MAT 377 University of Toronto, Fall 2021
MAT 377H1F Mathematical Probability, Fall 2021
Almut Burchard, Instructor
How to reach me: Almut Burchard, BA 6234, 9783318.
 almut @math ,
www.math.utoronto.ca/almut/
 Lectures
Mon 1011 and Wed 910 in SS 2127; Fri 1011 in SS 1084
Tutorial Mon 910 in SS 2127
If you cannot attend in person please email me
 Office hours TBA (Fridays)
Teaching assistant: Jack Montemurro,
jack.montemurro @mail.utoronto.ca .
Textbook:
Dmitry Panchenko,
Introduction to Probability Theory
Course content: Introduction to Probability Theory for
Math, CS and Physics specialists. We will not dwell on definitions
and standard techniques too much (which means you will have to
digest them quickly), and will try to learn important ideas
through examples from computer science, random graphs, statistics,
geometry, and number theory.
 Introduction: The language of probability.
Discrete probability spaces and distributions.
Bernoulli, binomial, geometric, exponential, negative binomial
distributions. Independence; conditional distributions.
 Second moment calculus: Variance and covariance;
the Law of Large Numbers. Applications to Bernstein polynomials,
ErdősRenyi graphs, and the HardyRamanujan theorem.
 Exponential inequalities: Hoeffding inequality, JohnsonLindenstrauss
lemma, HoeffdingChernoff inequality, Azuma inequality; applications.
 Gaussian distributions: Gaussian distributions on
R and R^{n}.
Central Limit Theorem. Gamma, Chisquared, F and
Studentst distributions. Linear regression.

(time permitting) Finite State Markov chains:
Definitions and basic properties.
Stationary distributions and convergence;
reversible Markov chains.
Homework. All exercises in the textbook are part of homework.

Quizzes. Onceaweek short quiz based on one
homework problem will be given in the tutorials (Mondays 911am).
Quizzes are marked 0, 1, 2 (20% of final mark; drop 1))
 Two midterm tests:
October 18, November 22i, inclass (20% each)
 Final assessment: (40%)
 Plus up to 5 bonus points for feedback and participation.
Attendance is expected; if you can't be at a lecture
in person please follow along in the book; use my office hours
for questions.
Missed term work: If you miss a quiz or midterm, please
let me know ASAP what happened. A Verofication of Illness
or Injury is currently not required. See the University's policy on
Verification of Illness or Injury.
Academic integrity statement:
Consult the website
Student Academic Integrity for information.
Schedule:
First lecture: September 10
Chapter 1  Introduction (Section 1)
 F: A motivating example
Week 1 (September 1317)
Chapter 1  Introduction (Sections 24)
 M:
First tutorial and quiz.
Discrete probability spaces and distributions. Expectation;
change of variables
(Recording failed)
 W: Random variables and distributions
(Benoulli B(p), Binomial B(n,p), geometric (p))
Recording (passcode a5BgTf8^M2)
 F: Independence. Poisson distribution
Recording (passcode g.6Ww*g.Z!)
Exercises 1.1.11.1.5 and 1.2.11.2.16
Week 2 (September 2024)
Chapter 1  Introduction (Sections 45)
 M:
Tutorial and quiz.
Independent random variables. Convergence of
Binomial to Poisson
Recording (passcode Tdgb&^4&61)
 W: Independence
and conditional distributions
Recording (passcode esnR7Cb^Vr)
 F: Fubini's theorem
Recording (passcode 2W*#DQp+8w)
Exercises in Sections 1.21.4.
Week 3 (September 27October 1)
Chapter 1  Examples (Section 5)
 M:
Tutorial and quiz.
Method of indicators; inclusionexclusion; random maps on {1, ..., n}
Recording (passcode &s&57AX^88)
 W: Random maps from {1, .. n} to {1, ..., n}; random permutations
Recording (passcode b?kD!w23Fz)
 F: Chebychev (Markov) inequality
Recording (passcode BCpG=05gY@)
Exercises in Sections 1.41.5
Week 4 (October 48)
Chapter 2  Second moment calculus (Sections 14)
 M:
Tutorial and quiz.
Cliques in the ErdősRényi graph. Variance (definition,
and the classical Chebychev inequality)
Recording (passcode aju&7P^pdU)
 W: Covariance and correlation.
Jensen's inequality. Law of Large Numbers
Recording (passcode J9!azx4khx)
 F: Bernstein polynomials.
Recording (passcode E+R!g&UcV5)
Exercises in Sections 2.12.3.
Week 5 (October 1115)
Chapter 2  Second moment calculus (Conclusion, Sections 45)

 M: Thanksgiving holiday
 W:
Cliques in the ErdősRényi graph, revisited
Recording (passcode rSj4!W^7a1)
 F: HardyRamanujan theorem
Exercises in Sections 2.32.4.
Week 6 (October 1822)
Chapter 3  Exponential inequalities (Sections 13)
 M: Midterm 1 (9:1011am, in class)
Week 7 (October 2529)
Chapter 3  Exponential inequalities (Sections 45)

Week 8 (November 15)
Chapter 4  Gaussian distributions (Sections 14)
Reading Week (November 812)
Week 9 (November 1519)
Chapter 4  Gaussian distributions (Sections 35)
Week 10 (November 2226)
Chapter 4  Linear regression (Section 5)
 M: Midterm 2 (in class)
Week 11 (November 29  December 3)
Chapter 5  Markov chains (Sections 12)
Week 12 (December 610)
Chapter 5  Markov chains (Sections 34)
TBD (after classes end)

 Final assessment
>
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