STP 421 – Probability

Fall 2013

Instructor: Dr. Jay Taylor, office: PSA 447; phone: 965-2641; e-mail: jtaylor@math.asu.edu
Time: Mondays and Wednesdays 3:00-4:15
Location: PABLO 105
Office Hours: Wednesdays 12:00-2:00 in PSA 447; also by appointment.
Text: Principles of Uncertainty by Joseph Kadane (Chapman & Hall, 2011). A pdf copy of this text is available (legally) at the following link. You can also purchase a hard copy through Amazon.com or CRC Press.

Course Description: This course will provide an introduction to probability theory that emphasizes the statistical applications of this subject, particularly those relevant to Bayesian inference. Topics covered will include coherence and the laws of probability, conditional distributions and Bayes' formula, random variables, discrete and continuous distributions, stochastic simulation, univariate and multivariate normal distributions, and the fundamental limit theorems of probability theory. Where possible, examples of applications of probability to real-world problems will be given.

Prerequisites: Three semesters of calculus, including vector calculus. Although we will cover some proofs in lectures, it is not necessary to have taken MAT 300 (Mathematical Structures) to do well in this course. However, you should be familiar with elementary set theory (unions, intersections, complements, etc.) and it is essential that you be able to carry out the kinds of calculations covered in college algebra and calculus courses. Please review these subjects if your mastery of them is rusty.

Problem Sets: These will be announced in class and posted on the course web page at the following link. Solutions will also be posted on the course web page a few days after each set is collected. (Note: Problem sets will usually not be accepted after the solutions have been posted except when there are extenuating circumstances such as illness.) You are welcome to work in groups, but you should write up your solutions individually and you should always give due credit if your solution came from another source, such as a text book, from an online resource, or from one of your classmates. Solutions should either be typed or neatly written. Because mathematics is best learned by attempting to solve problems, it is very important that you attempt every problem set, no matter how difficult, and turn in solutions, even if they are partial or incorrect.

Quizzes: There will be weekly quizzes, usually consisting of a single question/problem and lasting 5 minutes or less. These will be unannounced and cannot be made up except in the case of an emergency. However, I will drop your four lowest scores, which means that you can miss four of these without incurring a penalty.

Exams: There will be two exams, including a take-home midterm that will be distributed in early October and an in-class final exam (date TBA).

Grading and Assignments: Grades will be based on your cumulative score on the exams (100 points each), quizzes (100 points in total), and problems sets (200 points in total). Of the 500 points possible, 450 points and above will guarantee an A, 400 points and above will guarantee a B, 350 points and above will guarantee a C, and 300 points and above will guarantee a D.

Syllabus:

Date

Topic

Reading

26 Aug

Overview; Coherence and probability

Kad 1.1-1.2; slides

28 Aug

Probabilities of unions

Kad 1.3

2 Sept

Labor Day – no class


4 Sept

Random variables with finitely many values

Kad 1.4-1.5

9 Sept

Properties of expectations; Coherence prevents sure loss

Kad 1.6-1.7

11 Sept

Conditional probability; The birthday problem

Kad 2.1-2.2; slides

16 Sept

Simpson's Paradox; Bayes' theorem and Bayesian statistics

Kad 2.3-2.4

18 Sept

Independence

Kad 2.5

23 Sept

Gambler's Ruin

Kad 2.7

25 Sept

Iterated expectations; Independence of RVs

Kad 2.8

30 Sept

Binomial, multinomial and hypergeometric distributions

Kad 2.9-2.10

2 Oct

Variance and covariance

Kad 2.11

7 Oct

Weak Law of Large Numbers

Kad 2.13

9 Oct

Random variables with countably many values; countable additivity

Kad 3.1-3.3; slides

14 Oct

Fall Break – no class


16 Oct

Properties of expectations; Midterm distributed

Kad 3.4

21 Oct

Probability generating functions; Geometric distribution; Midterm due (solutions)

Kad 3.6-3.8

23 Oct

Poisson distribution

Kad 3.9

28 Oct

Genealogies and coalescents

slides

30 Oct

Continuous distributions

Kad 4.1; slides

4 Nov

Measure theory and the Lebesgue integral

instructor's notes

6 Nov

Expectations and integrals

Kad 4.4-4.5

11 Nov

Veteran's Day – no class


13 Nov

Exponential and gamma distributions

Instructor's notes

18 Nov

Joint distributions

Kad 4.2

20 Nov

Conditional distributions

Kad 4.3

25 Nov

Transformations of random variables

Kad 5.1-5.3; slides

27 Nov

Moment generating functions; characteristic functions

Kad 6.1-6.3

2 Dec

Normal distributions

Kad 6.9

4 Dec

The Central Limit Theorem

Kad 6.11