**Course Details:**

Time: T,Th 9:30am-10:45am

Room: Van Vleck B119

Office hours: Now by appointment

Instructor: Saverio Spagnolie

Office: Van Vleck 505

Course website: http://www.math.wisc.edu/~spagnolie/Courses/MATH331/index.html

**Piazza:**

The following page has been created on piazza.com for group discussion!

https://piazza.com/wisc/spring2013/math331/home

Rather than emailing questions, I encourage you to post your questions on Piazza. Bonus points for students who respond to those questions!

**Course syllabus:**

**Textbook:**

Fundamentals of Probability, with Stochastic Processes, 3rd ed. by S. Ghahramani

**Course Content:**

Math 331 is an introduction to the basic concepts of probability theory, the mathematical discipline for analyzing and modeling uncertain outcomes. The course concentrates on discrete models in probability, and beyond basic introduction to the subject, it also presents material on Markov chains. We will cover most of the textbook with an emphasis on discrete random variables.

**Grading:**

Your final grade will be determined by scores on weekly homework assignments and in-class quizzes (30%), on two midterm exams (40%), and on a final exam (30%). Your lowest homework score or quiz score will be dropped.

**Exam schedule**

- Midterm exam 1: Tuesday, Mar. 5 (in class)
- Midterm exam 2: Thursday, Apr. 18 (in class)
- Review session / office hours: Tuesday May 14, 9:30am, VV B203
- Final exam:
*Saturday*, May 18th, 10:05am-12:05pm**(Special Room: SOC SCI 6102)**

**Homework sets:**

Due Tues. Jan 29, *Pg. 9*: 1, 2, 3, 9, 10, 13, 20 (do not turn in; will be on Tuesday's quiz)

Due Tues. Feb. 5, *Pg. 23*: 5, 10, 13, 17, 25; *Pg. 34*: 4, 5 (do not turn in; will be in Tuesday's quiz).

Due Tues. Feb. 12, *Pg. 44*: 1, 3, 5, 14, 15, 17; *Pg. 50*: 1, 6, 10, 13, 15; *Pg. 63*: 1, 2, 7, 13, 19, 30.

Due Tues. Feb. 19, *Pg. 82*: 2, 3, 8, 9, 18; *Pg. 87*: 1, 6, 10; *Pg. 96*: 1, 7, 9, 13; *Pg. 105*: 1, 2, 8; *Pg. 119*: 2, 4, 15, 17, 29.

Due Tues. Feb. 26, *Pg. 150*: 1, 4, 5, 6, 7, 16; *Pg. 157*: 1, 3, 4, 7, 8, 10; *Pg. 173*: 2, 3, 7, 11, 12.

Due Tues. Mar. 5, *Pg. 182*: 3, 4, 5, 7, 8; *Pg. 185*: 2.

Due Thurs. Mar. 14, *Pg. 196*: 1, 3, 4, 8, 13, 20; *Pg. 211*: 3, 7, 11, 15; *Pg. 224*: 3, 5, 9, 15, 16.

Due Thurs. Mar. 21, *Pg. 238*: 1, 4, 10; *Pg. 254*: 4, 5, 6, 13 (13c is extra credit); *Pg. 266*: 5, 6, 8, 17 (17 is extra credit); *Pg. 281*: 1, 2, 7, 12, 16.

Due Tues. Apr. 9: *Pg. 325*: 1, 2, 3, 4; *Pg. 339*: 1, 2, 3, 5, 7, 8; *Pg. 353*: 1, 3, 5 (just the discrete case), 13; *Pg. 383*: 1, 3, 5.

Due Tues. Apr. 16: *Pg. 412*: 3, 5, 11, 13; *Pg. 424*: 4, 5, 7, 9; *Pg. 433*: 1, 5, 6; *Pg. 444*: 3, 4, 11, 14.

Due Tues. Apr. 30: *Pg. 465*: 1, 3, 6, 7, 9, 17; *Pg. 474*: 2, 6, 8; *Pg. 484*: 1, 2, 3, 4, 5, 11, 16.

Due Tues. May 14, 9:30am: Markov Chain HW set

**Schedule:**

- Week 1. Sections 1.1-1.2: Definition of the sample space, operations on sets (and events)
- Week 2. Sections 1.3-2.2: Axioms of probability, generalized counting principle
- Week 3. Sections 2.2-3.1: Counting techniques, permutations, and combinations, conditional probability
- Week 4. Sections 3.1-3.5: Conditional probability, law of multiplication, law of total probability, Bayes formula, independence
- Week 5. Sections 4.1-4.4: Random variables, distribution function, probability mass function, expectation of discrete random variables
- Week 6. Sections 4.5-5.1: Expectation and variance of discrete random variables, Bernoulli and Binomial random variables
- Week 7. Sections 5.1-5.3: Poisson random variable, Poisson approximation, Poisson process, geometric random variables
- Week 8. Sections 5.3-7.2: Negative binomial and hypergeometric random variables, continuous random variables, uniform and normal random variables
- Week 9. Sections 7.2-8.3: Normal and exponential random variables, joint distributions, expectation, independence, conditional distributions
- Week 10. Sections 9.1-10.1: Joint distribution of more than two random variables, expected values of sums of variables
- Week 11. Sections 10.2-10.4: Covariance and correlation of random variables, conditional expectation
- Week 12. Section 11.1: Moment-generating functions
- Week 13. Sections 11.1-11.3: Moment-generating functions, sums of independent random variables, Markov inequality
- Week 14. Sections 11.3-12.3: Chebyshev's inequality, estimating tail probabilities of random variables, Law of Large Numbers, Central Limit Theorem, Markov chains
- Week 15. Section 12.3: Markov chains, irreducible and regular Markov chains, transition probability matrix, stationary distribution
- Week 16. Review