EE24: Probabilistic Systems Analysis

Spring 2019

Overview

The goal of this class is the development of basic analytical tools for the modeling and analysis of random phenomena and the application of these tools to a range of problems arising in engineering, manufacturing, and operations research. The first portion of this class will cover introductory probability theory including sample spaces and probability, discrete and continuous random variables, conditional probability, expectations and conditional expectations, and derived distributions. The balance of the class will be concerned with statistical analysis methods including hypothesis testing, confidence intervals and nonparametric methods.

Staff

Lecturer:

Prof. Eric Miller
Office: Room 101A, Halligan Hall
Email: eric.miller@tufts.edu

TA:

Debojyoti Seth
Office: Halligan 137
email: Debojyoti.Seth@tufts.edu 

Schedule

• Lectures: Tuesdays and Thursdays, 10:30AM – 11:45 AM, Halligan Hall, Room 111B
• Prof. Miller's Office Hours
  • Official:
    • Tuesdays, 4:00 PM – 5:00 PM, Halligan Hall, Room 127
    • Wednesdays, 10:00 AM - 11:00 AM, Halligan Hall, Room 127
    • Thursdays, 2:00 PM - 3:00 PM,  Halligan Hall, Room 127
  • Email Prof. Miller for an appointment
  • Stop by Prof. Miller’s Office
• Mr. Seth's Office Hours
  • Mondays and Wednesdays, 11:00 AM - 12:00 PM in Halligan Hall Room 137

Pre-requisites

Officially: Math 42

Practically: 

• Knowledge of differentiation and integration of single and multi-variable (mostly two variables) functions
• There will be regular computational assignments, so it would be helpful to have had some experience with a tool such as Matlab or Mathematica.  Low level languages such as C or Python are fine to use as well.

Use of Piazza

This term we will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza

Find our class page at: https://piazza.com/class/jlcohws8peklh

Course Requirements

Problem sets:

• There will be 10 or 11 problem sets.
• They will count for 10% of the grade.
• The lowest problem set grade will be dropped if a course evaluation receipt is emailed to Prof. Miller
• All problem sets should be handed into the EE 24 box located in Halligan 101 by 5PM on the due date
• No late assignments will be accepted

​Computational assignments

• There will be 4-5 assignments in addition to the problem sets designed to provide experience with "processing" data using probabilistic and statistical computational methods 
• These will count for 10% of the grade

In-class exams:

• Two in class exams will be given each worth 25%

Final exam:

• A last exam worth 30% of the grade will be given during finals week 

Requests for re-grades

• Students have one week from the time an assignment or exam is returned to dispute a grade.  All such requests must be made in writing and submitted to Prof. Miller along with the material to be reviewed.

Academic Integrity

Tufts holds its students strictly accountable for adherence to academic integrity. The consequences for violations can be severe. It is critical that you understand the requirements of ethical behavior and academic work as described in Tufts’ Academic Integrity handbook. If you ever have a question about the expectations concerning a particular assignment or project in this course, be sure to ask me for clarification. The Faculty of the School of Arts and Sciences and the School of Engineering are required to report suspected cases of academic integrity violations to the Dean of Student Affairs Office. If I suspect that you have cheated or plagiarized in this class, I must report the situation to the dean.

Texts

Required text

• Probability and Stochastic Processes, A Friendly Introduction for Electrical and Computer Engineers, by Roy Yates and David Goodman, 3rd Edition, Wiley, 2014.

Other references

• Probabilistic Methods of Signal and System Analysis, by George R. Cooper and the late Clare D. McGillem, Oxford, 1998
• Introduction to Probability, by Dimitri P. Bertsekas and John N. Tsitsiklis, Athena Scientific, 2002
• Probabilistic Systems and Random Signals by A. H. Haddad, Prentice Hall, Upper, 2006
• Probability and Random Processes for Electrical Engineering by Leon Garcia, Addison Wesley, 1994
• First Course in Probability, by Sheldon Ross, Prentice Hall, 2005
• Probability, Random Processes, and Statistical Analysis  by  Hisashi Kobayashi, Brian L. Mark, and William Turin, Cambridge University Press, 2011.

Tentative Schedule

Date	Topic
Jan 17	Introduction and basic set theory
Jan 22	Axioms of probability and basic models
Jan 24	Conditional probability and Bayes Law
Jan 9	Bayes Law and Independence
Jan 31	Combinatorics
Feb 5	Discrete random variables (RVs), Probability mass functions (PMFs), Bernoulli process
Feb 7	Poisson RV, Cumulative distribution functions (CDFs), expectation, mean, and variance for discrete RVs
Feb 12	Probability density functions (PDFs), CDFs, and expectation for continuous RVs
Feb 14	Expectations for continuous RVs, common PDFs, Gaussian RVs
Feb 19	First In-Class Exam (tentative)
Feb 21	No class: Monday schedule
Feb 26	Jointly distributed discrete and continuous RVs, marginal distributions
Feb 28	Expected values for pairs of RVs, Covariance, Correlation, and Independence
Mar 5	Bivariate Gaussian models, Multivariate Probability Models
Mar 7	Bivariate Gaussian models, Multivariate Probability Models
Mar 12	Derived distributions for pairs of discrete random variables, derived distribution for continuous random variables
Mar 14	Derived distribution for pairs of continuous random variables
Mar 19	No class: Spring Break
Mar 21	No class: Spring Break
Mar 26	Conditioning a single random variable on an event and conditional expectations
Mar 28	Conditioning pairs of random variables on an event and conditioning on a random variable
Apr 2	Conditioning pairs of random variables on an event and conditioning on a random variable
Apr 4	Second in class exam (tentative)
Apr 9	Conditional expected value and bivariate Gaussians revisited.  Sums of random variables
Apr 11	Central limit theorem and applications
Apr 16	Properties of the sample mean, probability inequalities
Apr 18	Properties of point estimators, weak law of large numbers, Confidence intervals
Apr 23	Introduction to hypothesis testing and significance testing
Apr 25	Binary hypothesis testing