MAT 1600 Probability, University of Toronto 2020
 

MAT 1600 Mathematical Probability I, Fall 2020

Almut Burchard, Instructor

Please register in advance for zoom lectures. After registering, you will receive a confirmation email containing information about joining the meeting.
How to reach me: Almut Burchard, BA 6234, 978-3318.
almut @math , www.math.utoronto.ca/almut/
Lectures M 12:10-12PM, W 12-10-1pm, online
Office hours TBA
Teaching assistant: David Pechersky, david.pechersky @mail.utoronto.ca .
Textbook:   Dmitry Panchenko, Lecture Notes on Probability
  • Measure Theory: abstract measures and σ-algebras, Monotone Class Theorem, outer measures and Caratheorodry's theorem, Borel sets and Lebesgue measure
  • Integration: convergence theorems, Fubini's theorem, change of variables and polar coordinates in Rn
  • Lebesgue Differentiation: Hardy-Littlewood maximal function, density points, Radon-Nikodym theorem
  • Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, Lp -spaces
    Hölder and Minkowski inequalities
A good secondary source is Probability: Theory and Examples, by Rick Durrett.
Evaluation:
40% : Homework: weekly exercises (due Wednesdays, on crowdmark)
20% : Midterm test (Wednesday November 5, online)
40% : final examination (online)
Remarks. Please discuss lectures and homework problems among yourselves and with me, and consult other sources. But write up your assignments in your own words, and be ready to defend them! Your work will be judged on the clarity of your presentation as well as correctness and completeness.

Tentative sschedule:

First lecture (September 10-14)
Meeting the students; brief overview
Week 1 (September 14-18)
Chapter 1 -- Introduction (Sections 1-2)
M: Probability spaces
W: Random Variables
Assignment 1 [pdf, tex] (due September 23)
Week 2 (September 21-25)
Chapter 1 -- Introduction (Section 3)
M: Kolmogorov's extension theorem   (recording)
W: Proof of the extension theorem   (recording)
Assignment 2 [pdf, tex]; (due September 30)
[Outline of a solution of Problem 2]
Week 3 (September 28 - October 2)
Chapter 1 -- Introduction (Section 3) and Chapter 2 -- Laws of Large Numbers (Section 1)
M: Conditional expectation   (recording)
W: Inequalities for sums of independent random variables   (recording)
Assignment 3 [pdf, tex] (due October 7)
Week 4 (October 5-9)
Chapter 2 -- Laws of Large Numbers (Sections 2-3) Chapter 2 -- Laws of Large Numbers (Section 1)
M: Variance and covariance   (recording)
W: Weak law of large numbers. Approximation of Binomial by Poisson   (recording)
Assignment 4 [pdf, tex] (due October 14)
Week 5 (October 12-16)
Chapter 2 -- Laws of Large Numbers (Section 4)
M: Thanksgiving holiday
W: Borel-Cantelli lemmas. Statement of the Strong Law   (recording)
Assignment 5 [pdf, tex] (due October 21)
Week 6 (October 29-23)
Chapter 2 -- Laws of Large Numbers (Section 2-3)
M: Lecture rescheduled for Fields Medal Symposium
W: Proof of the Strong Law. 0-1 laws,
Assignment 6 [pdf, tex] (due October 28)
Week 7 (October 26-30)
Chapter 2 -- Laws of Large Numbers (Section 3-5)
M: Kolmogorov's inequality and Convergence of random series. Stopping times.
W: Wald's identity and strong Markov property. Azuma inequality. (Two hour lecture!)
Week 8 (November 2-7)
Chapter 3 -- Central Limit Theorem (Sections 1-2)
M: Convergence of laws. Characteristic functions
W: Midterm test (2 hours within a 48-hour period; online) [2016 Midterm]
Reading Week (November 9-13)
Week 9 (November 16-20)
Chapter 3 -- Central Limit Theorem (Section 3-4 )
Week 10 (November 23-27)
Chapter 4 -- Metrics on Probablity Spaces (Sections 1-2)
Week 11 (November 30 - December 4)
Chapter 4 -- Metrics on Probablity Spaces (Sections 3-4)
Week 12 December 7-9)
Chapter 4 -- Metrics on Probablity Spaces (Section 5)

The University of Toronto is committed to accessibility. If you require accommodations for a disability, or have any accessibility concerns about the course, the classroom or course materials, please contact Accessibility Services as soon as possible: disability.services@utoronto.ca, or http://studentlife.utoronto.ca/accessibility