections 4.12, 3.7, 4.6
Week 12.
Law of rare events, Conditional distribution, conditional expectation Sections 5.1, 5.7
Week 13.

Conditional distribution, conditional expectation,
Generating functions of random variables
Sections 5.1, 5.3, 5.4, 5.7
Week 14.

Generating functions of random variables
Applications: sums of independent random variables, random walks, branching processes
Sections 5.9, 5.10
Week 15.
Generating functions and limits in distribution