Math 331 - Introduction to Probability and Markov Chain Models
Please read the course syllabus
for the rules and requirements.
Meetings: TR 11:00-12:15, Van Vleck B211
Office: 409 Van Vleck
Email: valko at math dot wisc dot edu
Tuesday 13:30-14:30 or by appointment
I will use the
class email list to send out corrections, announcements, please check your
wisc.edu email from time to time.
Textbook: Fundamentals of Probability, with stochastic processes, 3rd ed., by S. Ghahramani
Prerequisites: The basic skills needed for Math 331 are
calculus and basic set theory. Success in this class will also require
the ability to think/reason abstractly. 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.
Evaluation: Course grades will be based on homework assignments,
quizzes, two midterm exams and the final exam.
REMINDER: Our first midterm exam will take place on October
13, Thursday (in class). It will cover everything we discussed in class
up to Section 4.3. A more detailed description of the topics you are
expected to know will be posted here.
REMINDER: Our second midterm exam will take place on November
19, Thursday (in class). It will cover everything we discussed in class
up to Chapter 10. A more detailed description of the topics you are
expected to know will be posted here.
The final exam will take place on Thursday, December 17 from 7:45AM to 9:45AM
at VV B211. It will cover everything we discussed in class
during the semester. A more detailed description of the topics you are
expected to know is posted here.
Lecture notes on Markov Chains (written by Prof. D. Anderson):
1. quiz: September 10, Thursday.
2. quiz: September 17, Thursday.
3. quiz: September 24, Thursday.
4. quiz: October 13, Thursday.
First midterm: October 1, Tuesday. Solutions of the first midterm
5. quiz: October 22, Thursday.
6. quiz: October 29, Thursday.
7. quiz: November 5, Thursday.
8. quiz: November 12, Thursday.
- Week 1. Definition of the sample space, operations on sets (and events), Sections 1.1-1.2
- Week 2. Axioms of probability, simple properties of probability, Continuity of probability, probabilities 0 and 1,
Generalized counting principle, Section 2.2
- Week 3. Counting techniques, Permutations, Combinations, Sections 2.2-2.4
Conditional probability, Section 3.1
- Week 4. Conditional probability, law of multiplication, law of total probability, Bayes formula, independence, Sections 3.1-3.5
- Week 5.
Random variables, distribution function, probability mass function,
expectation of discrete random variables, Sections 4.1-4.4
- Week 6. Expectation and variance of discrete
random variables, standardization, Sections 4.5-4.6, Bernoulli and
Binomial random variables, Section 5.1
- Week 7. Poisson random variable, Poisson approximation, Poisson process, Geometric random variables, Section 5.2-3
- Week 8. Negative binomial and hypergeometric random variables, Section 5.3
Continuous random variables, Chapter 6
Uniform and normal random variables, Sections 7.1-7.2
- Week 9.
Normal and exponential random variables, Sections 7.2-7.3
Joint distributions of two random variables, joint distribution
function, joint probability mass function, marginal probability mass
functions, expectation of function of two variables,
independent random variables, conditional distribution of random
variables, Sections 8.1-8.3
- Week 10.
Joint distribution of more than two random variables, Section 9.1
Expected values of sums of variables, computing expectations with the indicator variable trick, Section 10.1
- Week 11.
Covariance and correlation of random variables, Sections 10.2-10.3
- Week 12.
Moment-generating functions, 11.1
- Week 13.
Moment-generating functions, sums of independent random variables, Markov inequality, 11.1-11.3
- Week 14.
Chebysev's inequality, estimating tail probabilities of random variables, Law of Large Numbers, Central Limit Theorem, 11.3-11.5
Markov chains, transition probability function and matrix, computing
n-step transition probabilities by taking the powers of the transition
probability matrix12.3 + extra lecture notes
- Week 15. Markov chains, irreducible and regular
Markov chains, limit theorem for the powers of the transition
probability matrix for regular MC's, stationary distribution, 12.3 +
extra lecture notes
- Week 16. Review