數理統計課程內容涵蓋相當的廣度與深度，讓學生能夠瞭解基本的理論統計概念，課程範圍包含：
1. 機率：排列組合、條件機率、貝氏定理、隨機變數、分布函數、期望值、條件期望值、動差
2. 機率分配計算方法：變數變換、動差母函數、摺積
3. 分布族：常見離散型與連續型分布，其特性及之間的關係
4. 位置族、尺度族、指數族
5. 聯合分佈、邊際分佈、條件分佈、條件期望值、共變異數、相關係數、混合分佈
6. 不等式：Chebyshev、Jensen、Hölder
7. 收歛概念與極限分配：順序統計量、機率收斂、分布收斂、中央極限定理、Delta方法
8. 點估計方法：MLE、method of moments
9. 點估計量性質：不偏性、一致性、有效性、充分性、完備性
10. 點估計評估準則：MSE、CRLB、MVUE、Rao-Blackwell
11. 區間估計方法與評估準則
12. 檢定方法：LRT
13. 檢定統計量評估準則: Neyman-Pearson Lemma, UMP test, MLR
14. 貝氏統計

The mathematical statistics course content covers considerable breadth and depth, allowing students to understand basic theoretical statistical concepts. The course scope includes:
1. Probability: permutation and combination, conditional probability, Bayesian theorem, random variables, distribution function, expected value, conditional expected value, dynamic difference
2. Probability distribution calculation methods: variable transformation, dynamic difference generating function, convolution
3. Distribution family: common discrete and continuous distributions, their characteristics and the relationship between them
4. Position family, scale family, index family
5. Joint distribution, marginal distribution, conditional distribution, conditional expected value, covariance, correlation coefficient, mixed distribution
6. Inequality: Chebyshev, Jensen, Hölder
7. Convergence concept and limit allocation: sequential statistics, probability convergence, distribution convergence, central limit theorem, delta method
8. Point estimation method: MLE, method of moments
9. Properties of point estimators: impartiality, consistency, validity, adequacy, and completeness
10. Point estimation evaluation criteria: MSE, CRLB, MVUE, Rao-Blackwell
11. Interval estimation methods and evaluation criteria
12. Calibration method: LRT
13. Evaluation criteria for test statistics: Neyman-Pearson Lemma, UMP test, MLR
14. Bayesian Statistics

數理統計為統計理論的核心課程，課程主要教授統計理論與概念，為其他統計相關課程的理論基礎，並培養學生修習更高深與統計相關之課程的能力。修課學生應具備微積分的基礎。本課程藉由定義解析、定理推導與例題講解過程，學習機率論與統計推論的基本理論與概念。機率論課程內容包括機率基本概念與定義、分配、不等式、收歛概念與極限分配；統計推論課程內容包括點估計方法、區間估計方法、檢定方法等不同的方法與理論。

Mathematical statistics is the core course of statistical theory. The course mainly teaches statistical theories and concepts, which serves as the theoretical basis for other statistics-related courses and cultivates students' ability to take more advanced statistics-related courses. Students taking this course should have a foundation in calculus. This course learns the basic theories and concepts of probability theory and statistical inference through the process of definition analysis, theorem derivation and example explanations. The content of the probability theory course includes the basic concepts and definitions of probability, distribution, inequalities, convergence concepts and limit distribution; the content of the statistical inference course includes different methods and theories such as point estimation method, interval estimation method, and verification method.

1. George Casella and Roger L. Berger (2002) Statistical Inference (2/E), Duxbury Press.
2. R.V. Hogg, E.A. Tanis (2005) Probability and Statistical Inference (7/E), Prentice Hall.
3. Dennis Wackerly, William Mendenhall, Richard L. Scheaffer (2001) Mathematical Statistics with Applications (Mathematical Statistics) (6/E), Duxbury Press.

1. George Casella and Roger L. Berger (2002) Statistical Inference (2/E), Duxbury Press.
2. R.V. Hogg, E.A. Tanis (2005) Probability and Statistical Inference (7/E), Prentice Hall.
3. Dennis Wackerly, William Mendenhall, Richard L. Scheaffer (2001) Mathematical Statistics with Applications (Mathematical Statistics) (6/E), Duxbury Press.