課程資訊

Introduction to Probability Theory

101-2

MATH2501

201 31700

01

1.學士班二年級必修課。 2.內容含馬可夫鏈與泊松過程導論。

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1012IntroProb1

1. Basics
2. Conditional Probability: It includes independence, conditional probability,
and Bayes formula.
3. Distributions: It includes asymptotic approximation such as Poisson
approximation to Binomial, density and distribution functions, joint
distributions, marginal distributions, independence, and conditional
distributions
4. Expected Value: It includes moments, generating functions, expectation,
variance and covariance, correlation, and conditional expectation.
5. Limit Theorems: It includes laws of large numbers, the central limit theorem,
confidence intervals, and hypothesis testing.
6. 馬可夫鏈與泊松過程導論

The students should become familiar with basic probability and stochastic process, toward interest in random phenomena. Also, they can master the language and tool of probability to do quantitative reasoning in the field of economics, management science, and statistics. The objective of this course is to provide students having a good calculus background with a solid mathematical treatment of the fundamental concepts and techniques of probability theory. It is fundamentally important
for understanding the commonly observed random phenomena.

Prerequist: Calculus and one-semester linear algebra or matrix operation.

Office Hours

R. Durrett: The Essentials of Probability
R. Durrett: Essentials of Stochastic Processes (Chapters 1 and 3)
S. Ross: A First Course in Probability, the 5th or newer Edition (the newest is the 8th).
H. Tijms: Understanding probability : chance rules in everyday life (本校電子書, 例子與生活相關且深刻)

(僅供參考)

 No. 項目 百分比 說明 1. Homework 20% 2. Quizzes 20% 3. Midterm Exam 30% 4. Final Exam 30%

 課程進度
 週次 日期 單元主題 第1週 2/19, 2/21 Probability models and axioms (Sections 1.1-1.2) Conditional and Bayes' rule (Sections 1.3-1.4) 第2週 2/26 Independence (1.5) ，週四和平紀念日 第3週 3/05, 3/07 Independence and conditional independence (Section 1.5) , Counting (Section 1.6), Discrete random variables; probability mass functions 第4週 3/12, 3/14 Expectations, Joint PMFs of Multiple RVs, Conditioning, Independence (Sections 2.4 - 2.5); 週四第六堂習題課, Go over Problem 9(HW5 Q5), Problem 10, Problem 37, Problem 42, Problem 45, Problem 57 (Chapter 1) 第5週 3/19, 3/21 Multiple discrete random variables: expectations, conditioning, and independence (Sections 2.5 - 2.7) , Continuous RV, pdf, and cumulative distribution function (Sections 3.1-3.2). 第6週 3/26, 3/28 Review cumulative distribution function and teach Derived distributions (Sections 4.1), Normal RV (Section 3.3), Thursday: Go over joint PDFs of multiple RVs (Section 3.4) and Conditioning (Sections 3.5), Quiz 1 (Thursday, 2:15-3:10). 第7週 4/02 joint PDFs of multiple RVs (Section 3.4), 週四溫書假 第8週 4/09, 4/11 Conditioning, Continuous Bayes rule (Section 3.6), convolution; covariance and correlation (Section 4.2) Derived distributions; Iterated expectations (Sections 4.2-4.4); sum of a random number of random variables, 4/09 習題課 (14:20-15:10), 4/11 Quiz 2 (14:15-15:10) 第9週 4/16, 4/18 Derived distributions; Covariance, Iterated expectations (Sections 4.2-4.3); Review for midterm 第10週 4/23, 4/25 Midterm期中考 (4/23 範圍Chapter 1 to 4.1, 含covariance的定義; 題組一:定義、名詞、基本演算 (200 points) 共5題 題組二:應用 (75 points) 共2題 題組三: (120 points) 共3題) ; Convolution (Chapter 4.1), Covariance and Correlation (Chapter 4.2), Conditional Expectation and Variance Revisited. (Chapter 4.3) 第11週 4/30, 5/02 Conditional Expectation and Variance Revisited. (Chapter 4.3), Transforms (Section 4.4) 第12週 5/07, 5/09 Sum of a Random Number of Independent Random Variables (Section 4.5), Markov and Chebyschev Inequality (Section 5.1) 週四期中考特，該卷的總分數為180分, 期中考成績記分方式修正如下 1. 期中考成績不到180分者, 期中考成績將為"期中考成績及特考成績較高者" 2. 期中考成績超過180分者, 期中考成績將為"期中考成績加上15分" 週二將會很快的討論期中考考題 3. 部份特考考題以選擇題方式呈現, 但得分仍依所呈現的過程決定, 且有少數題目的正確選項為以上皆非 第13週 5/14, 5/16 Markov and Chebyschev Inequality, Weak law of large numbers, Convergence in Probability, and Central limit theorem (Sections 5.1-5.4), 第14週 5/21, 5/23 Review of Convergence in Probability, and Central limit theorem in Tuesday, Convergence with probability 1 (Borel Cantelli lemma) and Strong Law of Large Numbers 及習題課 第15週 5/28, 5/30 週四第五節: Quiz 3 (Chapters 4 and 5, 涵蓋中央極限定理的應用, Markov & Chebyschev inequalities, 變數變換 Jacobian, 大數法則及其證明) Bernoulli process (Section 6.1), Poisson process – I (Section 6.2) 第16週 6/04, 6/06 Tuesday: Review Quiz 3 and Poisson Process; Thursday: Finish up Poisson Process and Random Incidence Paradox. Review of Homework 第17週 6/11, 6/13 Tuesday: Wrap up and Review What we have learned after Midterm. Thursday: Quiz 4 (13:20-14:10), 習題課 (14:20-15:10) 第18週 6/18 Final Exam on 6/20: Cover Chapters 4, 5, and 6. 第六章的份量約為十分之一