Math 431 - Introduction to Probability Theory

Spring 2009

Meetings: TR 9:30-10:45, Van Vleck B115
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: Tuesday 15:00-16:00 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.

Please read the course syllabus for the rules and requirements.

Textbook: A First Course in Probability (Seventh Edition) by Sheldon Ross

You may use other editions of the textbook, but the homework will be assigned from the seventh edition.

Prerequisites: The basic skills needed for Math 431 are calculus (including multivariable calculus) and basic set theory. Success in this class will also require the ability to think/reason abstractly.

Course Content: Math 431 is an introduction to the basic concepts of probability theory, the mathematical discipline for analyzing and modelling uncertain outcomes. We will cover chapters 1-8 of the text which include the following topics:

Methods of counting
Axioms of probability
Conditional probability and independence
Discrete, continuous and multivariate random variables
Expectation, variance, covariance
Limit Theorems: law of large numbers and the central limit theorem

Evaluation: Course grades will be based on homework assignments, quizzes, two midterm exams and the final exam.

Homework assignments

Assignment 1 Due date: February 5 Solution 1
Assignment 2 Due date: February 12 Solution 2
Assignment 3 Due date: February 19 Solution 3
Assignment 4 Due date: February 26 Solution 4
Assignment 5 Due date: March 5 Update: you do not need to hand in the theoretical exercise 14 (b) (you can do it for extra credit) Solution 5
Assignment 6 Due date: March 12 Solution 6
Assignment 7 Due date: March 26 Update: you do not need to hand in the theoretical exercise 1, it will be moved to Assignment 8Solution 7
Assignment 8 Due date: April 2 Solution 8
Assignment 9 Due date: April 9 Solution 9
Assignment 10 Due date: April 23 Solution 10
Assignment 11 Due date: April 30 Solution 11
Assignment 12 Due date: May 7 Solution 12

List of marks in decresing order

Instructions for the homework assignments:
Homework must be handed in by the due date by the beginning of the class. It can be handed in by person in class or in the instructor's mailbox. Late submissions will not be accepted. Please read the syllabus for further instructions.
Bonus problems have no deadlines and they are to be submitted separately. All such problems must be handed in by the next to last week of the semester.

More information on the final exam.

More information on the second midterm. Second midterm solutions.

More information on the first midterm. First midterm solutions.

Reminder: the final exam will take place on May 13, Wednesday, 5:05-7:05 pm at VV B239. I will have extra office hours on May 11, Monday from 3 pm at VV B135.

Schedule:

Week 1. Counting techniques (the basic principle of counting, permutations, combinations, the binomial coefficient), Sections 1.2-1.4
Week 2. Counting techniques (multinomial coefficient), Axioms of probability (sample space, events, axioms, the basic properties, computing probability in simple examples, the inclusion-exclusion formula), Sections 1.5, 2.2-2.5
Week 3. Computing the probability of events by counting, Conditional probability, Sections 2.3-2.5, 3.2
Week 4. Properties of conditional probability, Bayes formula, Independence, Sections 3.2-3.4
Week 5. Examples on independence, Random variables, Probability mass function, Sections 3.4-3.5, 4.1-4.2
Week 6. Discrete random variables, cumulative distribution function, indicator variable, expectation, expectation of a function of a random variable, Sections 4.1-4.4
Week 7. Variance, notable discrete random variables (Bernoulli, Binomial, Hypergeometric, Poisson), Sections 4.5-4.8
Week 8. Notable discrete random variables (Poisson, Geometric, Negative Binomial), Continuous Random Variables, probability density function, Sections 4.5-4.8, 5.1-5.2
Week 9. Continuous Random Variables: density and distribution function, expectation, uniform random variable, normal random variable 5.2-5.4
Week 10. Continuous Random Variables: approximation of the binomial random variable with a normal (the De Moivre-Laplace Central Limit Theorem), other examples for continuous random variables (exponential, Gamma, Cauchy and Beta), hazard rate function 5.4-5.6 expectation, uniform random variable, normal random variable 5.2-5.4
Week 11. Computing the density of a function of a random variable, jointly distributed random variables (joint distribution function, joint density, joint probability mass function) 5.7, 6.1
Week 12. Independent random variables, factorization of the joint distribution, probability mass and density functions, discrete and continuous examples 6.2
Week 13. Sums of independent random variables (specific examples: Gamma, normal, Poisson, exponential, chi-squared random variables) 6.3
Week 14. Conditional distributions (discrete and continuous cases), Joint distribution of functions of random variables, Expectation of a function of random variables, expectation of sums of random variables, computing expectation of a random variable using linearity (the indicator variable trick) 6.5, 6.7, 7.1, 7.2
Week 15. Moments of the number of events that occur, Covariance and variance of sums of random variables, the correlation function, moment generating functions, the Central Limit Theorem and the Weak Law of Large Numbers 7.3, 7.4, 8.2, 8.3