# Department of Electrical &amp; Computer Engineering University of Tennessee

```Department of Electrical &amp; Computer Engineering
University of Tennessee
Course Title: Probability and Random Variables
Course Code: ECE 313
Credit Hours: 3
Pre-requisite: Math 241
Instructor: Seddik M. Djouadi
Office # Ferris 307
Tel: (865) 974-5447
Office Hours: TTH 12:30pm– 2:00pm or by Appointment
I Course Description :
This course is a basic core course in electrical and Computer Engineering, which is a necessary
background for courses such as Communications, Control, Computer Networks, etc.
The course will include an in-depth introduction to probability theory, sample spaces, set operations,
joint, conditional and total probabilities, independence, Bayes’ theorem, examples of important
probability laws, random variables, distributions, densities, Gaussian random variables, conditional
densities, operations on random variables, expectation, moments, characteristic function, conditional
expectation, multiple random variables, joint distributions and densities, sum of mutually independent
random variables, random signals continuous and discrete, correlation, power spectral density,
autocorrelation function, stationarity and ergodicity, cross-correlation, noise models, minimum mean
square estimation, linear estimation, MLE, RSL, linear systems fundamentals, random signal response of
linear systems.
Pre-requisite: MATH 241 Calculus III.
II Course Objective:
Introduce the students to probability concepts, and random processes as a tool to model and solve
practical problems, with application in Engineering.
III Text Book:
Probability, Random Variables and Random Signal Principles, by Peyton Z. Peebles, JR., McGraw Hill
Publishing, 2000.
VI Reference:
Probability Methods of Signal and System Analysis, by George R. Cooper and Clare D. McGillem,
Oxford University Press, 1999.
The following schedule is approximate and subject to change. You will be responsible for
any changes that are announced in class, and for all the material covered in class.
V Tentative Schedule:
1
Week 1
Set Operations, random experiments, Sample spaces, and events.
Probability Definition and Axioms.
Examples of simple experiments and their sample spaces.
Week 2
Joint, conditional and total probabilities; Independence
Bayes’ theorem. Sequences of independent events.
The binomial probability law.
Week 3
Introduction. Definition of a Random Variable
Probability Distribution Function. Probability Density Function
Continuous, Discrete and Mixed Random Variables
Week 4
The Gaussian Random variables. Other Density Examples: Poisson, Uniform,
Exponential. Conditional Distributions and Properties. Density Functions
Week 5
Density Functions and Properties.
Test # 1
Week 6
Operation on Random Variables. Expectation (Averages), Moments
Expectation of a function of Random Variables. Characteristic Function.
Transformation of a Random Variable.
Week 7
Examples of expectations, moments of particular random variables
Multiple Random Variables Introduction and Notation
Joint Distributions and Densities. Conditional Distrubution and Density
Week 8
Statistical Independence.
Sum of Mutually Independent Random Vectors.
The Central Limit Theorem.
Week 9
Operations on Multiple Random Variables.
Expectations and covariance matrices. Gaussian random vectors
Characteristic Functions of Random Variables.
Test #2
Week 10
Introduction. Stationarity and Independence.
Correlation Functions.
Gaussian Random Processes. Poisson Random Process.
2
Week 11
Random Processes – Spectral Characteristics.
Introduction. Power Spectral Density and its Properties.
Relationship between Power Spectrum and Autocorrelation Functions.
Week 12
Cross-Power Spectral Density and its Properties.
Examples of Noise Models.
Linear Systems with Random Inputs.
Week 13
Linear System Fundamentals.
Random Signal Response of Linear Systems.
Week 14
Spectral Characteristics of System Response.
Test #3
VI Assignments, Evaluation Procedures:
1. Homework assignments will be given periodically to supplement the lectures and the course
2. 3-in-class examinations will be given in the semester, to test the students on material covered.
The exams will be closed book and class notes.
(No late assignment will be accepted unless you have a
legitimate excuse)
Homework 20%
Exam 1
Exam 2
Exam 3
25%
25%
30%