Download Probability and Mathematical Statistics I

Probability and Mathematical Statistics I
Department of Statistics 36-625, Fall 2007
Lectures: MWF 2:30-3:20 PM; WEH 5312
Instructor: Ann B. Lee ([email protected])
Baker Hall 229 J; Phone: 268-7831
Office hours: Tu 4-5 PM
Teaching assistants: Erich Huang ([email protected])
Office hours: Th 5-7 PM in the FMSB building, Room 320.
Chris Neff ([email protected])
Office hours: Wed 6-7 PM in the FMSB building, Room 320.
Main text: Larry Wasserman, All of Statistics: A Concise Course in Statistical Inference,
Springer 2004 (Corrected second printing 2005).
This course is a fast-paced and intense introduction to the foundations of probability
theory and statistical inference. It is primarily intended for graduate students in computer
science and honors undergraduates in mathematics, statistics, and computer science. The
class is also useful for students beginning graduate work in statistics who need to fill in
their background on mathematical statistics. The class assumes a solid background in multivariable calculus and linear algebra. Previous exposure to probability and statistics is helpful
but not a formal requirement. A natural continuation of 36-625 is 36-626 in the spring which
will cover advanced topics in modern statistics.
Course Work: There will be weekly homework assignments, two midterms, and a final
exam. Homework assignments will be a mixture of theory and practical exercises; you may
have to read ahead to do some of the problems. I encourage you to discuss homework
problems with other students but do not copy other students’ assignments. In other words,
work together but write up your solution on your own.
Some problems will involve computing. I recommend that you use a high-level computer language such as R or Matlab. Some basic R documentation has been posted on the blackboard
site.
Grading policy: The term grade will be based 25% on the homework, 35% on the midterm
and 40% on the final. Homework assignments are due on Fridays in class; if you are unable
to make it to class, please slip your homework under my door before the class starts. No late
homework is accepted; your lowest HW grade will however be discarded. Make-up exams
and extensions will only be given with a note from your advisor.
Tentative Course Schedule
Probability Theory
(1) Week of 8/27
Basic theory: Sample spaces, events, and the axioms of probability.
Independence and conditional probability. Bayes’ Theorem.
Random variables and distribution functions.
(2) Week of 9/3
Labor day. No class on Monday.
Discrete vs continuous random variables. Examples.
Multivariate distributions. Marginal and conditional distributions.
(3) Week of 9/10
Transformations of random variables.
Expectation of a random variable. Variance and covariance.
(4) Week of 9/17
Conditional expectation. Moment generating functions.
Inequalities. Convergence of random variables.
(5) Week of 9/24
More on convergence. Introduction to large sample theory.
Law of large numbers. Application of LLN to Monte Carlo integration.
(6) Week of 10/1
Central limit theorem. Taylor expansions and the delta method.
Statistical Inference
(7) Week of 10/8
Basic concepts and terminology in statistical inference.
Midterm 1 (Th 10/11, 2 hrs)
(8) Week of 10/15 Non-parametric inference: Estimating the cdf and statistical functionals.
Bootstrap for variance estimation and confidence intervals.
Mid-semester break. No class on Friday.
(9) Week of 10/22 Parametric inference: Method of moments and maximum likelihood.
Properties of maximum likelihood estimators.
Fisher information and asymptotic normality. Efficiency.
(10) Week of 10/29 More on maximim likelihood estimators. Cramer-Rao inequality.
The delta method revisited. Parametric bootstrapping.
(11) Week of 11/5 Topics (e.g. numerical methods and EM) as time permits.
(12) Week of 11/12 Hypothesis testing and p-values.
Midterm 2 (Th 11/15, 2 hrs)
(13) Week of 11/19 Examples and more on hypothesis testing.
Thanksgiving holiday. No classes on Wednesday and Friday.
(14) Week of 11/26 Topics (e.g. statistical decision theory) as time permits.
(15) Week of 12/3 Review and wrap-up.