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課程資訊

課程名稱

機率導論
Introduction to Probability Theory

開課學期

109-2

授課對象

理學院數學系

授課教師

王振男

課號

MATH2502

課程識別碼

201 49740

班次

學分

4.0

全/半年

半年

必/選修

必帶

上課時間

星期二6,7(13:20~15:10)星期四6,7(13:20~15:10)

上課地點

新203新303

備註

限本系所學生(含輔系、雙修生)
總人數上限：70人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1092MATH2502_pr

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課程概述

Probability theory, originated in the consideration of games of chance, is the language to study commonly observed random phenomena. It has become a fundamental tool used by nearly all scientists, including engineers, econometricians, industrialists, jurists, medical practitioners, physicists, statisticians, etc. This course is intended as an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and sciences (including social sciences). We will take a non-measure theoretical approach in this course that is suitable for students who possess only the knowledge of elementary calculus. The goal is to present not only the mathematics of probability theory but also numerous applications of the subject.

Contents
Combinatorial analysis, Axioms of probability, Conditional probability, Independence, Discrete and continuous random variables, Jointly distributed random variables, Expectation, Moment generating functions, Limit theorems, Stochastic processes, and Markov chains.

課程目標

Students are able to "think probabilistically" and apply probability theory to diverse disciplines.

課程要求

First year calculus and basic Mathematical analysis.

預期每週課後學習時數

Office Hours

每週一 10:00~12:00

指定閱讀

Robert Ash, Basic Probability Theory. The book can be downloaded from https://faculty.math.illinois.edu/~r-ash/BPT.html.

參考書目

1. Sheldon M. Ross, A First Course in Probability, 10th Edition (Global Edition), Pearson.
2. Sheldon M. Ross, Introduction to Probability Models, Academic Press.
3. Patrick Billingsley, Probability and Measure, Wiley series in probability and mathematical statistics.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	作業	40%	週作業 (請用文書編輯軟體轉成pdf檔案繳交）
2.	期中考	30%
3.	期末考	30%

課程進度

週次	日期	單元主題
第1週	2/23,2/25	Axioms of probability, sample space, sigma-algebra, events, probability measure, probability space, Borel algebra, discrete probability, probability mass function, counting, ordered sample with replacement, ordered sample without replacement, unordered sample without replacement, unordered sample with replacement, binomial and multinomial coefficients, independence.
第2週	3/02,3/04	Independence, Dynkin's pi-lambda lemma, Extension of independence to the smallest sigma-algebra, Conditional probability, Total probability, Bayes' theorem, Applications of Bayes' theorem.
第3週	3/09,3/11	Applications of Bayes' theorem, Genetic counseling, Updating information sequentially, Random variables, Probability distribution function, Probability mass function, Expected value, Variance, Bernoulli distribution, Binomial distribution, Geometric distribution, Poisson distribution, Negative binomial distribution.
第4週	3/16,3/18	Hypergeometric random variable, Estimate of a population, Maximum likelihood estimate, Probability distribution functions, Construction of a random variable with prescribed distribution function, Absolute continuity, Probability density functions, Uniform distribution, Normal distribution, Exponential distribution, Poisson process, Memoryless property, Gamma distribution, Beta distribution, Cauchy distribution.
第5週	3/23,3/25	Functions of one random variable, Jointly cumulative probability distribution functions, Joint probability mass functions, Marginal probability functions, Joint probability density functions, Marginal probability density functions, Probability density function of independent random variables, Functions of several random variables, Expected value of sum of random variables, Variance of sum of independent random variables.
第6週	3/30,4/01	Sum of independent random variables, Sum of uniform distributions, Sum of Gamma distributions, Sum of exponential distributions, Chi-squared distributions, Sum of normal distributions, Sum of Poisson distributions, Sum of binomial distributions.
第7週	4/06,4/08	Joint probability density of a function of random variables, Order statistics, Simulation of two independent standard normals.
第8週	4/13,4/15	4/13 期中考。Covariance, Correlation coefficient, Uncorrelated, Positively and negatively correlated, Conditional probability mass functions, Conditional distribution, Conditional expectation, Conditional density.
第9週	4/20,4/22	Conditional expectation, Theorem of total probability, Theorem of total expectation, General concept of conditional expectation, Sub sigma-field, Mean square error.
第10週	4/27,4/29	The minimizer of the mean square error, Conditional variance, Moment generating functions (MGF), MGF uniquely determines the probability distribution, Characteristic functions, Fourier transform of the probability measure.
第11週	5/04,5/06	Characteristic function and moments, Convergence almost surely, Convergence in probability, Events occur infinitely often, Borel-Cantelli's lemmas, Cauchy in probability, Markov's inequality, Chebyshev's inequality, Weak law of large number.
第12週	5/11,5/13	DeMoivre-Laplace limit theorem, Approximate binomial by normal distribution, Central limit theorem, Weak convergence, Convergence in distribution or in law, Skorohod's theorem, Equivalence definition of weak convergence, Convergence in probability implies convergence in distribution, Central Limit Theorem, Proof of CLT.
第13週	5/18,5/20	Lindeberg's theorem and its proof, Strong law of large number, Proof of SLLN, One-sided Chebyshev's inequality, Chernoff's bounds.
第14週	5/25,5/27	自主學習週
第15週	6/01,6/03	Markov chains, Transition probabilities, Stationary transition probabilities, Transition matrix, Stochastic matrix, Higher order transitions, Chapman-Kolmogorov identity, Persistent and transient states, Polya's theorem (classification of states for symmetric random walks in any dimension).
第16週	6/08,6/10	Irreducible, period, aperiodic, stationary distribution, convergence to the stationary distribution, positive persistence, null persistence, simulations, Markov Chain Monte-Carlo method, detailed balance equation, Metropolis-Hastings algorithm.
第17週	6/15,6/17	Final exam on 6/17.