- STAC62: Stochastic Processes

STAC62F - Probability and Stochastic Processes I

Announcements

(Dec. 11) Office hours
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(Dec. 6) Office hours
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(Dec. 4) I will put up a link to my office hours here on Wednesday, Dec. 6 at 1pm and Monday, Dec. 11 at 1pm. The office hours will last for 2 hours or until I have answered all the questions.
(Nov. 30) Final Exam Dec. 13, 14:00-17:00 in HW216. The exam is open book with the only restriction being no computers and no devices allowing for communication (simple calculators are fine). There are 18 questions each worth 5 marks and one question worth 10 marks. The 5 mark questions cover the material in Chapters 1-3 while the 10 mark question is from Chapter 4.
(Nov. 27) Office hours
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(Nov. 22) Office hours
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(Nov. 20) Office hours
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(Nov. 18) Solutions to 2020 final
(Nov. 18) I uploaded a corrected version of the solution to midterm 2. The arithmetic in Question 2(d) in the last 3 lines had an error but I don't believe anyone lost marks because of this as I checked this, but let me know if you think you did. Also, in the plot for question 3(b) I plotted the sample function at omega=-1 rather than omega=0 which is what the question asked for. I'm elminating this part of the question from the evaluation as I'm treating these 5 marks as part of the 17 marks added to everybody.
(Nov. 15) The marks for midterm have been distributed. The class average was about 48 and I have recorded this mark plus 17 as your grade so the overall average is 65. Some people got over 100 and, based on what I know, they did the Exercises and spent time on them. Without spending time on the Exercises you cannot expect to get a good mark as that is how you learn this material. If some of your work was not graded or you want a regrade on a question email me identifying the relevant questions.
(Nov.15) Office Hours s Join Zoom Meeting
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(Nov. 13) Office Hours Join Zoom Meeting
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(Nov. 12) Solutions to Midterm2 here.
(Nov. 8) Office hours
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(Nov. 6) Office hours
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(Nov. 3) Office hours
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(Nov. 2) I will post a Zoom link here tomorrow (Friday, Nov. 3) at 1pm for office hours.
(Oct. 30) Here is a second midterm from a previous year.
(Oct. 30) I have to cancel the class (and office hours) today. I have a pinched nerve in my back and can't drive my car! My apologies for the late notice.
(Oct. 29) Midterm 2 will cover all the material in lectures 1-12. Online: Wednesday, November 8, 17:00-18:00. You will receive an email from Crowdmark at 4:55 pm giving you access to the midterm. You have until 6:15 pm to submit your answers.
(Oct. 25) Final Exam December 13, 14:00-17:00 in HW215, HW216.
(Oct. 14) The midterm is marked and I will release the marks on Oct. 16. Generally it was well done. There were some people who did not upload properly so their tests or some questions are not yet marked. After the marks are released and, if there is no mark registered for you on certain questions, then email me and I will mark these provided you had sent me your solutions via email on the day of the test. If you have queries or disagreements with your mark, then please see me about this during office hours. The solutions for Midterm 1 can be found here.
(Sept. 30) Midterm 1 On Wednesday, Oct. 4 you will receive an email from Crowdmark at 4:55 pm giving you access to the midterm. You have until 6:15 pm to submit your answers.
(Sept. 24) All students enrolled in the course should receive an email from Crowdmark at 12 noon on Monday, Sept. 25. If you don't receive this please contact me.
(Sept. 23) The midterm on October 4 will cover material up to the end of Lecture 5. A previous midterm with solutions can be found here but note that question 4 is based on material in Lecture 6 so this question is not entirely relevant to the midterm.
(Sept. 22) Dates of the Midterms
Midterm 1 - Wednesday, October 4, 17:00-18:00
Midterm 2 - Wednesday, November 8, 17:00-18:00

The midterms are online using Crowdmark and the "honor code" is in play. So any materials can be used to help write the exam except you cannot consult with others concerning your answers. The final is an in-person 3 hour exam during the exam period in December.
(Sept. 21) Lectures 3 and 4 have been replaced with versions where the subsections are now correctly numbered.
(Sept. 17) Exercise 1.4.5. I've added the specification that x_0=(0,0)'.
(Sept. 15) The office hours on Monday have been changed to 1-4, so another hour has been added.
(Sept. 15) Solutions to the Exercises in Lecture 1 are now posted beside the link to Lecture 1 below.
Note the website address was incorrect previously as it is
http://www.utstat.utoronto.ca/mikevans/stac62/staC622023.html
The videos are available on Quercus under Media Gallery.

Announcement for Data Science and Statistics Society 2023-2024

(Sept. 21, 2023)

Title: Exciting opportunity to 10x your resume with DS3

Join the DS3 academic team for an exciting opportunity!

Consultant: Dive into diverse research projects across various fields like healthcare, finance, and politics. Advise on study design, data collection methods, visualization, and interpretation. Develop both technical and soft skills while delivering actionable insights to clients. Academic Representative: Showcase your passion and academic expertise. Lead review seminars, host paper reading groups, and practice technical writing for our monthly newsletters on Medium.

Fill out the Google Form here to apply: Link to Google Form

Instructor

Professor Michael Evans
Office: IC498
Tel: (416)287-7274
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 12-1pm in SW319 and Wednesday 12-2 in HW 216.

Website

http://www.utstat.utoronto.ca/mikevans/stac62/staC622023.html

Office Hours

The in-person office hours will be right after class in my office, Monday 1-4pm and Wednesday 2-3pm.

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. The lecture slides correspond to the videos on Quercus. A * by the lecture means it has been updated for 2023.

1. Basic Probability

Lecture 1 *, Solutions to Exercises - what does probability mean, sample space, sigma algebra, probability measures and models
Lecture 2 *, Solutions to Exercises - Borel sets, ellipsoidal regions
Lecture 3 *, Solutions to Exercises and Solution to Exercise 1.5.3 - limit of a sequence of sets, continuity of P, Boole's inequality, Borel-Cantelli lemma, conditional probability
Lecture 4 *, Solutions to Exercises - statistical independence

2. Random Variables and Stochastic Processes

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

3. Expectation

Lecture 13 *, Solutions to Exercises - 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
Lecture 15 *, Solutions to Exercises - convergence with probability 1, monotone and dominated convergence theorems, computing expectations
Lecture 16 *, Solutions to Exercises Part 1, Solutions to Exercises Part 2 - properties of mean vectors and variance matrices
Lecture 17 *, Solutions to Exercises - mean, autocovariance and autocorrelation functions for stochastic processes, random walks
Lecture 18 *, Solutions to Exercises - Markov inequality, Chebyshev inequality, Chernoff bounds, Cauchy-Schwartz inequality, best affine predictor
Lecture 19 *, Solutions to Exercises - Jensen's inequality, Kullback-Leibler distance
Lecture 20 *, Solutions to Exercises - conditional expectations, martingales
Lecture 21 *, Solutions to Exercises - probability and moment generating functions, characteristic functions

4. Convergence

Lecture 22 * - convergence in distribution, weak law of large numbers, Central Limit Theorem
Lecture 23 *, Solutions to Exercises - convergence in probability, convergence in mean of order r, relationships among the modes of convergence, geometry of L^2

5. Gaussian Processes

Lecture 24 * - strictly stationary processes, discrete time, autoregressive of order 1 Gaussian process
Lecture 25 * - Wiener process (Brownian motion)

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.

Evaluation

There will be two tests of 1 hour each and a final exam of 3 hours worth 25%, 25% and 50%, respectively. The two 1 hour tests will be online and the final will be in-person.