Meetings: MWF 1:20PM - 2:10 PM. Virtual classroom. |

Instructor: David Anderson |

Office: Van Vleck 617 |

Instructor office hours: Mondays and Thursdays, 9am -
10am |

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%).

- Exam 1 Tuesday, March 2nd, evening exam.
7:15PM - 9:15PM (Madison time)

- Exam 2 Tuesday, April 13th, evening exam.
7:15PM - 9:15PM (Madison time)

- Final exam: Tuesday, May 4, 2021, 7:45AM - 9:45AM. (I
did not pick this time, and there is nothing to be done about
it.)

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

Homework
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.

- Homework will be due at 4pm on Fridays.

- No late assignments will be collected.

- Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions. You are certainly encouraged to discuss the problems with your fellow students, but in the end you must write up and hand in your own solutions.
- Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answers from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.
- As always in mathematics, numerical answers alone carry no credit. It's all in the reasoning you write down.
- Be neat.

- The use of LaTeX
or Overleaf is strongly
suggested. Please ask me for help with LaTeX if you have
interest.

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.

Week |
Topics Covered |
Sections in text(which you should read beforehand) |

1Jan. 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
1.5. |

2Feb. 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. |

3Feb. 8 - 12 |
Finish continuity of
probability measures Conditional probability independence of 2 events |
Sections 1.2, 1.3,
beginning of 1.4. |

4Feb. 15 - 19 |
independent events, random
variables probability mass function for discrete random variables, cumulative distribution function |
Sections 1.4, and 2.1 |

5Feb. 22 - 26 |
Probability density
function for absolutely continuous random variables, RVs
that are neither discrete nor continuous. Random vectors |
Sections 2.1 and 2.2. |

6March 1 - 5 |
Notions of equality. Functions of random variables and vectors Independence of random variables. MIDTERM 1 |
Sections 2.3 and 2.4.
Section 3.1 |

7March 8 - 12 |
Independent random
variables and independent trials. |
Sections 3.1 and 3.2. |

8March 15 - 19 |
Independent trials and sums
of random variables--convolutions (and lots of named
distributions) Sums of absolutely continuous random variables |
Sections 3.2 and 3.3. |

9March 22 - 26 |
exchangeable random
variables (read) random walks (Monday and Wednesday) Expectations (Friday) |
Sections 3.4 (read on own),
and 3.5. Section 4.1 |

10March 29 - April 2 |
Expectations |
Sections 4.1 - 4.3 |

11April 5 - 9 |
More expectations |
Sections 4.1 - 4.3 |

12April 12 - 16 |
Laws of large numbers, Markov and Chebyshev inequalities Borel Cantelli MIDTERM 2 |
Sections 5.1 - 5.4 |

13April 19 - 23 |
Convergence in distribution Central limit theorem |
Chapter 6 |

14April 26 - 30 |
Confidence intervals Conditional expectations |
Chapters 7 and 8. |