|Meetings: MWF 1:20PM - 2:10 PM. Virtual classroom.
|Instructor: David Anderson|
|Office: Van Vleck 617|
|Instructor office hours: Mondays and Thursdays, 9am -
This is the course homepage. Part of this information is repeated in the course syllabus that you will find on Canvas. Here you will find our weekly schedule and updates on scheduling matters. The Mathematics Department also has 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 the natural sciences, social sciences, 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, the 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.
Gradescope is an online grading system. You will upload
your HW, quizzes, and exams to Gradescope when they are completed.
Course grades will be based on quizzes and home work (20%), two midterm exams (25% each) and the cumulative final exam (30%).
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.
and reading assignments will be posted on the Canvas site of the
course. The assignments will be usually due on Fridays at 4pm.
You will submit your assignment to Gradescope.
Homework assignments will contain exercises related to the covered material, and material from sections of the textbook that have been assigned, but not explicitly covered in class.
We will have a few quizzes throughout the semester. These
will be done online using Honorlock. Each quiz is worth (in
terms of grading) the same as a homework assignment.
Here we record the topics covered. Section numbers refer to
the lecture notes on Canvas. The specific dates for the
topics and readings are tentative and subject to slight changes.
||Sections in text
(which you should read
Jan. 25 - 29
|Monday: Warm-up: random
experiments with equally likely outcomes. The birthday
problem, Buffon's needle problem.
Wednesday and Friday: The probability space. Probability spaces with equally likely outcomes. Sigma-algebras and Borel sets.
|Warm Up, Sections 1.1 and
Feb. 1 - 5
Quiz #1. Will be 4:15pm on Tuesday or 9am on
Wednesday. Whichever time you can make.
Monday: equally likely outcomes (examples), and infinite sequences of rolls of a die (i.e., Examples 1.8, 1.12, 1.13).
Wednesday and Friday: Properties of probability measures, inclusion-exclusion, bound on probabilities, continuity of probability measures.
|Sections 1.1, 1.2.
Feb. 8 - 12
|Finish continuity of
independence of 2 events
|Sections 1.2, 1.3,
beginning of 1.4.
Feb. 15 - 19
|independent events, random
probability mass function for discrete random variables, cumulative distribution function
|Sections 1.4, and 2.1
Feb. 22 - 26
function for absolutely continuous random variables, RVs
that are neither discrete nor continuous.
|Sections 2.1 and 2.2.
March 1 - 5
|Notions of equality.
Functions of random variables and vectors
Independence of random variables.
|Sections 2.3 and 2.4.
March 8 - 12
| Independent random
variables and independent trials.
||Sections 3.1 and 3.2.
March 15 - 19
|Independent trials and sums
of random variables--convolutions (and lots of named
Sums of absolutely continuous random variables
|Sections 3.2 and 3.3.
March 22 - 26
random walks (Monday and Wednesday)
|Sections 3.4 (read on own),
March 29 - April 2
||Sections 4.1 - 4.3
April 5 - 9
||Sections 4.1 - 4.3|
April 12 - 16
|Laws of large numbers,
Markov and Chebyshev inequalities
|Sections 5.1 - 5.4
April 19 - 23
|Convergence in distribution
Central limit theorem
April 26 - 30
|Chapters 7 and 8.