- STAC62: Stochastic Processes

STAC62F - Probability and Stochastic Processes I

Announcements

The videos are available on Quercus under Media Gallery.
Sept. 6 - review Chapter 1 in the textbook from STAB52.
Sept. 6 - I was asked to post the following.
Participate in a Study on Machine Learning and Understanding
The ELO Lab at UofT is recruiting participants for a study on machine learning and understanding. You’ll read a poem and a short story, answer questions about each, and have your facial expressions and audio recorded over Zoom. Participants can win $500. To participate or for more information, contact Milan at steven.lazic@mail.utoronto.ca.
Sept. 15 - review Chapter 2 in the textbook from STAB52.
Sept. 20 - Midterm Friday, October 10, 19:00 to 21:00 in Room IA1150.
Sept. 24 - For the lecture on Sept. 24 the first hour of the video lacks audio.
Sept. 26 - In 2023 STAC62 had two one hour midterms. Here are the tests and the solutions.
Midterm 1, Midterm 2
Oct. 1 - The midterm will cover the material in Lectures 1-10. At least 85% of the marks will be allocated to the material in Lectures 1-8. For the remaining marks, there will be a question on the multivariate normal material covered in Lecture 9 and there may be a question on stochastic processes as covered in Lecture 10. The midterm is open book and you can bring any notes and books that you wish to have. No electronic equipment (phones, watches, computers, etc.) is permitted.
Oct. 6 - Extra Office Hours this week The TA Yuxiang Jia will offer office hours Thursday and Friday 13:00-15:00, in IA3100.

Instructor

Professor Michael Evans
Office: IA4026
email: mevansthree.evans@utoronto.ca

Time and Place

Three hours of lectures per week with a video posted of each lecture. The classes are: Monday 12-13 in IAB1100 and Wednesday 12-14 in SW319.

Website

http://www.utstat.utoronto.ca/mikevans/stac62/staC622025.html

Office Hours

The in-person office hours will be right after class in my office IA4026, Monday 13-15 and Wednesday 14-15.

Evaluation

There will be a midterm of 2 hours and a final exam of 3 hours worth 40% and 60%, respectively. Both the midterm and the final will be open book (paper only, no electronics).

Course Description

STAC62 is a theoretical course. It is concerned with the mathematics of probability theory. The course material is difficult and somewhat abstract. You have to expect to work fairly hard to learn it effectively. A good understanding of the topics covered is necessary for many applications like mathematical finance, statistical computation, machine learning, statistical inference, etc.

The following topics will be covered.

1. Basic Probability
2. Random Variables and Stochastic Processes
3. Expectation
4. Convergence
5. Gaussian Processes

Lecture Notes and Texts

The course will be based on the class notes as posted here. You can print out the notes and follow along in class or with the videos. The lectures will follow the notes fairly closely.

The notes will contain Exercises which you are required to do. Solutions to the Exercises will be periodically posted typically a week after the relevant class. If you cannot do the Exercises, then you need to review the Lecture Notes until you can, otherwise you have not understood the material. I will also post some additional Exercises from time to time.

If you do not spend time doing the Exercises you will very likely do poorly in the course. It is the only way to learn this material.

The first four chapters and Chapter 11 of the online textbook from STAB52 are also relevant to the course. You are required to review this material. Some problems for the Exercises will be taken from this book. The text Probability and Random Processes: by Grimmett and Stirzaker may also prove to be helpful but it is generally above the level of this course.

The lecture slides will be posted below ahead of each class and correspond to the videos.

1. Basic Probability

Lecture 1 - what does probability mean, sample space, sigma algebra, probability measures and models
Solutions to Exercises
Lecture 2- Borel sets, ellipsoidal regions
Solutions to Exercises
Lecture 3 - limit of a sequence of sets, continuity of P, Boole's inequality, Borel-Cantelli lemma, conditional probability
Solutions to Exercises
Lecture 4 - statistical independence
Solutions to Exercises

2. Random Variables and Stochastic Processes

Lecture 5 - random variables, inverse images, marginal probability model, random vectors
Using the Good Sets Principle
Solutions to Exercises
Lecture 6 - cumulative distribution functions, discrete distributions, multinomial distribution
Solutions to Exercises
Lecture 7 - absolutely continuous distributions, density functions, the standard multivariate normal distribution
Solutions to Exercises
Lecture 8 - change of variable discrete case and marginal distributions
Solutions to Exercises
Lecture 9 - change of variable absolutely continuous case case and marginal distributions
Solutions to Exercises
Lecture 10 - definition of a stochastic process, Kolmogorov Consistency Theorem, Gaussian processes
See Solutions for Lecture 11 for Solutions to Exercises for Lecture 10.
Lecture 11 - mutually statistically independent random variables
Solutions to Exercises
Lecture 12 - conditional distributions, discrete and absolutely continuous cases, marginals and conditionals of the multivariate normal
Solutions 1 to Exercises
Solutions 2 to Exercises

3. Expectation

Lecture 13 - simple functions, expectation of a simple function, positive and negative parts of a r.v., general definition of expectation of a r.v.
Lecture 14 - properties of E