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.
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)
2. 作業成績 40%
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