課程大綱

課程資訊

課程名稱

機率方法
Probabilistic Methods in Engineering

開課學期

110-2

授課對象

工學院機械工程學系

授課教師

林以凡

課號

ME5057

課程識別碼

522EU6330

班次

學分

3.0

全/半年

半年

必/選修

選修

上課時間

星期二3,4,5(10:20~13:10)

上課地點

綜502

備註

本課程以英語授課。
總人數上限：55人

課程簡介影片

核心能力關聯

本課程尚未建立核心能力關連

課程大綱

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課程概述

ME 5057 is an introductory probability and random process course for engineering students. Our syllabus has a strong emphasis on the theoretical foundations. ME 5057 is different from the statistics courses you took in high school. For example, you probably know about how to calculate the “mean” and “standard deviation” of some data, but have you thought about how to use the mean and standard deviation to classify objects in an image? We will not go deep into image classification in this course, but we will teach you a set of basic concepts in probability theory which will eventually allow you to study these problems in the future.

課程目標

The objective of this course is that by the end of the semester, you will have
• a solid background in probability and random processes that can help you take advanced courses;
• an ability to formulate engineering problems using a probabilistic approach;
• an ability to analyze large-scale systems using statistical methods;
• an ability to identify the concept of random variables and properties of common types of random variables, and how to solve probabilistic problems;
• experience in using computers to solve probability problems.
Also, you will be able to
• use set-theoretic notation to describe events and compute probabilities;
• compute and interpret conditional probability, total probability, and describe Bayes' theorem;
• test for independence of events or of random variables;
• describe different types of discrete random variables and solve problems with important distributions such as Bernoulli, binomial, geometric, and Poisson distributions;
• identify continuous random variables and solve problems with important distributions such as uniform, normal, and exponential distributions;
• define what expectation and variance mean and be able to compute them;
• calculate moments of random variables and derive the distributions of functions of random variables;
• compute the covariance and correlation between jointly distributed variables;
• identify random process and what wide sense stationary process mean;
• compute power spectral density through LTI system.

課程要求

Linear Algebra, Differential Equations, and Signals and Systems [may be taken concurrently]
ME 5057 is perhaps the most mathematically challenging course in the ME curriculum. We will use all you have learned in calculus and linear algebra to do something you have never seen before. Therefore, we require all students to be fluent in calculus and linear algebra. If you are uncertain whether you are ready to take ME 5057, please talk to Dr. Lin.
We will try to minimize the dependency of the course “Signals and Systems” in the first half of the semester. However, we expect you to be prepared for the Fourier Transforms when we start to discuss about moment generating functions (MGF).

預期每週課後學習時數

Office Hours

另約時間備註： By email appointment with ME 5057 in the Subject line

指定閱讀

參考書目

D. P. Bertsekas and J. N. Tsitsiklis, Introduction to Probability, Athena Scientific, 2nd Ed, 2008.
The book by Bertsekas and Tsitsiklis is an excellent text on probability. It contains many good examples, and problems with solutions. It also provides many pictorial illustrations. Sometimes these pictures can help you understand the material better.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Fianl	30%	The final is closed-book, closed- note. No electronics, including calculators, cell phones, and smart watches are allowed. The final is cumulative and may include the combinations of true & false, multiple choices, and work-out problems. Some formulas and tables may be provided.
2.	Mid-terms	40%	(20% each). There are two mid-term exams in this course. Each mid-term is for 90 minutes. The mid terms are scheduled on – Mid-term 1: Oct 28 in class. – Mid-term 2: Dec 09 in class. Mid-terms are closed-book, closed-note. No electronics, including calculators, cell phones, and smart watches are allowed. Mid-terms will be used to access demonstration of the learning objectives and may include the combinations of true & false, multiple choices, and work-out problems. Some formulas and tables may be provided.
3.	Computer Assignment	10%	(5% each). There will be two computer assignments in this course. You are required to use MATLAB or Python to solve the questions. The assignments should be done individually.
4.	Homework	0%	Homework will not be graded, but the some questions in the homework will be in the quiz.
5.	Quiz	20%	(4% each). There will be a 20-minute quiz in class. In each quiz, you will solve several questions chosen from homework. Quizzes are closed-book, closed-note. No electronics, including calculators, cell phones, and smart watches are allowed. Some formulas and tables may be provided.

課程進度

週次	日期	單元主題
Week 1	2/15	Introduction, Set Theory, Probability Model, Conditional Probability, Bayes' Theorem
Week 2	2/22	Total Probability, Independence, Discrete RV, PMF, Expectation
Week 3	3/1	Variance, Bernoulli, Binomial, Geometric RV, Poisson, Continuous RV, PDF, CDF
Week 4	3/8	Expectation, Variance, Uniform, Exponential, Gaussian, Function of One Random Variable
Week 5	3/15	Midterm I
Week 6	3/22	Function of One RVs, Multiple RVs, Joint CDF, Joint PDF, IID, Joint Expectation
Week 7	3/29	Conditional RV, Conditional PMF of Two RVs Covariance, Correlation, Conditional RV
Week 8	4/5	No Class -- Holiday
Week 9	4/12	Conditional Expectation Examples, Sum of Two RVs, Moment Generating Function, Characteristic Function, Joint Characteristic Function
Week 10	4/19	Two Functions of Two RVs, Random Vectors, Joint Gaussian, Estimation, MLE
Week 11	4/26	Midterm II
Week 12	5/3	MAP, MMSE, Limit Theorems, WLLN
Week 13	5/10	WLLN, CLT, Random Process, Mean Function, WSS, Auto Correlation function, Stationary Process
Week 14	5/17	Power Spectrum, Random Process through LTI System, Cross Correlation through LTI System
Week 15	5/24	Review
Week 16	5/31	Final Exam