MAT 1600 University of Toronto Fall 2016
 

MAT 1600 Mathematical Probability I, Fall 2016

Almut Burchard, Instructor

How to reach me: Almut Burchard, BA 6234, 978-3318
almut @math.toronto.edu, www.math.utoronto.ca/almut/
Lectures MWF 12:10-1PM, BA 6183 .
Office hours Mondays 3-4:30, BA 6234
Teaching assistant: Malors Espinosa Lara, srolam.espinosalara @mail.utoronto.ca
Textbook:   R. Durrett, Probability: Theory and Examples. Fourth edition, Cambridge University Press (*earlier editions will do*)
We will cover at least Chapters 2 and 3 (on the Law of Large Numbers and the Central Limit Theorem)
Evaluation:
40% : Homework: weekly exercises (due Wednesdays in class. Late submissions will be marked at the end of the term)
20% : midterm test (Wednesday October 26, 5:00-6:30pm in GB 304 on 35 St. George Street )
40% : final examination (Tuesday, December 13, 2-5pm)
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.

Schedule:

Week 1 (September 12-16)
Sections 1.1-1.3 -- Probability spaces
M: Overview and basic definitions
W: Infinite sums
F: Monotone class theorem (see Lieb & Loss, Section 1.3)
Assignment 1 (due September 21)
Corrections and comments
Week 2 (September 19-23)
Sections 1.4-1.7 -- Random variables
M: Random variables and their distributions.
W: Expectation
F: Independence
Assignment 2 (due September 28)
Outline of the proof of Problem 12
Week 3 (September 26-30)
Section 2.1 -- Independent random varables
M: Independence, distribution, and expectation
W: Sums of independent random variables
F: Kolmogorov's extension theorem
Assignment 3 (due October 5)
Minor correction
Week 4 (October 3-7)
Section 2.2 -- Weak laws of large numbers
M: L2 weak laws.
Office hours are canceled this week
W: Triangular arrays
F: Truncation
Assignment 4 (due October 12)
Week 5 (October 10-14)
Section 2.3 -- Borel-Cantelli lemmas (Durrett Section 2.3)
M: Thanksgiving Holiday
Office hours are moved to Ocober 11
W: The lemmas
F: Some applications
Assignment 5: Durrett Section 2.3 # 2, 3, 6, 12, 17, 20 (due October 19)
Week 6 (October 17-21)
Section 2.3 -- Examples
M: A simple case of the Strong Law
W: The coin toss: Waiting times, head runs
F: Random permutations
No assigmnent
Week 7 (October 24-28)
Section 2.4 -- Strong law of large numbers
M: Records
W: No lecture.
5:00-6:30pm Midterm test, GB 304 on 35 St. George Street
F: The strong law
Assignment 6: Please re-work the midterm test (due November 4. Marks will be averaged with the in-class results)
Week 8 (October 31-November 4)
Section 2.4 -- Strong law of large numbers, cont'd
M: The maximal inequality
W: Kolmogorov's 3-series theorem
F: Examples
Week 9 (November 7-11)
Section 3.1 and 3.2 -- Weak convergence;
W: The de Moivre-Laplace theorem
F: Some examples and useful tools
Week 10 (November 14-18)
Section 3.3 -- Characteristic functions and the CLT
M:
W:
F:
Week 11 (November 21-25)
Section 3.6 Poisson convergence
M:
W:
F:
Week 12 (November 28-December 2)
Sections 3.5 and 3.7 -- Local limit theorems; stable laws
M:
W:
F:
Last days (December 5-7)
M:
W: Makeup day

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