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課程名稱 |
機率論一
Probability Theory (Ⅰ) |
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開課學期 |
111-1 |
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授課對象 |
理學院 應用數學科學研究所 |
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授課教師 |
林偉傑 |
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課號 |
MATH7509 |
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課程識別碼 |
221EU3410 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期二6,7(13:20~15:10)星期四7(14:20~15:10) |
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上課地點 |
天數101天數101 |
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備註 |
本課程以英語授課。
總人數上限：60人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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課程概述 |
This course is the first-half of a yearly sequence of measure-theoretic probability theory. Tentative topics include: basics of probability, measure theory, random variables, distribution, Poisson processes, limit theorems. |
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課程目標 |
Students will understand the basics of measure-theoretic probability. |
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課程要求 |
Students should be familiar with elementary analysis. Knowledge on undergraduate probability or measure theory is not required, but it will be helpful. If students have not learned measure theory, they are strongly recommended to take real analysis at the same time. |
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預期每週課後學習時數 |
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Office Hours |
每週四 15:10~16:10 |
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指定閱讀 |
Here are some reference books:
Probability and Measure, by P. Billingsley, 3rd edition, Wiley, 1995.
Probability: Theory and Examples, by R. Durrett. 5th edition. Cambridge U. Press 2019.
Probability with martingales, by D. Williams. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991.
Probability theory lecture notes by D. Panchenko. |
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參考書目 |
See "Designated Reading". |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
60% |
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2. |
Midterm |
20% |
10/25 |
3. |
Final |
20% |
12/20 |
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週次 |
日期 |
單元主題 |
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第1週 |
9/6, 9/8 |
9/6: Algebra, \sigma-algebra, probability measures, basic properties.
9/8: Lebesgue measure |
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第2週 |
9/13, 9/15 |
9/13: Caratheodory extension
9/15: Uniqueness of extension, monotone class theorem, limit sets, first Borel-Cantelli, fair coin flips |
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第3週 |
9/20, 9/22 |
9/20: Independence, conditional probability, Kolmogorov 0-1 law, second Borel-Cantelli, random variables
9/22: Independence of random variables, expectation |
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第4週 |
9/27, 9/29 |
9/27: Variance, inequalities, first and second moment methods, Bernoulli bond percolation, almost sure convergence, convergence in probability, bounded convergence theorem
9/29: Weak and strong laws of large numbers, gambling systems/random walks |
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第5週 |
10/4, 10/6 |
10/4: Betting strategies, gambling policies, stopping time, moment generating functions
10/6: Cramer's theorem, measures |
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第6週 |
10/11, 10/13 |
10/11: n-dimensional Lebesgue measure, distribution functions, measurable functions, push forward, exponential distribution
10/13: integration, Fatou's lemma, monotone convergence theorem |
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第7週 |
10/18, 10/20 |
10/18: Dominated convergence theorem, uniform integrability, densities, change of variable
10/20: Product measure, Fubini's theorem |
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第8週 |
10/25, 10/27 |
10/25: Midterm
10/27: Integration by parts |
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第9週 |
11/1, 11/3 |
11/1: General random variables, independence, convolution, existence of independent random variables, moments
11/3: Moment generating functions for general random variables |
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第10週 |
11/8, 11/10 |
11/8: Weak law of large numbers, coupon collector's problem, Kolmogorov's maximal inequality
11/10: One series theorem, Etemadi's inequality |
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第11週 |
11/15, 11/17 |
11/15: Proof of the strong law of large numbers, Poisson process
11/17: Poisson point process, weak convergence |
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第12週 |
11/22, 11/24 |
11/22: Skorohod's theorem, mapping theorem, the Portmanteau theorem, Helly's selection theorem, tightness
11/24: Characteristic functions, Fourier inversion |
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第13週 |
11/29, 12/1 |
11/29: Continuity theorem, central limit theorem, Lindeberg CLT for independent random variables
12/1: Lindeberg CLT, Lyapunov CLT, central limit theorem for the number of cycles in a random permutation |
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第14週 |
12/6, 12/8 |
12/6: Weak convergence in R^d, Cramer-Wold device, Gaussian vectors, d-dimensional CLT
12/8: Accompanying laws, limit theorem for triangular array |
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第15週 |
12/13, 12/15 |
12/13: Infinitely divisible distributions, stable laws
12/15: The method of moments |
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第16週 |
12/20 |
12/20: Final exam |
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