Winter 2017, Sections 1 and 2

Professor Roman Vershynin, Department of Mathematics, University of Michigan

** Office** 3064 EH

**Email** romanv "at" umich "dot" edu

**Section 1:** Tu, Th 10:10 - 11:30 am in 3088 EH

**Section 2:** Tu, Th 1:10 - 2:30 pm in 1505 CCL

**Office hours:** M, W 1:10 - 2:30 pm in 3064 EH

**Course description:** This is a fairly rigorous introduction to probability theory
with some emphasis given to both theory and applications,
although a knowledge of measure theory is not assumed.
Topics covered are: probability spaces, conditional probability,
discrete and continuous random variables, generating functions,
characteristic functions, random walks, limit theorems,
and some more advanced topics.

**Prerequisites:** Math 451 required.

**Textbook:** John B. Walsh, Knowing the odds: an introduction to probability, ISBN: 9780821885321

The course grade will be determined as follows:

**Homework:**20%. Homework problems will be assigned every class. Solutions will be collected every Tuesday at the beginning of the class. Late homework will not be accepted. You are welcome and encouraged to discuss homework with other students, but you must write your solutions individually. One homework with the lowest score will be dropped.**Quizzes:**15%, every Thursday. One quiz with the lowest score will be dropped.**Midterm Exam 1:**20%, Thursday, February 16, in class.**Midterm Exam 2:**20%, Thursday, March 23, in class.**Final Exam:**25%, Thursday, April 27, 1:30-3:30 pm, AHG115 (Angell Hall, ground floor, room 115).

There will be no make-up for the quizzes or exams for any reason. A missed midterm exam counts as zero points, with the following exception. If you miss a midterm exam due to a documented medical or family emergency, the exam's weight will be added to the weight of the final exam.

If you think you need an accommodation for a disability,
please let me know as soon as possible.
In particular, a Verified Individualized Services and Accommodations (VISA) form
must be provided to me at least *two weeks* prior to the need for a test/quiz accommodation.
The Services for Students with Disabilities (SSD) Office
(G664 Haven Hall; http://ssd.umich.edu/) issues VISA forms.

#### Thursday, January 5

Read Sections 1.1-1.3. Class notes.*[Corrections to class notes: in Definition on p.4, the first axiom should be P(\Omega)=1. In Proposition on p.6, the first part should be P(\emptyset) = 0.]*Homework 1 (due January 17, part 1): prove that a sigma-algebra is closed under countable intersections, 1.3, 1.18, 1.22, 1.23.#### Tuesday, January 10

Read Sections 1.4-1.6. Class notes.*[Corrections to class notes: in Warning on p.14, the example should involve rolling two dice, and event C = {sum of the two dice=7}.]*Homework 1 (due January 17, part 2): 1.24, 1.27, 1.29, 1.32, 1.33, 1.40, 1.44 (try Bayes rule in 1.44).#### Thursday, January 12

Read Sections 1.7, 2.1. Class notes.Homework 2 (due January 24, part 1): 1.86, 1.90, 2.4, 2.5, 2.6, 2.13, 2.16.#### Tuesday, January 17

Read Section 2.1 (starting with Proposition 2.6) and Sections 2.2, 2.3, 2.4. Class notes.Homework 2 (due January 24, part 2): 2.22, 2.26, 2.27, 2.34.#### Thursday, January 19

Quiz 1 (with solutions). Read Section 2.5. Class notes.Homework 3 (due January 31, part 1): 2.28, 2.31, 2.37, 2.41 (Correction: Replace problem 2.41 with with the following: "Pick n independent points at random and uniformly from [0,1]. Find the densities of their maximum and their minimum.")#### Tuesday, January 24

Read Section 2.6 (variance only), Section 2.8 (Bernoulli and Binomial). Class notes.Homework 3 (due January 31, part 2): 2.45, 2.47, 2.54, 2.55.#### Thursday, January 26

Quiz 2 (with solutions). Read Section 2.8 (Poisson and geometric). Class notes.Homework 4 (due February 7, part 1): 2.59, 2.65.#### Tuesday, January 31

Read Sections 3.1, 3.2, 3.5.a Class notes.Homework 4 (due February 7, part 2): 3.2, 3.4, 3.7, 3.16. See clarifications sent as an announcement in Canvas.#### Thursday, February 2

Quiz 3 (with solutions). Read Section 3.5.b, 3.5.c, 3.5.d. Class notes.Homework 5 (due February 14, part 1): 3.8, 3.11 a, c (without computing the moment generating function), 3.14 (just the variance).#### Tuesday, February 7

Read Section 3.6 (joint and marginal distributions), Example 2.24.3, Section 3.7.1. Class notes.Homework 5 (due February 14, part 2): 3.21, 3.22, 3.24, 3.28 (a,b,c), 3.30 (skip the median), 3.41.Practice problems for Midterm Exam 1:sample-exam-1, sample-exam-2, practice-midterm1a and its solutions, practice-midterm1b, practice-quiz1, Q1-solutions, Q2-solutions, Q3-solutions, Q4-solutions,#### Thursday, February 9

Quiz 4 (with solutions). Read Section 3.3 (Markov and Chebyshev's inequalities). Class notes.Homework 6 (due March 7, part 1): 3.27.#### Tuesday, February 14

Read Section 3.3 (Jensen's and Lyapunov's inequalities). Class notes.Homework 6 (due March 7, part 2): 3.19. Stay tuned: there will be other parts of this homework.#### Thursday, February 16

Midterm Exam 1.Solutions.#### Tuesday, February 21

Read Sections 3.6.1, 3.7.2 (covariance matrix). Class notes.Homework 6 (due March 7, part 3): 3.43. Additional problems.#### Tuesday, March 7

Read Sections 3.7.2 (multivariate Gaussian distribution). Class notes.Homework 7 (due March 14, part 1): 3.38, 3.39, 3.40.#### Thursday, March 9

Quiz 5 (with solutions). Read Sections 2.6 (moment generating functions). Class notes.Homework 7 (due March 14, part 2): 3.13, 3.14, 3.15; compute the MGF of Exp(\lambda).#### Tuesday, March 14

Read Section 3.7 (conditional distributions and expectations). Class notes.Homework 8 (due March 21, part 1): Walsh 3.28 (d), 3.34, 3.35, 3.36; Ross from Chapter 3: 2, 5 (first part), 12, 15.#### Thursday, March 16

Quiz 6 (with solutions). Read Walsh Section 3.7 (Law of total expectation: Proposition 3.41); Ross Section 3.4, 3.5. Class notes.Homework 8 (due March 21, part 2): Ross from Chapter 3: 21, 25, 52.Practice problems for Midterm Exam 2:sample-exam-1 problems 3, 4, 5, 6; sample-exam-2 problems 2(b), 6; practice-midterm1a problems 3, 4, 5(a), 7 and their solutions; practice-midterm1b problems 3, 4; 525f08quiz3_sol-1 problem (b); 525f08quiz5_sol.#### Tuesdaty, March 21

Read Ross 3.4, 3.5. Class notes.#### Thursday, March 23

Midterm Exam 2.It will cover everything starting from joint and marginal distributions (February 7) through conditional distributions and expectations (March 16). Solutions.#### Tuesday, March 28

Read Walsch parts of Sections 4.1, 5.1, Ross 2.7 (Weak and strong laws of large numbers, central limit theorem, modes of convergence). Class notes.Homework 9 (due April 4): download here. No additional homework will be assigned on Thursday.#### Thursday, March 30

Applications and extensions of law of large numbers and central limit theorem. Class notes.#### Tuesday, April 4

Introduction to Markov chains. Lecture follows Ross Sections 2.8, 4.1.Homework 10 (due April 11): Ross after Chapter 4: problems 1, 2, 3, 5, 6, 7, 8, 13. No additional homework will be assigned on Thursday.#### Thursday, April 6

Quiz 7 (with solutions). Chapman-Kolmogorov equations. Lecture follows Ross Section 4.2.#### Tuesday, April 11

Read Ross Section 4.3. Class notes.Homework 11 (due April 18, part 1): Ross after Chapter 4: problems 10, 14 (all four). More problems will be assigned on Thursday.#### Thursday, April 13

Quiz 8 (with solutions). Limiting probabilities, time reversal in Markov chains. Read Ross Sections 4.4, 4.8. Class notes.Homework 11 (due April 18, part 2): Ross after Chapter 4: problems 18, 20, 22, 23, 26.#### Tuesday, April 18

Markov Chain Monte Carlo methods: random walks on graphs, PageRank, Hastings-Metropolis algorithm, Gibbs sampler. Read Ross Sections 4.9. Class notes.Practice problems for Final Exam: try all problems fromsample-exam-1, sample-exam-2, practice-midterm1a and its solutions, practice-midterm1b, practice-quiz1, Q1-solutions, Q2-solutions, Q3-solutions, Q4-solutions, Q6-solutions, Q7-solutions-1, Q8-solutions, Q9-solutions, 525f08quiz3_sol-1, 525f08practicemidterm2-2 except problem 7, 525f08quiz5_sol, hw10, midterm, problems-in-probability (these tend to be on the easier side), midterm.

**Course webpage:** http://www-personal.umich.edu/~romanv/teaching/2017-18/525/525.html

**Canvas webpage (your grades):** https://umich.instructure.com/courses/121676/