機率

Probability is a course to learn how to deal with the issues of uncertainty with broad applications in the areas of engineering, science, finance, etc. This course will guild you to view the world with probabilistic thinking and learn the skills to solve the engineering problems subject to the impacts of uncertainty, such as noise. This course will establish your mathematical skills for handling the issues of Big Data analytics in the future, and learn MATLAB or R programming skills.

微積分

無備註

課程時間為週二56節+週五第2節,週五第2節為助教時間與考試安排。請同學於每週二12:30提前至教室(工四館103)開始上課。

Quiz: 30% Mid Term:30% Final exam: 40%

• UNIT I: PROBABILITY MODELS AND DISCRETE RANDOM VARIABLES

  This unit covers the basic framework of probability theory: probabilistic models, conditional probabilities, independence, the Bayes’ rule, and counting methods. In addition, it introduces discrete random variables and the concept of the Probability Mass Function (PMF) used to describe the probability distribution of one or several random variables. Finally, it defines the concepts of expectation and variance, and their basic properties. Lecture 1: Probability Models and Axioms Lecture 2: Conditioning and Bayes' Rule Lecture 3: Independence Lecture 4: Counting Lecture 5: Discrete Random Variables; Probability Mass Functions; Lecture 6: Discrete Random Variable Examples; Joint PMFs Lecture 7: Multiple Discrete Random Variables

  講授：

  12

• UNIT II: GENERAL RANDOM VARIABLES

  This unit introduces continuous random variables and develops their properties in a manner that parallels the development for the discrete case. In addition, it covers a few more advanced topics such as the derivation of the distribution of a function of a random variable, covariance and correlation, and a more abstract view of the conditional expectation. Lecture 8: Continuous Random Variables Lecture 9: Multiple Continuous Random Variables Lecture 10: Continuous Bayes' Rule; Derived Distributions Lecture 11: Derived Distributions; Convolution; Covariance and Correlation Lecture 12: Iterated Expectations; Sum of a Random Number of Random Variables

  講授：

  12

• UNIT III: RANDOM PROCESSES

  This unit provides an introduction to some simple classes of discrete random processes. This includes the Bernoulli and Poisson processes that are used to model random arrivals and for which we characterize various associated random variables of interest and study several general properties. It also includes Markov chains, which describe dynamical systems that evolve probabilistically over a finite state space. We present the general structure of Markov models and study both their long-term and transient behavior. Lecture 13: Bernoulli Process Lecture 14: Poisson Process - I Lecture 15: Poisson Process - II Lecture 16: Markov Chains - I Lecture 17: Markov Chains - II Lecture 18: Markov Chains - III

  講授：

  12

• UNIT IV: LAWS OF LARGE NUMBERS AND INFERENCE

  In this section, we start with a discussion of limit theorems: the weak law of large numbers and the central limit theorem. We then introduce the subject of inference (estimation and hypothesis testing), from two alternative viewpoints: first, Bayesian inference, which relies on a prior distribution for unknown quantities and on the Bayes rule to incorporate new evidence; and, second, classical inference, in which no probabilistic assumptions are made on the unknown quantities and instead relies heavily on the laws of large numbers to provide statistical guarantees, e.g., in the form of confidence intervals. Lecture 19: Weak Law of Large Numbers Lecture 20: Central Limit Theorem Lecture 21: Bayesian Statistical Inference - I Lecture 22: Bayesian Statistical Inference - II Lecture 23 Classical Statistical Inference - I Lecture 24: Classical Inference - II Lecture 25: Classical Inference - III

  講授：

  12

  備註：\n

Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability (2nd edition), 2008 Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science (1st edition), Cambridge University Press (September 27, 2018) Joseph K. Blitzstein and Jessica Hwang, Introduction to Probability (Second Edition), Chapman and Hall/CRC (February 8, 2019)

地點

工程四館 804

時間

Friday, 13:00 to 14:00

聯絡方式

wang@nycu.edu.tw