Fall 2024, Instructor
數理統計一 (Mathematical Statistics I) (QF 314800)
Course Description & Audience: This course concentrates on theoretical statistics using the first principles of probability theory. Our aim is to introduce many fundamentals and lay a solid foundation for students to explore various areas such as quantitative finance, data science, statistical signal processing, machine learning, financial mathematics, and econometrics in the future. Many of the results will be delivered in a definition-theorem-proof manner. The prerequisite is one year of calculus. To succeed in this course, some mathematical maturity is expected. The intended topics to cover are listed below:
Introduction to Statistics
Probability Theory
Transformation and Expectations
Common Families of Distributions
Multiple Random Variables
Sampling Distributions and Random Sample
Limiting Behaviors and Central Limit Theorem
Elementary Statistical Inferences
Prerequisites: Students planning to take this course should be fairly familiar with multivariate calculus. As mentioned previously, some mathematical maturity is expected to succeed in this course.
Time and Place: Lectures are at T7T8T9, Room 204 TSMC Building.
Office Hours: The instructor's office is open Monday from 12:00 to 13:00 in Room 608 of the TSMC Building (台積館). Meetings are also possible at other times by appointment.
Textbooks & References: Students will be provided with significant handout material to support the lectures at no cost. The material is mainly drawn from the following recommended textbooks.
G. Casella and R. L. Berger, Statistical Inference, Cengage Learning, 2001.
R. Hogg, J. McKean, A. Craig, Introduction to Mathematical Statistics, Pearson, 2018.
J. A. Rice, Mathematical Statistics and Data Analysis, Cengage Learning, 2006
D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Mathematical Statistics with Applications, Thomson Brooks/Cole, 2008.
Teaching Method: Lecture.
Teaching Assistant: 甘容 (Rong Gan), ganzong@gmail.com.
Homework: Approximately weekly.
Grading: The grade will be based on one midterm test (30%), homework (30%), and a final exam (40%). The instructor may exercise discretion up to 10% in each grading category.
Course Material
Week 01: 09/03
First day of class.
Basic Probability Theory I
Week 02: 09/10
Basic Probability Theory II (CDF)
Week 03: 09/17 (No Class)
Mid-Autumn Festival
Week 04: 09/24
Transformation and Expectation I (CDF, PDF, Transformation of Random Variables)
Week 05: 10/01
Transformation and Expectation II (Expectation, Variance, Moments)
Week 06: 10/08
Transformation and Expectation III (MGF)
Multiple Random Variables I (Joint Distributions, Joint Density, Marginal Distributions)
Week 07: 10/15
Multiple Random Variables II (Conditional Distributions, Expectations, Variances)
Week 08: 10/22
Multiple Random Variables III (Conditional Expectations, Independence, Bivariate Transformation)
Week 09: 10/29
Midterm 1
Week 10: 11/05
Multiple Random Variables IV
Random Samples I
Week 11: 11/12
Random Samples II (Sampling from Normal Distribution)
Week 12: 11/19
Random Samples III (Order Statistics)
Week 13: 11/26
Limiting Behavior I (Convergence in Probability, WLLN, Continuity Theorem)
Week 14: 12/03
Limiting Behavior II (Almost Sure Convergence, SLLN, and Convergence in Distribution)
Course Evaluation
Week 15: 12/10
Limiting Behavior III (CLT, Slustky Theorem)
Course Evaluation
Week 16: 12/17
Final Exam
Course Evaluation
Assignments
Exam
Midterm: Everything up to Chapter 4.5.
Final Exam: Chapter 1 to Chapter 6.
Supplementary Documents
Mathematics Premier for Introduction to Mathematical Statistics by Prof. J. McKean
ChatGPT, https://openai.com/blog/chatgpt/ OpenAI