ECS315: Probability and Random Processes

Everything we do, everything that happens around us, obeys the laws of probability. We can no more escape them than we can escape gravity... "Probability," a philosopher (Bishop Butler) once said, "is the very guide of life." We are all gamblers who go through life making countless bets on the outcome of countless actions.

Every field of science is concerned with estimating probability. A physicist calculates the probable path of a particle. A geneticist calculates the chances that a couple will have blue-eyed children. Insurance companies, businessmen, stockbrokers, sociologists, politicians, military experts - all have to be skilled in calculating the probability of the events with which they are concerned.

[Gardner, 1986]

Synopsis

Probability theory is the branch of mathematics that tells us how to estimate degrees of probability. If an event is certain to happen, it is given a probability of 1. If it is certain not to happen, it has a probability of 0.

This course introduces the principles of probability and random processes to undergraduate students in electronics and communication. The topics to be covered include random experiments, events, probability, discrete and continuous random variables, probability density function, cumulative distribution function, functions of random variables, expectations, law of large numbers, central limit theorem, introduction to random processes, Gaussian random process, autocorrelation and power spectral density.

Announcements

• Note: In case the course website is unavailable, files for the current year are backed up on Google Drive.
• The (second) online self-evaluation form is available now.
• Information regarding the final exam
• Check this course website regularly for breaking news about the final.
• Date: December 11, 2018
  
• TIME: 9:00-12:00
  
• ROOMs: BKD 3207, 3214, and 3215
• Basic calculators, e.g. FX-991MS, are permitted.
• 8 pages + one cover page + one page for the Phi/Q tables
• To save time, read the cover page to be posted here before going into the exam room.
  
• 16+1 = 17 questions. (9+4+2+3+3+6+4+21+12+3+8+3+4+8+8+1+1)
• Closed book. Closed notes.
• One A4 sheet allowed.
  • Use the returned study sheet from the midterm exam.
  • If you haven't picked up your A4 sheet yet, please get it from P'Ann.
    
  • Add more formulas or information on the remaining side (or in the remaining space(s)).
    
    • Basically, you have one whole page for the final exam.
    • Q: Can I make a brand new study sheet?
      A: No.
      However, if you think the old one is in a bad shape (falling apart), you can use a photocopy of your own. (Reduction in size is not allowed though)
    • Q: What can I do if I lose my study sheet?
      A: Dr.Prapun may be able to print out a scanned copy for you. (The1 writing won’t be as crisp as the original one but it’s better than nothing.) Otherwise, see below.
    • Q: I think the midterm study sheet (and the thing I wrote on it) is cursed (making me get poor score); I don’t want to see or use anything on it anymore. What can I do?
      A: You may bring a brand new sheet for the final exam. However, you can only use one side of it.
  • Basically, same rules as the midterm:
    • Must be hand-written in your own handwriting.
    • No small pieces of paper notes glued/attached on top of it.
    • Indicate your name and ID on the upper right corner of the sheet (in portrait orientation).
    • Do not modify (,e.g., add/underline/highlight) content on the sheet inside the exam room.
    • Submit your A4 sheet with your exam.
  • Violating the above instructions may cost you upto 10 pt.
• Cover all the materials that we discussed in class and practice in the exercises and HWs.
• Material Distribution (score-wise): 32 (Ch 7-8) + 20.5 (Ch 9) + 33.5 (Ch 10) + 13 (Ch 11)
• For your studying pleasure....
• All post-midterm annotated notes combined in one pdf file.
• All post-midterm HWs and their solutions
  • Checked HWs can be picked up in front of the EC office on the 6th floor.
• All post-midterm exercises and their solutions
• Graded exercises are posted on the SIIT Lecture Note System
• If you have valid reason for missing class on the day that we have exercise, please indicate the date, exercise number, and the reason in the (second) online self-evaluation form. Make sure that you also submit/email supporting document/evidence to Dr.Prapun (if you haven't done so).
• All post-midterm slides
• See some old exams.
• Information regarding the midterm exam [Posted @ 9AM on Sep 24; Last updated @ 10AM on Sep 24]
  • Check this course website regularly for breaking news about the midterm.
  • Date: October 4, 2018 (Thursday)
    
  • TIME: 9:00-11:00
    
  • ROOMs: BKD 2501-2, 2401
    
  • Information about the midterm exam:
  • 8 pages (including the cover page)
  • To save time, read the cover page (to be posted) here before going into the exam room.
  • 14+1 = 15 questions. (6+1+5+4+7+4+4+1+1+8+8+3+5+9+1 = 67 pt)
  • Cover all the materials that we discussed in class and practice in the HWs.
  • Material Distribution (score-wise): 8 (CH1-2) + 19 (CH3-4) + 15 (CH5) + 25 (CH6)
  • Closed book. Closed notes.
  • (1 pt) One A4 page allowed.
    • Must be hand-written in your own handwriting.
    • No small pieces of paper notes glued/attached on top of it.
    • Indicate your name and ID on the upper right corner of the sheet (in portrait orientation).
    • Do not modify (,e.g., add/underline/highlight) content on the sheet inside the exam room.
    • Make sure that another side is blank. This will be used for the final exam.
    • Submit your A4 sheet with your exam. (You will get it back before the final exam.)
    • Q: I don't need any formulas. What should I do?
      A: Bring in and submit a blank sheet of paper with your name and ID. Note that you can still only use one side on the final exam.
      
    • Violating the above instructions will cost you 10 pt.
  • Basic SIIT-approved calculators, e.g,. FX-991MS, are permitted, but borrowing is not allowed.
  • For your studying pleasure....
  • All pre-midterm annotated notes combined in one pdf file.
  • All pre-midterm HWs and their solutions
  • All pre-midterm exercises and their solutions
  • Graded exercises are posted on the SIIT Lecture Note System
  • All pre-midterm slides
  • For those who have already obtained the textbook [Y&G], this table gives the corresponding positions (if any) in the textbook that correspond to the material discussed in class.
• Note that we also share the tutorial/make-up session with ECS332. See Google calendar below.
• This site can be accessed via ecs315.prapun.com.

• Welcome to ECS315! Feel free to look around this site.

General Information

• Instructor: Asst. Prof. Dr.Prapun Suksompong (prapun@siit.tu.ac.th)
  • Office: BKD, 6th floor of Sirindhralai building
• Office Hours: See Google calendar below.
• Lectures: See Google calendar below.
• Course Syllabus [Posted @ 10AM on Aug 9]
• Textbook: [Y&G] R. D. Yates and D. J. Goodman, Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 2nd ed., Wiley, 2004.
  • Call No. QA273 Y384 2005. ISBN: 978-0-471-27214-4
  • Student Companion Site
• References
• Draft of the Lecture notes [Posted @ 12PM on Aug 9]
• Caution: The lecture notes will still be updated throughout the semester. Therefore, do not rely on this version for use in class. The arrangement/inclusion/exclusion of topics may be different from the ones posted below and the versions available at the copy center.
• Probability and probabilistic reasoning for electrical engineering / Terrence L. Fine. Call No. QA273 F477 2006
• Probability and random processes for electrical engineering / Alberto Leon-Garcia. Call No. TK153 L425 1994
• Free textbook: Introduction to Probability by Charles M. Grinstead and J. Laurie Snell
• Henk Tijms. Understanding Probability: Chance Rules in Everyday Life. Cambridge University Press, 3rd edition, 2012. Call No. QA273 T48 2012
• Probability, random variables, and stochastic processes / Athanasios Papoulis, S. Unnikrishna Pillai. Call No. QA273 P2 2002
  • Probability, random variables, and stochastic processes / Athanasios Papoulis. Call No. QA273 P2 1991
• A first course in probability / Sheldon Ross. 6E Call No. QA273 R83 2002
  • A first course in probability / Sheldon Ross. Call No. QA273 R83 1976
  • A first course in probability / Sheldon Ross. 9E
• Probability models, introduction to / Sheldon M. Ross. 10E Call No. QA273 R84 2010
• Probability models, introduction to / Sheldon M. Ross. 6E Call No. QA273 R84 1997
• Probability models, introduction to / Sheldon M. Ross. 11E

Handouts and Course Material

• Slides: Course Introduction [Posted @ 11AM on Aug 10]
  • Full version [Posted @ 5PM on Aug 14]
• Part I: Introduction, Set Theory, Classical Probability theory, and Combinatorics [Posted @ 11AM on Aug 9]
  • Chapter 1: Probability and You
  • Annotated version [Posted @ 5PM on Aug 16; Updated @ 2PM on Aug 21]
  • Slides [Posted @ 5PM on Aug 16]
  • Excercise 1 Solution [Posted @ 5:30PM on Aug 30]
  • Chapter 2: Set Theory
  • Annotated version [Posted @ 5PM on Aug 16; Updated @ 2PM on Aug 21 and @9PM on Aug 23]
  • Slides [Posted @9PM on Aug 23]
  • Excercise 2 Solution [Posted @ 11AM on Sep 2]
  • Chapter 3: Classical Probability
  • Annotated version [Posted @ 5PM on Aug 16; Updated @9PM on Aug 23]
  • Slides [Posted @9PM on Aug 23; Updated @ 9PM on Aug 28]
  • Chapter 4: Enumeration / Combinatorics / Counting
  • Annotated version [Posted @ 5PM on Aug 16; Updated @9PM on Aug 23 and @ 5:30PM on Aug 30]
  • Slides [Posted @ 5PM on Aug 16; Updated @9PM on Aug 23 and @ 5:30PM on Aug 30]
  • Excercise 5 Solution [Posted @ 11AM on Sep 2]
• Part II: Kolmogorov's Formal Probability Theory and Event-Based Probability Theory [Posted @ 9AM on Aug 27]
• Chapter 5: Probability Foundations 
• Annotated notes [Posted @ 9PM on Aug 28; Updated @ 5:30PM on Aug 30]
• Slides [Posted @ 9PM on Aug 28; Updated @ 5:30PM on Aug 30]
• Excercise 3 Solution [Posted @ 11AM on Sep 2]
• Excercise 4 Solution [Posted @ 11AM on Sep 2]
• Section 6.1: Conditional Probability
  • Annotated notes [Posted @ 5PM on Sep 4; Updated @ 10PM on Sep 6, @ 3:30PM on Sep 11, @9:30PM on Sep 13, and @ 5:30PM on Sep 18]
  • Slides [Posted @ 5PM on Sep 4; Updated @ 9:30PM on Sep 13 and @ 5:30PM on Sep 18]
  • Excercise 6 Solution [Posted @ 12PM on Sep 9]
  • Excercise 7 Solution [Posted @ 12:30AM on Sep 16]
  • The Theory That Would Not Die, How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy" (Yale University Press, 2011)
  • A book review (The New York Times)
  • An article (Scientific American)
  • An interview (BBC Radio Wales' Sunday Supplement)
    
• Section 6.2: Event-based Independence
• Annotated notes [Posted @ 5:30PM on Sep 18; Updated @ 5PM on Sep 20]
• Slides [Posted @ 5PM on Sep 27]
• Excercise 8 Solution [Posted @ 5PM on Sep 27]
• Section 6.3: Bernoulli Trials
• Annotated notes [Posted @ 5PM on Sep 20; Updated @ 4PM on Sep 25 and @ 5PM on Sep 27]
• Slides [Posted @ 4PM on Sep 25]
• Excercise 9 Solution [Posted @ 8PM on Sep 27]
• Part III: Discrete Random Variables
• Chapter 7: Random Variables [Posted @ 9AM on Sep 18]
• Annotated notes [Posted @ 5:30PM on Oct 9; Updated 5PM on Oct 11]
• Slides [Posted @ 5:30PM on Oct 9]
• Excercise 10 Solution [Posted @ 9PM on Oct 22]
• Sections 8.1-8.2: Discrete RV: PMF and CDF [Posted @ 9AM on Sep 18]
• Annotated notes [Posted @ 5PM on Oct 11; Updated @ 5PM on Oct 18]
• Slides [Posted @ 9:00PM on Oct 16]
• Excercise 11 Solution [Posted @ 9PM on Oct 22]
• Excercise 12 Solution [Posted @ 9PM on Oct 22]
• Section 8.3-8.4: Families of Discrete Random Variables [Posted @ 5PM on Oct 14]
• Annotated notes [Posted @ 5PM on Oct 18; Updated @ 10PM on Oct 25, @ 5:30PM on Oct 30, and @ 9PM on Nov 1]
• Slides [Posted @ 9PM on Nov 1]
• Excercise 13 Solution [Posted @ 3PM on Nov 13]
• Excercise 14 Solution [Posted @ 3PM on Nov 13]
• Chapter 9: Expectation and Variance [Posted @ 5PM on Oct 14]
• Annotated notes [Posted @ 9PM on Nov 1; Updated @ 4:30PM on Nov 6 and @ 4PM on Nov 14]
• Excercise 15 Solution [Posted @ 4:30PM on Nov 5]
• Excercise 16 Solution [Posted @ 3PM on Nov 13]
• Slides [Posted @ 4:30PM on Nov 6]
• References: [Y&G] Chapter 2
  
• Part IV: Continuous Random Variables
• Chapter 10 [Posted @ 10AM on Nov 5]
• Annotated notes for Sections 10.1-10.3 [Posted @ 4PM on Nov 14; Updated @ 11PM on Nov 19 and @ 9AM on Nov 22]
• Annotated notes for Sections 10.4 [Posted @ 9AM on Nov 22; Updated @ 11AM on Nov 25]
• Slides [Posted @ 9AM on Nov 20; Updated @ 9AM on Nov 22]
• Phi/Q tables [Posted @ 9AM on Nov 19]
• Some of the examples require
• the use of integration by parts which is reviewed in Appendix A3
• the application of l'Hôpital's rule
• Excercise 17 Solution [Posted @ 9AM on Nov 29]
• Excercise 18 Solution [Posted @ 9AM on Nov 29]
• Excercise 19 Solution [Posted @ 9AM on Nov 29]
• References
• From Discrete to Continuous Random Variables: [Y&G] Sections 3.0 to 3.1
  
• PDF and CDF: [Y&G] Sections 3.1 to 3.2
  
• Expectation and Variance: [Y&G] Section 3.3
  
• Families of Continuous Random Variables: [Y&G] Sections 3.4 to 3.5
  
• Part V: Multiple Random Variables
• Chapter 11 [Posted @ 11PM on Nov 19]
• Annotated notes [Posted @ 11:30PM on Dec 1]
• Excercise 20 Solution [Posted @ 11:30PM on Dec 1]
• References:
• A Pair of Random Variables: [Y&G] Sections 4.1 to 4.3 and Section 4.10
• Extending the Definitions to Multiple RVs: [Y&G] Sections 5.1 to 5.4
• Function of Discrete Random Variables: [Y&G] Section 4.6 (Theorem 4.9)
• Expectation of function of discrete random variables: [Y&G] Sections 4.7 and 6.1
• Linear Dependence: [Y&G] Section 4.7
• Part VI: Limiting Theorems
  • Chapter 12 (LLN and CLT) [Posted @ 9AM on Nov 26]
• Part VII: Additional Topics
• Chapter 13 Three Types of Random Variables [Posted @ 9AM on Nov 26]
• Appendix
• Appendix A3 [Posted @ 10AM on Nov 15]
• Video: Tabular Method for Integration by Parts
• Video: Tabular Integration being referred to as the Tic-Tac-Toe method in the movie Stand And Deliver

Problem Set

1. HW 1 (Due: Aug 28)
   • Annotated version [Posted @ 2PM on Aug 21]
   • Solution [Posted @ 9AM on Sep 3]
2. HW2 (Due: Sep 4)
   • Solution [Posted @5PM on Sep 18]
3. HW3 (Due: Sep 11)
   • Solution [Posted @5PM on Sep 18]
4. HW4 (Due: Sep 18)
   • Annotated version [Posted @ 10PM on Sep 13]
   • Solution [Posted @5PM on Sep 18]
5. HW5 (Due: Sep 25)
   • Solution [Posted @8PM on Sep 27]
6. HW6 (Not Due)
   • Solution [Posted @5PM on Sep 18]
7. Self-Evaluation Form (1) (Due: Oct 4)
8. HW7 (Due: Oct 25)
   • Annotated version [Posted @ 4:30PM on Oct 18]
   • Solution [Posted @ 10PM on Oct 27]
9. HW8 (Due: Nov 6)
   • Solution [Posted @ 9AM on Nov 15]
10. HW9 (Due: Nov 13)
    • Solution [Posted @ 9AM on Nov 15]
11. HW10 (Due: Nov 22)
    • Solution [Posted @ 1:30PM on Nov 25]
12. HW11 (Due: Not Due)
    • Solution [Posted @ 9AM on Nov 22]
13. HW12 (Due: Not Due)
    • Solution [Posted @ 11PM on Nov 25; Updated @ 10AM on Nov 29]
    • P_XY_jointfromMarginal_indp.m
    • P_XY_marginal_2.m
    • P_XY_EVarCov.m
14. Self-Evaluation Form (2) (Due: Dec 11)

Calendar

Reading Assignment

1. Section 1.2 in the lecture notes
2. Section 2.5 in the lecture notes

Old Exams

• 2017 Midterm Exam
• 2016 Midterm Exam
• Annotated version from 2017
• The solutions of the in-class exercises in 2017 also contain the solutions for many of these problems.
• 2013 Midterm Exam
  • Annotated version from 2017
  • Annotated version from 2015
  • Annotated version from 2014
• 2010 Midterm Exam
  • Annotated version from 2015
  • Annotated version from 2014
  • Solution
• 2013 Final Exam [Link fixed @ 11PM on Dec 9]
  • You may skip the following parts because they are not covered this year:
  • 3b, 3cii-iii, 3d
  • 6, 7, 8
  • Annotated version from 2015
  • Annotated version from 2014
• 2011 Final Exam
  • We covered quite a different range of topics there because the curriculum was different. However, you can find some relevent problems in there as well.
  • Many of the relevant problems have already been included as exercises in the current lecture notes.
• 2010 Final Exam
  • We covered quite a different range of topics there because the curriculum was different. However, you can find some relevent problems in there as well.

More References

• Older version of the textbook: Probability and stochastic processes : a friendly introduction for electrical and computer engineers / Roy D. Yates, David J. Goodman. Call No. QA273 Y384 1999
• 10-page probability cheatsheet compiled from Harvard's Introduction to Probability course
• Allen B. Downey, Think Stats: Exploratory Data Analysis, O'Reilly Media, 2014 (free book)
• Allen B. Downey, Think Bayes: Bayesian Statistics in Python, O'Reilly Media, 2013 (free book)
• Encyclopædia Britannica Online: Probability Theory
• Random signals for engineers using MATLAB and Mathcad / Richard C. Jaffe. Call No. TK5102.9 J34 2000
• Davenport, W.B., Probability and Random Processes, McGraw-Hill, New York, 1970. (Excellent introductory text.)
• Feller, W., An Introduction to Probability Theory and its Applications, Vols. 1, 2, John Wiley, New York, 1950. (Definitive work on probability—requires mature mathematical knowledge.)
  • Call No. QA273 F37 1966
• Peter Olofsson, Probabilities The Little Numbers That Rule Our Lives, Wiley, 2006
  
• Stochastic processes / Sheldon M. Ross. Call No. QA274 R65 1996
• Stochastic processes / Emanuel Parzen. Call No. QA273 P278 1962
• MATLAB Primer, 8th edition T. A. Davis. CRC Press, 2010.
  • Seventh Edition by T. A. Davis and K. Sigmon: Call No. QA297 D38 2005
  • Third Edition by K. Sigmon (Free)
  • Second Edition by K. Sigmon (Free)

Misc. Links

• Video: Probability 101
• More information about theMonty Hall Problem
  • Video: The Monty Hall Problem Explained
• Video: The Monty Hall Problem
• Video: Monty Hall Problem: Numb3rs and 21
• Paper: Monty Hall, Monty Fall, Monty Crawl
• Articles: How Random is the iPod Shuffle? [HowStuffWorks]; Is iTunes’ Shuffle Mode Truly Random?[About.com]; iTunes: Just how random is random?[CNET.com.au, 2007]; My IPod for a Random Playlist [wired.com, 2005];
• Video: It *could* just be coincidence
• MV: Bill Nye the Science Guy - "50 Fifty"
• Video: Chevalier de Mere's Scandal of Arithmetic
• Free educational software: Orstat2000
  • Originally developed to promote probability and operations research in the senior forms of Dutch high schools (and early college).
  • Contain modules for coin-tossing, central limit theorem, etc.
• Probability review from MATH REVIEW for Practicing to Take the GRE General Test
• Video: Mlodinow’s talk @ Google
• Video: How many ways can you arrange a deck of cards? (There Are More Ways To Arrange a Deck of Cards Than Atoms on Earth)
  • Provide nice animation explaining permutation and factorial.
  • "Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again."
• Video: The Binomial Distribution / Binomial Probability Function
• Article: Greenlighting Movies: A High-Risk Game
• Video: The Poisson Distribution
• If you want to experience probability theory at a more advance level, one standard textbook that you can refer to is "Probability: Theory and Examples" by Prof. Durrett. Currently, the 4th edition of the textbook is available online.
• Video: Peter Donnelly shows how stats fool juries (same clip on youtube)
• Video: Lies, damned lies and statistics (about TEDTalks): Sebastian Wernicke on TED.com
• Article about clinical/medical decision making: Jill G. Klein, "5 pitfalls in diagnosis and prescribing: psychological biases that can lead to poor judgement," 2005.
  • Related topics: Pitfall #1 (representatiove heuristic), Pitfall #2(availability heuristic), and Pitfall #5 (illusory correlation).
    
• The Median Isn't the Message by Stephen Jay Gould
• Video: Daniel Kahneman: The riddle of experience vs. memory
  • Happiness Can Be Bought For About $60,000 Per Year
• Articles on risk intelligence
  • Dylan Evans, How to Beat the Odds at Judging Risk, The Wall Street Journal, May, 2012
  • Alison George, What Gamblers and Weather Forecasters Can Teach Us About Risk: An interview with the creator of the "risk quotient" intelligence scale., pp 30-31, New Scientist, May 19, 2012
• Generation of random numbers
  • Article: Park, S.K., and K.W. Miller. "Random Number Generators: Good Ones Are Hard to Find." Communications of the ACM, 31(10):1192–1201. 1998.
  • Article: Tom McNichol, "Totally Random: How two math geeks with a lava lamp and a webcam are about to unleash chaos on the Internet"
  • Article: C. Moler, Random thoughts, "10^435 years is a very long time", MATLAB News and Notes, Fall, 1995
  • Article: Ziggurat algorithm generates normally distributed random numbersdescribing the ziggurat algorithm introduced in MATLAB version 5.
• Games of chance
  • Poker
    • Paper: Cheung, Y. L. "Why Poker is Played with Five Cards." Math. Gaz. 73, 313-315, 1989.
    • Tim Farajian's Texas Hold'Em Poker Analyzer in MATLAB
      • Allow a user to simulate different scenarios in a Texas Hold'Em game.
      • Automatically simulate as many hands as you would like, and display winning probabilities or expected returns.
  • Blackjack
    • Cleve Moler's Blackjack in MATLAB + article
      • Paper: Simulating Blackjack with MATLAB
    • Michael Iori's Blackjack in MATLAB
• Quotations about Statistics
• Video: Statistics - Dream Job of the next decade
• Virtual Laboratories in Probability and Statistics
• Google Calculator (Cheat Sheet)
• Sometimes the easiest way to get information on a counting problem is to compute a few small values of a function, then look for a match at the sequence server; if you find a hit, you can sometimes get citations to the literature.
• Prapun's Notes on Probability Theory (Cornell Version)
• MATLAB
  • MIT OpenCourseWare > Electrical Engineering and Computer Science > 6.094 Introduction to MATLAB (January (IAP) 2009)
  • Fundamentals: Academic Tutorial
  • Video: Controlling Random Number Generation
  • Free online book: Cleve Moler, Experiments with MATLAB, 2008
  • Free textbook: Cleve Moler,Numerical Computing with MATLAB, 2004
• Learn the Greek Alphabet in less than 10 minutes
• The Greek Alphabet Song