課程資訊

Introduction to Probability Theory

110-2

MATH2502

201 49740

4.0

The main objective of the course is to provide students a mathematical treatment of the fundamental concepts and techniques of elementary probability theory. We will take a non-measure theoretical approach in this course so that it 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.

Content:
Axioms of probability, conditional probability, independence, discrete and continuous random variables, jointly distributed random variables, expectation, moment generating functions, limit theorems.

If time allows, we will also talk about random walks, Markov chains, Poisson processes, and possibly their applications.

Students are able to understand the basics of elementary probability theory.

First year calculus and basic mathematical analysis.

Office Hours

Below are some reference books which are of suitable level.

1. Sheldon M. Ross, Introduction to Probability Models, Academic Press.
2. Geoffrey Grimmett and Dominic Welsh, Probability: An Introduction, 2nd Edition
3. Charles Grinstead and Laurie Snell, Introduction to Probability. See https://math.dartmouth.edu/~prob/prob/prob.pdf
4. David F. Anderson, Timo Seppäläinen and Benedek Valkó. Introduction to Probability, Cambridge University Press, 2018.

Textbook: Sheldon M. Ross, A First Course in Probability, 10th Edition (Global Edition), Pearson.

We will NOT follow the textbook closely. In particular, the materials and notations can be different from the textbook.

(僅供參考)

 No. 項目 百分比 說明 1. Homework 20% 2. Midterm exam 40% Midterm will be on 3/31. 3. Final exam 40% The final exam is cumulative. It will be on 6/2.

 課程進度
 週次 日期 單元主題 第1週 2/15,2/17 2/15: Motivations, sample space, event space, axioms of probability, basic properties of a probability. 2/17: Basic properties of a probability, finite sample space, examples. 第2週 2/22,2/24 2/22: Union bound, inclusion-exclusion principle, limit of sets, continuity of probability, the first Borel-Cantelli lemma. 2/24: Conditional probability, multiplication rule. 第3週 3/01,3/03 3/1: Law of total probability, Bayes' theorem, independence of events. 3/3: Probabilistic method, the second Borel-Cantelli lemma. 第4週 3/08,3/10 3/8: Conditional independence, random variables, distribution functions, discrete random variables. 3/10: Bernoulli distribution, binomial distribution, Poisson distribution, geometric distribution, negative binomial distribution, continuous random variables. 第5週 3/15,3/17 3/15: Uniform distribution, exponential distribution, memoryless property, function of a random variable, inverse distribution function. 3/17: Generating random variables from a uniform distribution, expectation of a discrete random variable. 第6週 3/22,3/24 3/22: Expectation of a continuous random variable, infinite and nonexistent expectation, expectation of a function of a random variable, variance. 3/24: Basic properties of variance. 第7週 3/29,3/31 3/29: Mean squared error, mean absolute error, normal distribution, joint distribution. 3/31: Midterm. 第8週 4/05,4/07 4/5: Holiday. 4/7: Independent random variables, conditional distribution. 第9週 4/12,4/14 4/12: Functions of two or more random variables, sum, maximum and minimum of independent random variables, gamma distribution. 4/14: Transformation of a joint density, beta distribution. 第10週 4/19,4/21 4/19: Expectation of functions of two or more random variables, covariance, correlation, conditional expectation. 4/21: Properties of conditional expectation, conditional variance and prediction. 第11週 4/26,4/28 4/26: Branching processes, moments, moment generating function. 4/28: Moment generating function characterizes distribution, bivariate normal distribution. 第12週 5/03,5/05 5/3: Markov's inequality, Chebyshev's inequality, convergence of random variables, weak and strong law of large numbers. 5/5: Central limit theorem. 第13週 5/10,5/12 5/10: Proof of the central limit theorem, Markov chains, Chapman-Kolmogorov equations. 5/12: Recurrence and transience. 第14週 5/17,5/19 5/17: One and two dimensional random walks, stopping time, strong Markov property, positive recurrent, null recurrent. 5/19: Perron-Frobenius Theorem. 第15週 5/24,5/26 5/24: Proof of the Perron-Frobenius Theorem, limiting probabilities, periodicity, ergodicity, PageRank, gambler's ruin. 5/26: Time reversibility, detailed balance condition, Markov chain Monte Carlo (MCMC). 第16週 5/31,6/02 5/31: No class. 6/2: Final exam.