Type of Credit: Elective
Credit(s)
Number of Students
近代機率理論的發展以測度理論為基礎, 依據測度理論而發展的期望值理論是機率計算及分析的基礎, 學習測度論需要有相當的數學分析基礎, 這個課程一方面讓學生有初步接觸測度論的機會, 學習實數線上的測度理論及期望值理論, 學生藉此可以加強相關的數學分析的能力, 有這樣的經驗對未來學習一般的測度理論將會有相當的幫助, 未來也可以銜接研究所機率論的學習打下良好的基礎. 另一方面討論機率空間的基本性質, 學生能夠學習機率論基本的數學語言, 以做為未來機率論繼續的學習建立基礎. 以實數的二進位展開為例, 探討數的隨機性質, 獨立性質, 及大數法則的極限定理.
如果你是碩士生, 你也有很紮實的數學分析能力, 你比較好的安排是上學期選修實變函數論, 同時複習高等微積分, 如此一來實變函數論你會學得很扎實, 高等微積分在國外有一說是Real Analysis(實變函數論的英文名)的前半部,因此高等微積分要學得好, 實變函數論你才會學得好. 上學期把高等微積分及實變函數論學好之後, (下學期)之後你要學習分析相關的課程, 機率論相關的課程, 你會學得比較輕鬆有感覺, 學習就會獲得樂趣.
能力項目說明
學生有這樣的基礎, 對於修課的同學, 未來在學習測度論就會相對地容易了.
| 教學週次Course Week | 彈性補充教學週次Flexible Supplemental Instruction Week | 彈性補充教學類別Flexible Supplemental Instruction Type |
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週次 Week |
課程主題
Topic |
課程內容與指定閱讀 Content and Reading Assignment |
教學活動與作業 Teaching Activities and Homework |
學習投入時間 Student workload expectation |
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課堂講授 In-class Hours |
課程前後 Outside- of-class Hours |
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1 |
Sigma-algeba and Measures |
Introduction, Set Operations, Sigma-algebra and Algebra, Borel Sigma-algebra on R^n, Measure Spaces, Probability Spaces |
講述教學法，並提出習題為作業 |
2 |
2 |
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2 |
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講述教學法，並提出習題為作業 |
2 |
2 |
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3 |
Sigma-algeba and Measures |
Caratheodory’s Extension Theorem |
講述教學法，並提出習題為作業 |
2 |
2 |
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4 |
Lebesgue Integration Theory |
Measurable Functions. Integral |
講述教學法，並提出習題為作業 |
2 |
2 |
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5 |
Lebesgue Integration Theory |
Limit Theorems Lebesgue Monotone Convergence Theorem Fatou’s Lemma The Lebesgue Dominated Convergence Theorem Chebyshev’s Inequality |
講述教學法，並提出習題為作業 |
2 |
2 |
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6 |
Lebesgue Integration Theory |
Types of Convergence for Measurable Functions |
講述教學法，並提出習題為作業 |
2 |
2 |
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7 |
Product Measure |
Product Sigma-algebra Monotone Class Theorem. |
講述教學法，並提出習題為作業 |
2 |
2 |
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8 |
Product Measure |
Product Measure.
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講述教學法，並提出習題為作業 |
2 |
2 |
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9 |
Product Measure |
Fubin’s Theorem. |
講述教學法，並提出習題為作業 |
2 |
2 |
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10 |
Random Variables |
Expectation of Random Variables. Distribution Measure of a Random Variable |
講述教學法，並提出習題為作業 |
2 |
2 |
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11 |
Random Variables |
Distribution Function of a Random Variable Borel Measurable Functions of a Random Variable.
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講述教學法，並提出習題為作業 |
2 |
2 |
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12 |
Random Variables |
Independence, Independent Sigma-algebras, Independent Events, Independent Random Variables, Construction of Independence Random Variables
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講述教學法，並提出習題為作業 |
2 |
2 |
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13 |
Classical Limit Theorems |
Bernoulli Trials Weak Law of Large Number Borel Theorem of Normal Numbers |
講述教學法，並提出習題為作業 |
2 |
2 |
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14 |
Classical Limit Theorems |
L^2 - Weak Law L^2 Convergence Convergence in Probability The Weierstrass Approximation Theorem |
講述教學法，並提出習題為作業 |
2 |
2 |
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15 |
Classical Limit Theorems |
First Borel Cantelli Lemma Second Borel Cantelli Lemma Strong Law of Large Number
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講述教學法，並提出習題為作業 |
2 |
2 |
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16 |
Conditional Expectation |
Conditional Probability with Respect to a Sigma-algebra, Sigma-algebra Generated by a Measurable Partition |
講述教學法，並提出習題為作業 |
2 |
2 |
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17 |
Conditional Expectation |
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講述教學法，並提出習題為作業 |
2 |
2 |
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18 |
Conditional Expectation |
Conditional Expectation with Respect to a Sigma-algebra L^1 random Variables |
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1. 考試成績: 60% (點名，課堂表現), 期中考30%, 期末考30%
2. 作業成績 40%
Reading:
Rodrigo Banuelos (2003), Lecture Notes Measure Theory and Probability
Additional Readings.
A. Grigoryan(2008), Measure Theory and Probability (Lecture Notes)
Santosh S. Venkatesh (2013), The Theory of Probability: Explorations and Applications, Cambridge University Press
Patrick Billingsley(1995), Probability and Measure.
M Caplinski & E Kopp (2003), Measure, Integral and Probability, 2nd Edition
J. Jacob and P. Protter (2004), Probability Essential, 2nd Edition
Robert B. Ash, Catherine A. Dleans-Dade, Probability and Measure Theorem, 2nd Edition