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Probability
Spring 2008


News

  • Final Grade: If you disagree with your grade or any of my corrections, please come to my office immediately, latest on Wednesday, 25 June, before 3 PM! Thereafter I will not be able to correct any mistake anymore! Thanks!
  • Final Exam: The final exam will take place on
    • Tuesday, 17 June, 10:10-13:00 (Note that this is one hour longer than usual!)
    Regulations:
    • open book: any book is allowed
    • not allowed are: any telecommunication devices like mobile phones, any laptop with wireless capabilities, any "friend", or any other help from outside...
    • covered material: everything covered in class
  • Class Evaluation: The class evaluation will be online between June 3 to June 20. I would very much appreciate your feedback, so please spend a couple of minutes to fill out the online form! Thanks!
  • Exercise 8 needs to be handed in only on 22 April and not as wronly stated on the Exercise on 15 April!
  • Mid-Term Exam: The mid-term exam will take place on
    • Tuesday, 15 April, 10:10-13:00 (Note that this is one hour longer than usual!)
    Regulations:
    • open-book: any book, notes, and exercises are allowed
    • not allowed are: any telecommunication devices like mobile phones, any laptop with wireless capabilities, any "friend", or any other help from outside...
    • covered material: everything covered in class from Chapter 1 to 3

Course Objectives

This course is an introduction to probability. Its goal is to give the students a profound knowledge about probability theory and some of its important applications. We will cover the following subjects:

  • Sample Space and Probability
    • probabilistic models and conditional probability
    • total probability and Bayes' Rule
    • independence
  • Discrete Random Variables (RV)
    • probability mass function (PMF)
    • transforming RVs
    • expectations
    • joint PMF
    • conditioning and independence
  • General Random Variables
    • probability density function (PDF) and cumulative distribution function (CDF)
    • joint PDF
    • Gaussian RVs
    • conditioning
    • transforming RV
    • sum of random number of independent RVs
    • least squares estimation
  • Stochastic Processes
    • Bernoulli process
    • Poisson process
    • Markov chains
  • Limit Theorems
    • Markov and Chebyshev inequalities
    • weak and strong law of large numbers
    • convergence in probability
    • central limit theorem

For more detail see the time table below.

We believe that this course is essential for any engineer and we very much hope that a student who finishes the course will feel comfortable with the theory and can apply it.

Prerequisites

The following lectures/topics are recommended:

  • basic math from high-school

Instructor

Prof. Stefan M. Moser
Engineering Building IV, Office 727
phone: 03-571 21 21 ext. 54548
e-mail:

Teaching Assistant

In case you would like to discuss some questions in Chinese, you may contact the TA of this class:

  • Liu Jen-Yang
    Email:
    Room: Engineering Building IV, Room 811 (ED811)
    Phone: 03-571 21 21 ext. 54571
    Office hours: Wednesday, 13:30-15:30, or according to prior appointments

Time and Place

The course is scheduled for 3 hours per week:

  • Tuesday, 10:10-12:00, Engineering Building IV, Room 111 (ED111)
  • Thursday, 9:00-9:50, Engineering Building IV, Room 111 (ED111)

The course starts on Tuesday, 19 February, and finishes on Thursday, 19 June.

Office Hours

NCTU requests that every teacher offers two hours per week where students may come to ask questions:

  • Tuesday, 15:30-17:30, Engineering Building IV, Office 727

However, we would like to encourage you to show up in the teacher's or teaching assistant's office at any time in case you have questions about the class or related subjects. Moreover, we are always available during and after classes.

Textbook

Dimitri P. Bertsekas, John N. Tsitsiklis: "Introduction to Probability," Athena Scientific, Massachusetts, 2002.

For certain topics there will be additional handouts during classes.

Exercises

Every week, an exercise will be distributed in class and also made available online for download. This exercise will consist of several problems that need to be solved at home and handed in during the class of the following week. A model solution will be distributed in class and made available online afterwards.

We believe the exercises to be extremely important and crucial to the understanding of the course. They also serve as a preparation for the mid-term and final exams and we therefore highly recommend to solve them. To pass the course you need to hand in at least 10 exercises.

Exams

There will be one mid-term and one final exam. Both exams are going to last three hours and be open-book. Details about the covered material will be published in due time.

Grading

The grade will be an average of

  • the homework (15%),
  • the midterm exam (35%), and
  • the final exam (50%).

The grade of the homework will not be based on the correctness of the answers, but rather on the effort the student shows in trying to solve them. This course is worth 3 credits.

Special Remarks

The lecture will be held in English.

Time Table

W Date Topic Handouts Exercise (due on) Solutions Comments
1 19 Feb. Introduction, set theory, probabilistic models, conditional probability Syllabus (Version 2) Exercise 1 (26 Feb.)    
  21 Feb. Conditional probability     -----  
2 26 Feb. Total probability theorem, Bayes' rule, independence   Exercise 2 (4 Mar.)    
  28 Feb. No lecture (Holiday)     -----  
3 4 Mar. Counting, discrete RV: PMF   Exercise 3 (11 Mar.) Solutions 1  
  6 Mar. Functions of RVs, expectations     Solutions 2  
4 11 Mar. Joint PMFs, conditioned PMFs   Exercise 4 (18 Mar.)    
  13 Mar. Independent RVs, continuous RVs: PDF     Solutions 3  
5 18 Mar. Continuous RVs: PDF, CDF, Gaussian RV   Exercise 5 (25 Mar.)    
  20 Mar. Continuous RVs: conditioning, multiple RVs     Solutions 4  
6 25 Mar. Continuous RVs: multiple RVs   Exercise 6 (1 Apr.)    
  27 Mar. Derived distributions     Solutions 5  
7 1 Apr. Derived distributions, transform (moment generating function)   Exercise 7 (8 Apr.)    
  3 Apr. No lecture (Holiday)     -----  
8 8 Apr. Transform (moment generating function), sum of independent RVs, conditional variance   Exercise 8 (22 Apr.) Solutions 6  
  10 Apr. Sum of random number of independent RVs     Solutions 7  
9 15 Apr. Mid-Term Exam   -----   ATTENTION: This is a 3 hours exam: 10:10-13:00
  17 Apr. Discussion mid-term exam     -----  
10 22 Apr. Covariance and correlation, MMSEE Handout about Gaussian RVs Exercise 9 (29 Apr.)    
  24 Apr. MMSEE, LMMSEE, covariance matrices     Solutions 8  
11 29 Apr. Covariance matrices, multivariate Gaussian distribution   Exercise 10 (6 May)    
  1 May Multivariate Gaussian distribution     Solutions 9  
12 6 May Multivariate Gaussian distribution, stochastic processes: Bernoulli process   Exercise 11 (13 May)    
  8 May Poisson process     Solutions 10  
13 13 May Poisson process   Exercise 12 (20 May)    
  15 May Poisson process     Solutions 11  
14 20 May Markov process   Exercise 13 (27 May)    
  22 May Markov process: steady state and stationarity     Solutions 12  
15 27 May Markov process: steady state and stationarity, long-term frequency interpretation   Exercise 14 (3 Jun.)    
  29 May Markov process: short-term transient behavior     Solutions 13  
16 3 Jun. Limit Theorems: Markov Inequality, Chebyshev Inequality, Chernoff bound, Jensen's bound, weak law of large numbers , convergence in probability   Exercise 15 (10 Jun.)   The class evaluation is online until June 20. Please take 5 minutes to fill it out!
  5 Jun. Convergence in probability, strong law of large numbers, almost sure convergence, Borel-Cantelli lemma     Solutions 14  
17 10 Jun. Borel-Cantelli lemma, central limit theorem, ergodicity   Exercise 16 (12 Jun.)    
  12 Jun. Ergodicity     Solutions 15,
Solutions 16
 
18 17 Jun. Final Exam   -----   ATTENTION: This is a 3 hours exam: 10:10-13:00
  19 Jun. Discussion of final exam     -----  

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Senior Researcher & Lecturer, ETH Zurich, Switzerland
_|_     -- --|-    _     /  /   Adj. Professor, National Chiao Tung University (NCTU), Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser-isi.ethz.ch/


Last modified: Tue Mar 3 16:47:20 UTC+8 2009