Mathematical Statistics I

My Current Teaching @ National Tsing Hua University

Fall 2025, 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 key fundamentals and lay a solid foundation for students to explore 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 multivariate calculus. To succeed in this course, a certain level of 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: 戴翔(dairyan930128@gmail.com) & 許允齊(star930302@gmail.com).  TA's office hours: TBA

Homework: Approximately weekly. 

Grading: The grade will be based on one midterm test (30%), homework (20%), and a final exam (50%). The instructor may exercise discretion up to 10% in each grading category.

AI Teaching Assistant:

Gemini Gems (Math Stats Tutor. test version) : https://gemini.google.com/gem/1QrhLPcZ6CkL6w-lZ7-3YMJYAzCunS2RK?usp=sharing

AI Teaching Assistant — Terms of Use: This link above is for enrolled students of this course only; please do not redistribute it. To use it, sign in with your Google account (your NTHU Google account is recommended). AI responses are for learning reference and may contain errors — the lecture notes and in-class material take precedence. Do not enter personal or sensitive information in your conversations.

Course Schedule

• Week 01 (09/08)

  • Review of Basic Probability Theory I

• Week 02 (09/15)

  • Review of Basic Probability Theory II

• Week 03 (09/22)

  • Transformation and Expectations I. CDF techniques

• Week 04 (09/29)

  • Transformation and Expectations II. Expected Value and Variance

• Week 05 (09/30)

  • Transformation and Expectation III: Moment Generating Functions

  • Multiple Random Variables I: Joint Distributions, Marginal Distributions, Expectations

• Week 06 (10/06)

  • Multiple Random Variables II: Conditional Distributions and Expectation

• Week 07: 10/13

  • Multiple Random Variables III

• Week 08: 10/20

  • Midterm 1

• Week 09: 10/27

  • Review of Midterm 1.

  • Multiple Random Variables IV

• Week 10: 11/03

  • Random Samples: 

• Week 11: 11/10

  • Random Samples II: Cochran Theorem; Order Statistics

• Week 12: 11/17

  • Limiting Behaviors I: Convergence in Probability; WLLN 

• Week 13: 11/24

  • Limiting Behaviors II: Almost Sure Convergence; SLLN 

• Week 14: 12/01

  • Limiting Behaviors III: Convergence in Distribution; CLT

• Week 15: 12/08

  • Limiting Behaviors IV.

  • Review the Course

  • Course Evaluation

• Week 16: 12/15 

  • Final Exam

Lecture Handouts

• Chapter 1: Basic Probability Theory 

• Chapter 2: Transformation and Expectations 

• Chapter 3: Common Families of Distributions

• Chapter 4: Multiple Random Variables 

• Chapter 5: Random Samples

• Chapter 6: Limiting Behaviors 

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

• Gemini, gemini.google.com/ Google