Math/Stat 431 Introduction to the Theory of Probability

Fall 2016 Lecture

Meetings: TuTh 9:30AM-10:45AM Van Vleck B123
Instructor: Benedek Valkó
Office: 409 Van Vleck.
Office Hours: W 3:30PM-4:30PM or by appointment
E-mail: valko at math dot wisc dot edu

Important deadlines.

Course description

Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

Math 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.

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.

To go beyond 431 in probability you should take next Math 521 - Analysis, and after that one or both of these: Math 632 - Introduction to Stochastic Processes and Math 635 - Introduction to Brownian Motion and Stochastic Calculus. Those who would like a proof based introduction to probability could consider taking  Math 531 - Probability Theory (531 requires a proof based course as a prerequisite).

Where is probability used?

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. Aside from being a beautiful subject in and of itself, is used throughout the sciences and industry.  For example, in biology many models of cellular phenomena are now modeled probabilistically as opposed to deterministically. As for industry, many models used by insurance and financial companies are probabilistic in nature. Thus, those wishing to go into actuarial science or finance need to have a solid understanding of probability. Probabilistic models show up in the study of networks, making probability theory useful for those interested in computer science and information technology.


To be technically prepared for Math 431 one needs to be comfortable with the language of sets and calculus, including multivariable calculus, and be ready for abstract reasoning. Basic techniques of counting is also useful, but we will review these along the way. Probability theory can seem very hard in the beginning, even after success in past math courses.


We will use the Learn@UW website of the course to post homework assignments and solutions. The lecture notes will also be posted there.


The course follows lecture notes by David Anderson, Timo Seppäläinen, and Benedek Valkó.  These will be provided to the students at no cost.  The textbook can be found at the Learn@UW website.  The following textbooks can be used as a resource for extra practice problems. They have been placed on reserve at the math library (floor B2 of Van Vleck).

We will be using Piazza for class discussion.  The system is catered to getting you help fast and efficiently from classmates and myself.  Rather than emailing math questions to me, I encourage you to post your questions on Piazza. The students (and instructors) from the other three sections of 431 will have access to the same page, and can ask and answer questions. If you have any problems or feedback for the developers, email

Find our class Piazza page at:

Course grades will be based on homework and  quizzes (15%), two midterm exams (2x25%), and a comprehensive final exam (35%).
Midterm exams will be in the evenings of the following dates.
No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper. 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.


To help prepare for the midterm exams we will have short in-class quizzes during the first couple of weeks.

There will be no more quizzes in the semester.


Homework assignments will be posted on the Learn@UW site of the course. Weekly homework assignments are usually due on Thursday at the beginning of the class. You can also submit your solution in an electronic form via Learn@UW, but it has to arrive by 9:30AM on the due date. Note that there is a (short) homework assignment due on the first Thursday (September 8).
Some homework assignments will contain bonus problems for those who would like extra challenge. The points from the bonus problems will be converted into extra credit at the end of the semester.

Instructions for homework

Weekly schedule

Here is a tentative weekly schedule, to be adjusted as we go. The numbers refer to sections in lecture notes that can be found at the Learn@UW website.

Covered topics
Suggested reading for next week
Week 1.
9/6, 9/8
Axioms of probability, sampling, review of counting, infinitely many outcomes, review of the geometric series (Sections 1.1–1.3).
Sections 1.4–1.5, 2.1
Week 2.
9/13, 9/15
Rules of probability, random variables (Sections 1.4–1.5, 2.1). Sections 2.2–2.4
Week 3.
9/20, 9/22
Conditional probability, Bayes formula, independence (Sections 2.1–2.3).
Sections 2.4–2.5, 3.1
Week 4.
9/27, 9/29
Independent trials, birthday problem (Sections 2.4–2.5).  Sections 3.1–3.2
Week 5.
10/4, 10/6
Probability distribution of a random variable, Expectation and variance (Sections 3.1–3.2).  Sections 3.3, 4.1
Week 6.
10/11, 10/13
Gaussian distribution, Normal approximation for the binomial distribution (Sections 4.1). Midterm 1 Sections 4.1–4.3
Week 7.
10/18, 10/20
Normal approximation and the law of large numbers, confidence intervals, the Poisson distribution.
  (Sections 4.1–4.3) 
Sections 4.3, 5.1
Week 8.
10/25, 10/27
Poisson approximation, Exponential distribution
Moment generating function, using the MGF to compute moments (Sections 4.3, 5.1). 
Sections 6.1–6.3
Week 9.
11/1, 11/3
Using the MGF to identify the distribution, the distribution of functions of random variables
Joint distribution of discrete random variables, the joint pmf
 (Sections 5.1-5.2, 6.1 ).
Sections 7.1–7.2
Week 10.
11/8, 11/10
Joint distributions of continuous random variables,
Sums of independent random variables, (Sections 6.1-6.3, 7.1). 
Sections 8.1–8.2
Week 11.
11/15, 11/17
Sums of independent random variables cont., Exchangeability
Expectations of sums and products (Sections 7.1-7.2, 8.1-8.2).
Section 8.2-8.3
Week 12.
Expectation and variance of the sample mean, coupon collector,
using MGF to compute convolution (Sections 8.2-8.3)
Week 13.
11/29, 12/1
Covariance and correlation (Section 8.4). Markov’s and Chebyshev’s inequalities,
Law of large numbers, central limit theorem (9.1–9.3).
Midterm 2
Sections 10.1–10.3
Week 14.
12/6, 12/8
Conditional distributions (Sections 10.1–10.3).
Sections 10.1–10.3
Week 15.
12/13, 12/15
Conditional distributions, review (Sections 10.1–10.3). 

The Math Club provides interesting lectures and other math-related events. Everybody is welcome.