**Instructor:**
Roman Vershynin

Office: 4844 East Hall

E-mail: romanv "at" umich "dot" edu

**Class meets:** MWF 2:10 - 3:00 in 1084 East Hall.

**Office Hours:** M 11:00-12:00, W 10:20-12:00 in 4844 East Hall.

**Prerequisites:**
Calculus sequence (Math 215, 255 or 285).

**Course Description:**
Basic concepts of probability are introduced, and applications to other sciences are noted.
The emphasis is on concepts, calculations, derivations and problem-solving, rather than on formal proofs.
Serious use is made of material from the calculus sequence (Math 116 and Math 215).
Prior knowledge of basic combinatorics is helpful but is not assumed.
Specific topics include methods of both discrete and continuous probability, conditional probability,
independent events, random variables, jointly distributed random variables,
expectations, variances, and limit laws.

**Text:**
Sheldon Ross, *A first course in probability*. Pearson Prentice Hall, 8th ed. (2010).
ISBN: 9780136033134.

The course covers most of Chapters 1--7 and a part of Chapter 8 (possibly also a part of Chapter 9).

**Grading:**
There will be two midterm exams and one final exam.
Homework will be assigned every class, and it will be collected **every Wednesday** before class.
You are welcome and encouraged to collaborate on homework, but please write down your solutions individually.
There will also be occasional pop quizzes. Exams and quizzes are closed book, without notes, and no calculator.
These activities contribute to the course grade as follows:

- Homework (20%). Scroll down to see it. To form study groups and for online collaboration, feel free to use the chat room and forums on the Ctools class page
- Quizzes (15%). One lowest quiz will be dropped.
- Midterm Exam 1 (20%). Wednesday, February 9, in class. Covers Chapters 1,2,3.
- Midterm Exam 2 (20%). Friday, March 18, in class. Covers Chapters 4,5, and a bit of 7.5 (conditional expectation).
- Final Exam (25%). Thursday, April 21, 1:30 - 3:30, in class. Covers the whole course.

**Missing/late work:**
There will be no make-up for the quizzes or exams for any reason.
A missed 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 other midterm exam.
If you miss both midterm exams, the course is considered failed.
Late homework will not be accepted. In extenuating circumstances,
you can e-mail me your scanned or typed homework (as a single pdf file) on the day
when the homework is collected.

**Lecture Schedule and Homework:**
It is a useful practice to read ahead the sections to be covered.
Self-test problems do not need to be turned in.
Working on them, whether individually or in a group, is a very good practice.
Solutions for the self-test problems are included in the end of the book.

- Wednesday, January 5
- Sections 1.1--1.3, beginning of 1.4.
- Self-test problems (p.20): 1, 2, 3, 6.
- Homework 1, due January 12 (p.16): 3, 15, 17. (p.17): 21.
- Friday, January 7
- Sections 1.4--1.5.
- Self-test problems (p.20): 4, 7, 9, 10, 11, 12.
- Homework 1, due January 12 (p.16): 9, 13.
- Solutions to Homework 1
- Monday, January 10
- Sections 2.1--2.2. Study the commutative, associative and distributive laws on p. 25. (Prove them). Prove the second De Morgan's law.
- Self-test problems (p.56), 1.
- Homework 2, due January 19 (p.50): 1, 3, 6.
- Wednesday, January 12
- Sections 2.3--2.4. Study the matching problem (p.41, Example 5m).
- Self-test problems (p.56), 2, 4.
- Homework 2, due January 19 (p.50-51): 9, 11, 12, 14.
- Solutions to Quiz 1
- Friday, January 14
- Sections 2.5. Find a small mistake in the final calculation for the matching problem, and correct it!
- Self-test problems (p.56): 6, 7, 8, 10, 12, 13, 20.
- Homework 2, due January 19 (p.50-51): 17, 21.
- Solutions to Homework 2
- Wednesday, January 19
- Sections 3.1--3.3 (beginning).
- Self-test problems (p.114-116): 2, 5, 9a, 11a, 12, 4.
- Homework 3, due January 26 (p.102-110): 16, 19b, 21.
- Solutions to Quiz 2
- Friday, January 21
- Sections 3.3--3.4.
- Self-test problems (p.114-116): 4, 6, 9b, 11b, 15.
- Homework 3, due January 26 (p.102-110): 15, 18, 19a, 26, 35, 90.
- Solutions to Homework 3
- Monday, January 24
- Sections 3.4, 4.1 (beginning).
- Self-test problems (p.114-116): 21.
- Homework 4, due February 2 (p.107): 64, 66(a) referring to Figure 3.4, (p.172-179): 1.
- Wednesday, January 26
- Sections 4.1, 4.2.
- Solutions to Quiz 3
- Homework 4, due February 4 (p.172-179): 13, 17, 19. Note: the textbook often calls CDF "the distribution function".
- Friday, January 28
- Sections 4.3, 4.4.
- Self-test problems (p.183-185): 4.1, 4.3, 4.6
- Homework 4, due February 4 (p.172-179): 25, 20, 30.
- Solutions to Homework 4
- Monday, Janunary 31
- Section 4.5.
- Self-test problems (p.183-185): 4.5.
- Homework 5, due February 9 (p.172-179): 21, 35, 37.
- Solutions to Homework 5.
- Wednesday, February 2
- Section 4.6.
- No additional homework (other than 3 problems assigned on Jan 31). Use this time to review the previous material for Exam 1.
- Friday, February 4
- Section 4.7.
- No additional homework (other than 3 problems assigned on Jan 31). Use this time to review the previous material for Exam 1.
- Practice Exam 1 (due to Prof. Montgomery): study problems 1, 5. Solutions.
- More exam problems (due to Prof. Montgomery): try problems 1, 2, 3, 5, 6, 9, 10, 12.
- Review Section 4.10, especially formula (8.1) and example 10a (which is similar to a homework problem).
- Solutions to Quiz 4
- Monday, February 7
- Section 7.5 (part of it), 4.8.1.
- No additional homework (other than 3 problems assigned on Jan 31). Use this time to review the previous material for Exam 1.
- Wednesday, February 9
**Midterm Exam 1.**In-class. Covers Chapters 1, 2, 3. Solutions to Midterm Exam 1.- Homework 6, due February 16 (p.172-179): 43, 48, 52, 53; (p.373-379): 8, 58.
- Friday, February 11
- Sections 5.1.
- Study Examples 1a - 1d.
- Self-test problems (p.229-231): 1, 2.
- Homework 6, due February 16 (p.224-227): 1, 4.
- Solutions to Homework 6
- Monday, February 14
- Sections 5.2-5.3.
- Self-test problems (p.229-231): 7.
- Homework 7, due February 23 (p.224-227): 11, 14 (should read "... by computing the distribution of X^n and then the expectation".)
- Wednedsay, February 16
- Section 5.4 (begin).
- Solutions to Quiz 5
- Self-test problems (p.229-231): 8, 9, 10, 11, 12.
- Homework 7, due February 23 (p.224-227): 15, 16, 22 (you may choose to wait until Friday to do these problems, or study Examples 4b, 4d, 4e first.)
- Friday, February 18
- Section 5.4 (end).
- At home: review Section 5.7.
- Homework 7, due February 23 (p.224-227): 18, 20, 28.
- Solutions to Homework 7
- Monday, February 21
- Section 5.5.
- Homework 8, due March 9 (p.224-227): 34, 37.
- Wednesday, February 23
- Section 6.1.
- Study Examples 1b, 1c, 1e in Section 6.1.
- Solutions to Quiz 6
- Homework 8, due March 9 (p.287-291): 6, 8, 10.
- Friday, February 25
- Section 6.2.
- Study Examples 2a, 2f, 2h in Section 6.2.
- Homework 8, due March 9 (p.287-291): 15, 16, 22, 27.
- Solutions to Homework 8
- Monday, March 7
- Section 6.3 (begin).
- Study Example 3a, Proposition 3.2, Example 3c.
- Self-test problems (p.293-296): 5, 6, 7, 13.
- Homework 9, due March 16 (p.287-291): 28(a), 30, 31.
- For Quiz on Wednesday: make sure to review 5.5 (the exponential distribution). Self-test problem (p.230-231): 13.
- For Midterm 2: make sure to Review Section 5.7, including the exercises there. Self-test problem (p.230): 16.
- Practice Exam 1 (due to Prof. Derksen): all problems.
- Practice Exam 2 and solutions (due to Prof. Montgomery): Problems 1, 2, 3 (all parts except b), 4.
- Practice Exam 3 (due to Prof. Khoury): Problems 1, 5, 9, 11.
- Practice Exam 4 (due to Prof. Khoury): Problems 1, 2, 3, 7, 9.
- Practice Exam 5 (due to Prof. Khoury): Problems 5, 9, 10, 11.
- More problems (due to Prof. Khoury): both problems.
- Make sure to study
**Practice Exams 1 and 2**, they are most representative of the material covered in our class. - Wednesday, March 9
- Section 6.3 (continued).
- Solutions to Quiz 7
- Friday, March 11
- Sections 6.3 (end).
- Homework 9, due March 16 (p.287-291): 28(b), 32.
- Solutions to Homework 9
- Monday, March 14
- Sections 6.4, 6.5.
- Self-test problems (p.293-296): 14.
- Homework 10, due March 23 (p.287-291): 40, 38, 41.
- This formula sheet will be available during the midterm exam. Note that this page does not specify the allowed values of k of PDFs of the discrete distributions.
- Wednesday, March 16
- Section 6.7 (begin).
- Study Examples 5a, 5b.
- Self-test problems (p.293-296): 3, 15.
- Homework 10, due March 23 (p.287-291): 42 (compute the conditional PDF of Y given X=x), 48, 52.
- Solutions to Homework 10
- Friday, March 18
**Midterm Exam 2.**In-class. See above for what it covers, and for sample problems. Solutions to Midterm Exam 2.- Monday, March 21
- Section 6.7 (end).
- Study Examples 7a, 7b, 7c.
- Self-test problems (p.293-296): 15.
- Homework 11, due March 30 (p.287-291): 54, 55.
- Wednesday, March 23
- Sections 7.1, 7.2 (begin).
- Study Examples 2a, 2c, 2e.
- Homework 11, due March 30 (p.373-379): 4, 5, 7.
- Solutions to Quiz 8
- Friday, March 25
- Section 7.2.
- Study Examples 2e, 2h, 2i, 2j, 2s, 3d (expected value only), 3f (expected value only).
- Homework 11, due March 30 (p.373-379): 11, 12(b), 21.
- Solutions to Homework 11
- Monday, March 28
- Section 7.4 (begin).
- Self-test problems (p.384-387): 1, 3, 9.
- Homework 12, due April 6 (p.373-379): 38.
- Wednesday, March 30
- Section 7.4 (continued).
- Study Example 4b
- Self-test problems (p.384-387): 12, 13.
- Homework 12, due April 6 (p.373-379): 6, 31, 37, 42.
- Solutions to Quiz 9
- Friday, April 1
- Section 7.4 (end).
- Study Examples 2c, 4a, 4b.
- Self-test problems (p.384-387): 17.
- Homework 12, due April 6 (p.373-379): 45, 63(a).
- Solutions to Homework 12
- Monday, April 4
- Section 7.5.
- Wednesday, April 6
- Section 7.7.
- Solutions to Quiz 10
- Homework 13, due April 13 (p.373-379): 48, 50, 56 (express the number of stops as a sum of N Bernoulli random variables. Compute the expected value of this sum by conditioning on the number of the people that enter on the ground floor.)
- Friday, April 8
- Section 8.3.
- Study Examples 3a, 3b, 3c, 3e.
- Homework 13, due April 13 (p.413-415): 5, 7, 8, 14.
- Monday, April 11
- Section 8.2.
- Self-test problems (p.415-416): 1, 2, 3.
- Wednesday, April 13
- Section 8.2 (the Weak Law of Large Numbers), Section 8.5 (Chernoff bounds).
- Practice Exam 0 (due to Prof. Khoury).
- Practice Exam 1 (due to Prof. Khoury).
- Practice Exam 2 (due to Prof. Khoury).
- Practice Exam 3 (due to Prof. Khoury).
- This formula sheet will be available during the exam. The MGF's that are not included there will not be needed.
- Wednesday, April 15
- Applications of probability: Monte-Carlo integration, volumes in high dimensions, graph theory.
- Scanned notes of the lecture.
- This material will not be tested in the Final Exam.
- Monday, April 18
- Review session.
- Please take notes. The problems discussed during this class will not be posted.
- Thursday, April 21, 1:30 - 3:30 pm.
**Final Exam.**In class. Covers the whole course. See above for sample exams.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2010-11/425/425.html

Also see the Ctools class page which contains the archive of e-mail messages to the class, forums and chat room.