數理統計為統計系學生學習統計理論的核心課程，提供統計相關課程的理論根基，並且培養未來修習與統計相關高階課程的能力，本學期課程主題以機率理論為主，範圍包含： 1.機率：條件機率、隨機變數、分配函數、期望值、條件期望值 2.尋找機率分配的技巧：變數變換、動差母函數 3.分配：離散與連續型分配其特性及分配之間的關係、位置與尺度族、t分配和F分配、混合分配 4.不等式：Chebyshev、Jensen、Hölder 5.隨機樣本收歛概念與極限分配：不偏性、一致性、機率收斂、分配收斂、中央極限定理、Delta方法。Mathematical statistics is a core course for students in the Department of Statistics to learn statistical theory, providing a theoretical foundation for statistics-related courses and cultivating the ability to take advanced courses related to statistics in the future. The subject of this semester's courses is mainly probability theory, and the scope includes: 1. Probability: Conditional probability, random variables, distribution function, expected value, conditional expected value 2. Techniques for finding probability distribution: variable transformation, dynamic difference generating function 3. Distribution: characteristics of discrete and continuous distribution and the relationship between distribution, position and scale family, t distribution and F distribution, mixed distribution 4. Inequality: Chebyshev, Jensen, Hölder 5. Random sample convergence concept and limit distribution: Unbiasedness, consistency, probabilistic convergence, distribution convergence, central limit theorem, delta method.
數理統計為統計系學生學習統計理論的核心課程，提供統計相關課程的理論根基，並且培養未來修習更高深與統計相關之課程的能力，課程內容涵概相當的廣度與深度，強調學生能夠了解基本的理論統計概念及在不同情況下的統計程序，為能獲得較佳的學習成效，學生應具備微積分與機率論的基礎，課程主題包括機率理論與統計推論，範圍包含：
1.機率：條件機率、隨機變數、分配函數、期望值、條件期望值
2.尋找機率分配的技巧：變數變換、動差母函數
3.分配：離散與連續型分配其特性及分配之間的關係、位置與尺度族、多變量常態分配、t分配和F分配、混合分配
4.不等式：Chebyshev、Jensen、Hölder
5.隨機樣本收歛概念與極限分配：不偏性、一致性、機率收斂、分配收斂、中央極限定理、Delta方法
6.隨機樣本：抽樣、單一樣本與兩樣本之信賴區間及假設檢定的常態理論及其相關的大樣本方法、順序統計量、生成隨機樣本技巧、拔靴法
7.最大概似法及其漸進理論：Cramér-Rao不等式、有效性、最大概似估計量、最大概似估計量的漸近性質、EM演算法
8.縮減資料：指數族、充分性、完備性、完備充分統計量、UMVUE、Rao-Blackwell定理、Basu定理
9.最適假設檢定：Neyman-Pearson引理、MP檢定、UMP檢定、MLR族、UMPU 檢定、LR檢定、sequential檢定
Mathematical statistics is a core course for students in the Department of Statistics to learn statistical theory. It provides the theoretical foundation for statistics-related courses and cultivates the ability to take more advanced statistics-related courses in the future. The content of the course covers considerable breadth and depth, emphasizing that students can understand the basics. The theoretical statistical concepts and statistical procedures in different situations. In order to obtain better learning results, students should have a foundation in calculus and probability theory. The course topics include probability theory and statistical inference, and the scope includes:
1. Probability: conditional probability, random variables, distribution function, expected value, conditional expected value
2. Techniques for finding probability distribution: variable transformation, dynamic difference generating function
3. Distribution: characteristics of discrete and continuous distribution and the relationship between distribution, position and scale family, multi-variable normal distribution, t distribution and F distribution, mixed distribution
4. Inequality: Chebyshev, Jensen, Hölder
5. Random sample convergence concept and limit allocation: impartiality, consistency, probability convergence, allocation convergence, central limit theorem, Delta method
6. Random samples: sampling, confidence intervals of single sample and two samples, normality theory of hypothesis testing and its related large sample methods, sequential statistics, techniques for generating random samples, and the boot method
7. Maximum likelihood method and its asymptotic theory: Cramér-Rao inequality, validity, maximum likelihood estimator, asymptotic properties of maximum likelihood estimator, EM algorithm
8. Reduction of data: exponential family, sufficiency, completeness, complete and sufficient statistics, UMVUE, Rao-Blackwell theorem, Basu theorem
9. Optimum hypothesis test: Neyman-Pearson lemma, MP test, UMP test, MLR family, UMPU test, LR test, sequential test
1. R.V. Hogg, J.W. McKean, A.T. Craig (2013) Introduction to Mathematical Statistics (7/E), Pearson
Prentice Hall.
2. L.J. Bain and M. Engelhardt (1992) Introduction to Probability and Mathematical Statistics (2/E), Duxbury.
3. R.J. Laesen, M.L. Marx (2006) An Introduction to Mathematical Statistics and Its Applications (4/E),
Pearson Education International
4. D.D.Wackerly, W.Mendenhall III, R.L.Scheaffer (2007) Mathematical Statistics with Applications (7/E),
Duxbury.
1. R.V. Hogg, J.W. McKean, A.T. Craig (2013) Introduction to Mathematical Statistics (7/E), Pearson
Prentice Hall.
2. L.J. Bain and M. Engelhardt (1992) Introduction to Probability and Mathematical Statistics (2/E), Duxbury.
3. R.J. Laesen, M.L. Marx (2006) An Introduction to Mathematical Statistics and Its Applications (4/E),
Pearson Education International
4. D.D.Wackerly, W.Mendenhall III, R.L.Scheaffer (2007) Mathematical Statistics with Applications (7/E),
Duxbury.
| 評分項目 Grading Method | 配分比例 Grading percentage | 說明 Description |
|---|---|---|
| Mid-term ExamMid-term Exam mid-term exam |
33 | |
| Final ExamFinal Exam final exam |
33 | |
| Quizzes and recitation classQuizzes and recitation class quizzes and recitation classes |
34 | 4-6 quizzes |
