機率學
Probability
| 節 | 週四 |
|---|---|
2 09:00–09:50 | 機率學 MB506（光復） 3 節連堂 |
3 10:10–11:00 | |
4 11:10–12:00 |
* 根據陽明交大上課時間表所列
This class mainly focuses on the theoretical concept in probability and measure theory for graduate students in the fields of operations research, statistics, management science, biology, and engineering. Several topics in the real analysis will also be taught as a preparation for students who haven't received a related mathematical training before. The objective of our class is to develop students with foundations to pursue advanced classes of stochastic processes, Markov decision process, queueing theory, stochastic programming, and statistics.
Calculus
無備註
Most of the references can be found in the e-library system. Also, there is a limited number of paper copies stocked in the school's library.
1.Assignments: Students will work 3~5 problems every week. The types of homework problem mainly include calculating, deriving, and proving. 2. Exams: (1) Two midterms (2) A final exam 3. Grading: (1) Homework:40% (2) In-class performance:10% (2) Midterm:30% (3) Final exam:20%
Unit 1: Elementary in real analysis
1. Set operations 2. De Morgan's Law 3. Functions 4.Supremem and infimum 5. Sequences of sets 6. Limits and convergences
- 講授：
- 12
- 實作：
- V
Unit 2: Probability and measure theory
1. Axioms of probability 2. Probability space and measures 3. Conditional probability 4. Random variables 5. Integration and expectation 6. Variance of a random variable
- 講授：
- 15
- 實作：
- V
Unit 3: Additional topics in probability
1. Important inequalities 2. Convergence of random variables 3. Limit theorems
- 講授：
- 9
- 實作：
- V
Unit 4: Extensions
1.Sampling and Monte Carlo simulation 2.Transforms in probability 3.Sum of random variables and convolution 4.Stochastic processes
- 講授：
- 9
- 實作：
- V
Unit 5: Exams
1. Midterm exams 2. Final exam
- 講授：
- 9
- 實作：
- V
| 週次 | 主題 |
|---|---|
| 第 1 週 | Class intro / Rudimentary real analysis |
| 第 2 週 | Set / Set operations / Sequences of sets and their limits / De Morgan's Law |
| 第 3 週 | Extended real number / Boundness of set / Supremum and infimum of sets |
| 第 4 週 | Sequences of real numbers / Subsequences / Bounded sequences / Convergent sequences |
| 第 5 週 | Limit superior and inferior / Fatous lemma |
| 第 6 週 | Midterm exam I |
| 第 7 週 | Limits of function / Continuous functions / Monotocity of functions / Convergences of a sequence of functions / Integrable functions |
| 第 8 週 | Fields / Borel σ-field / Measurable space/ Lebeque measure |
| 第 9 週 | Axioms of probability / Probability measure space |
| 第 10 週 | Independence / Conditional probability / Law of total probability |
| 第 11 週 | Distribution functions / Density functions |
| 第 12 週 | Midterm exam II |
| 第 13 週 | Random variables |
| 第 14 週 | Integration and expectation / Conditional distribution and expectations |
| 第 15 週 | Markov' inequality / Chebyshev's inequality / Jensen's inequality |
| 第 16 週 | Modes of convergence / Laws of large numbers / Limit theorems in probability |
| 第 17 週 | Sampling approaches / Transformation methods in probability / Sum of random variables and convolution |
| 第 18 週 | Final exam |
The class notes are prepared by the lecturer. References: (1) S. Resnick, A Probability Path, Birkhauser, Boston, 2001. (2) H. L. Royden, Real Analysis, Macmillan, New York, 1963. (3) P. Billingsley, Probability and Measure, Wiley & Sons, New York, 1995. (4) S. Ross, A First Course in Probability, 7th Edition, Pearson Prentice Hall, New Jersey, 2006.
- 地點
- MB513
- 時間
- By appointment
- 聯絡方式
- sichen@nctu.edu.tw