**Instructor:**
Roman Vershynin

Office: 4844 East Hall

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

**Class meets:** **Section 3** MWF 1:10 - 2:00 in 1068 East Hall;
**Section 7** MWF 2:10 - 3:00 in 1084 East Hall.

**Office Hours:** Tu 3:30-5:00 pm, W 10:30 - 12:00 am, F 10:00 - 11:00 am in 4844 East Hall.

**Prerequisites:** This course makes serious use of the material from the Calculus sequence (Math 215, 255 or 285).
You should have solid knowledge of **differentiation** and **integration**, in particular of the
geometric ideas behind these concepts (tangent lines, local extrema, areas, volumes).
This course requires a working knowledge of **multivariate** differentiation and integration (including change of variables).
If you do not have a strong background in Calculus, please do not take this course; we do not want you to fail because of your inadequate background.

Previous exposure to basic combinatorics (combinations and permutations) is generally expected. If you do not have this background and wish to take this course, please carefully study Sections 1.1-1.4 of the textbook and the self-test problems at the end of Chapter 1.

**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.
Topics include methods of both discrete and continuous probability, conditional probability,
independence, random variables, joint distributions, expectations, variances, covariances, 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.

**Grading:**
There will be one midterm exam and one final exam.
Homework will be assigned every class (scroll down to "Course Scehdule"),
and it will be collected **every Wednesday** before class.
You are encouraged to collaborate on homework, but please write down your solutions individually.
There will also be in-class quizzes every Wednesday. 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. One lowest homework will be dropped. To form study groups and for online collaboration, feel free to use the chat room and forums on the Ctools class page.
- Quizzes (20%). One lowest quiz will be dropped.
- Midterm Exam (25%). Wednesday, February 22, in class. Covers Chapters 1 - 4.
- Final Exam (35%). Both sections:
**Friday, April 20, 1:30-3:30 pm, in 1400 CHEM**. 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 final exam.
Late homework will not be accepted. In case of a medical or road emergency that prevents you from attending the class,
you can e-mail me your scanned or typed homework as a single pdf file, along with short explanation of what happened,
on the day when the homework is collected by 5 pm.

**Course Schedule:**
It is a useful practice to read ahead the sections to be covered. To see what material is coming next,
have a look at my previous Math/Stats 425 course.
After each class, the schedule will be updated with the description of the material we covered in class, examples and self-test problems, and new homework. It is a useful practice to work on the examples and self-test problems. Their solutions are included in the back of the textbook; do not turn them in as a part of homework.

- Wednesday, January 4
- Multiplication Principle (a.k.a. the basic principle of counting) (1.1). Permutations (1.3). Combinations (1.4 begin). Lecture notes
- All examples in Sections 1.2; Examples 3a, 3b, 3c; 4a, 4b, 4c. Self-test problems (p.20-21): 1, 2, 3, 4, 6.
- Homework 1, due January 11 (p. 16-17): 3, 5, 7(a,b), 15, 17, 21.
- Friday, January 6
- Combinations contd. (1.4). Binomial theorem. Multinomial coefficients (1.5). Lecture notes
- Examples 4d, 4e; 5a, 5b, 5c. Self-test problems (p. 20-21):5, 8, 9, 10, 11, 12.
- Homework 1, due January 11 (p. 16-17): 9, 13, 28.
- Monday, January 9
- Sample space and events. Operatoins on events (2.1-2.2). Lecture notes
- All Examples in 2.1-2.2. Self-test problems (p. 56-57): 1.
- Homework 2, due January 18 (p. 50-54): 1, 3, 6.
- Wednesday, January 11
- Quiz 1
- Axioms of probability (2.3). Properties of probability. Inclusion-Exclusion Principle (2.4 up to Example 4a). Lecture notes
- All examples in those sections. Self-test problems (p. 56-57): 2, 4, 14, 15.
- Homework 2, due January 18 (p. 50-54): 9, 11, 12 (hint: use Venn diagram for this last problem). (Do not do Problem 14 at this time; it will be assigned later in Homework 3.)
- Friday, January 13
- Inclusion-Exclusion Principle continued. Sample spaces having equally likely ourcomes (2.5 begins). Lecture notes
- Examples 5a-d, recall the Birthday Problem (Example 5i). Self-test problems (p. 56-57): 5, 6, 7, 8.
- Homework 2, due January 18 (p. 50-54): 8, 17, 21.
Monday, January 16: no class (Martin Luther King Jr. Day) - Wednesday, January 18
- Quiz 2
- Generalized Inclusion-Exclusion Principle (Proposition 4.4). The Matching Problem (Example 5m). More problems on equally likely outcomes. Lecture notes
- Examples 5e, l, n. Self-test problems (p.56-57): 10, 12, 13, 17, 18, 20.
- Homework 3, due January 25 (p. 50-54): 14, 23, 25, 28, 32.
- Friday, January 20
- Conditional Probabilities. The Law of Total Probability (3.1-3.2). Lecture notes
- Examples 2b, e, f, the first example on p.21 of the lecture notes. Self-test problems (p.114-116): 2, 3, 5, 9a.
- Homework 3, due January 25 (p.102-110): 16, 19b, 21.
- Monday, January 23
- Bayes Formula (3.3). Lecture notes
- Examples 3a, c, d (covered in class), e, f, k (covered in class), n.
- Homework 4, due February 1 (p.102-110): 15, 18, 19a, 26, 35, 90 (pass to the complement, use the inclusion-exclusion principle).
- Wednesday, January 25
- Quiz 3
- Independent events (3.4). Simple random walk. Lecture notes
- Examples 4b, c, g, h, i. Self-test problems (p.114-116): 4, 9b, 10, 11, 12, 15, 16 (should refer to problem 3.66b), 21, 23.
- Homework 4, due February 1 (p.102-110): 50, 57(a,b), 64, 66(a) (in some editions there is a typo -- this problem should refer to Figure 3.4).
- Friday, January 27
- Random variables (4.1). Discrete random variables (4.2). Probability mass function (pmf) and cumulative distribution function (cdf). Lecture notes
- Examples 4a, b, c, d.
- No additional homework.
- Monday, January 30
- Expected value (4.3). Examples: lottery; group testing. Lecture notes
- Examples 3a, b, c, d. Self-test problems (p.183-185): 1, 2, 3, 4 (if you dare), 6.
- Homework 5, due February 8 (p.172-179): 1, 13, 17, 19. Note: the textbook often calls CDF "the distribution function".
- Wednesday, February 1
- Quiz 4
- Expected value of a function of a random variable (4.4). Variance (4.5). Lecture notes
- Examples 4a, 5a. Self-test problems (p.183-185): 5.
- Homework 5, due February 8 (p.172-179): 21, 25, 30, 35, 37.
- Friday, February 3
- Example: the matching problem (expectation and variance). Bernoulli and binomial random variables (4.6). Lecture notes
- Examples 6a, b, c, e, f. Self-test problems (p.183-185): 9, 10, 11a, 12, 13.
- Homework 5, due February 8 (p.172-179): 42, 43, 48.
- Monday, February 6
- Poisson distribution (4.7). Example: People vs. Collins trial. Lecture notes
- Examples 7a, b, c. Self-test problems (p.183-185): 14, 15. More examples are added for Feb.3 class above
- Homework 6, due February 15 (p.172-179): 45, 52, 53, 54, 59.
- Wednesday, February 8
- Quiz 5
- Binomial and Poisson distributions (continued). Example: jury duty letters. Lecture notes
- Self-test problems (p.384-387): 3, 4.
- Homework 6, due February 15 (p.172-179): 64(a,b); (p.373-379): 7 (express the number of chosen objects as a sum of 10 indicators), 8.
- Friday, February 10
- Computing expectations by conditioning (from 7.5). Geometric distribution (4.8.1). Lecture notes
- Examples: from Section 7.5: 5c, h; from section 4.8: 8a, b. Self-test problems (p.384-387): 17; (p.183-185): 22.
- Homework 6, due February 15 (p.373-379): 48(a,b), 53, 58 (condition on the outcome of the first flip).
- Monday, February 13
- Continuous distributions (5.1). Uniform distribution (5.3). Lecture notes
- Examples: 1a, b, c (all very useful), 3c. Self-test problems (p.229-213): 1, 2, 7.
- Homework 7, due February 22 (p.224-227): 1, 4, 11, 13.
- Wednesday, February 15
- Quiz 6
- Transformations of random variables. Lecture notes
- Example from Section 5.2: 1d.
- Homework 7, due February 22: additional problems.
- Friday, February 17
- Expectation of continuous random variables (5.2). Expectation and variance of the uniform distribution (Ex.3a). Lecture notes
- Self-test problems (p.229-231): 3, 4, 5, 6.
- Homework 7, due February 22 (p.224-227): 7, 14.
- Monday, February 20
- Standard normal distribution (5.4). Lecture notes
- Wednesday, February 22
**Midterm Exam.**In class. Covers Chapters 1 - 4.- Sample problems:
- Practice Exam (due to Prof. Montgomery): problems 1, 5. Solutions.
- More exam problems (due to Prof. Montgomery): try problems 1(a), 2, 3, 5, 6, 9, 10, 12.
- Practice Exam (due to Prof. Derksen): problems 1, 2, 4, 5.
- Practice Exam (due to Prof. Montgomery): problems 3(except b), 4(except d,f). Solutions.
- Practice Exam (due to Prof. Khoury): problems 1, 5, 8, 9, 11.
- Practice Exam (due to Prof. Khoury): problems 3, 4a, 5, 9, 10, 11.
- Practice Exam (due to Prof. Khoury): problems 2, 3, 7.
- Practice Exam (due to Prof. Khoury): problems 2, 8(a,b).
- Practice Exam (due to Prof. Khoury): problems 1, 6, 9.
- Practice Exam (due to Prof. Khoury): problems 1, 3, 8.
- Homework 1: all problems, Homework 2: 9, 11, 8, 21; Homework 3: 14, 28, 32, 16, 19, 21; Homework 4: 15, 35, 57, 66a; Homework 5: 17, 19, 43, 37; Homework 6: all problems.
- Friday, February 24
- Normal distribution with general mean and varaince (5.4). Lecture notes
- Examples: 4b, c, d, e. Self-test problems (p.229-231): 8, 9, 10, 11.
- Homework 8, due March 7 (p.224-227): 15, 16, 17, 18, 22.
Winter break: February 27 - March 2 - Monday, March 5
- Normal approximation to the binomial distribution (5.4.1). Lecture notes
- Examples 4f, g, h, i. Self-test problems (p.229-231): 12, 17.
- Homework 9, due March 14 (p.224-227): 20, 27, 28.
- Wednesday, March 7
- Quiz 7
- Exponential distribution (5.5). Lecture notes
- Examples 5a, b, c, d. Self-test problems (p.229-231): 18, 19.
- Homework 9, due March 14 (p.224-227): 31, 32, 34, 39.
- Friday, March 9
- Joint distributions (6.1). Lecture notes
- Examples 1a, b, c, d, e. Self-test problems (p.293-296): 6.2, 6.3, 6.6(a,c), 6.7(b,c,d,e,f).
- Homework 9, due March 14 (p.287-291):6, 8, 9(a,b,c,e), 10.
- Monday, March 12
- Independent random variabes (6.2). Lecture notes
- Examples 2a, d, f.
- Homework 10, due March 21 (p.287-291): 15, 16, 20, 22, 23, 27.
- Wednesday, March 14
- Quiz 8
- Sums of independent random variables (6.3). Lecture notes
- No additional homework problems today.
- Friday, March 16
- Sums of independent random variables (6.3). Convolutions. Sums of independent normal random variables. Lecture notes
- Homework 10, due March 21 (p.287-291): 29, 30, 31.
- Monday, March 19
- Sums of independent random variables (Bimonial, Poisson, exponential). Gamma distribution (6.3 continued). Lecture notes
- Homework 11, due March 28 (p.287-291): 28, 33(a,b).
- Wednesday, March 21
- Quiz 9
- Conditional distributions (6.4-6.5). Lecture notes
- Homework 11, due March 28 (p.287-291): 40, 41, 42 (compute the conditional pdf of Y given X=x).
- Friday, March 23
- Conditional distributions (6.4-6.5 continued). Lecture notes
- Homework 11, due March 28 (p.287-291): 48, and an additional problem.
- Monday, March 26
- Transformations of joint distributions (6.7). Lecture notes
- Homework 12, due April 4 (p.287-291): 52, 54, 55, 56(a).
- Wednesday, March 28
- Quiz 10
- Properties of expectation (7.1). Expectation of sums of random variables (7.2-7.3). Lecture notes
- Homework 12, due April 4 (p.373-379): 4, 5, 11, 21.
- Friday, March 30
- Coupon collector's problem (Section 7, Examples 3d, 2i). Expectation of product of independent random variables (Proposition 4.1). Lecture notes
- Homework 12, due April 4 (p.373-379): 19(a).
- Monday, April 2
- Covariance, correlation (7.3). Lecture notes
- Homework 13, due April 11 (p.373-379): 50, 38, 40. Additional problem.
- Wednesday, April 4
- Quiz 11
- Properties of covariance. Variance of a sum. Lecture notes
- Homework 13, due April 11 (p.373-379): 6, 31, 36, 37, 42, 45, 63(a).
- Friday, April 6
- Applications to statistics: sample mean and sample variance (Ex. 4a). Lecture notes
- Homework 13, due April 11 (p.373-379): 65.
- Monday, April 9
- Moment generating functions (7.7). Lecture notes
- Wednesday, April 11
- Quiz 12
- Central Limit Theorem (8.3). Lecture notes
- Homework 14, will not be collected (p.413-414): 5, 7, 8, 14.
- Friday, April 13
- Markov's and Chebyshev's inequalities. Weak law of large numbers.
- Monday, April 16
- Review.
- Practice Exam 0
- Practice Exam 1
- Practice Exam 2
- Practice Exam 3
- Homework 7: 4, 7, 13, 14, additional problems 1, 2. Homework 8: 16, 22. Homework 9: 28, 32, 8, 9, 10. Homework 10: 16, 20, 22, 27, 31. Homework 11: 28, 41, 42, 48. Homework 12: 54, 56(a), 4, 5, 11, 21. Homework 13: 50, 31, 36, 37, 42, 63(a). Homework 14: 5, 7, 8, 14.
- This formula sheet will be available during the exam.
In addition, you may bring
**two index cards**(3x5 in) to the exam. You may write whatever you like on**one**side of each card. Please only write by hand on the cards; photocopyied or printed cards are not allowed. - Friday, April 20
**Final Exam.****1:30-3:30 pm, in 1400 CHEM.**- Covers the whole course.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2011-12/425/425.html

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