## EECS 495 - Probabilistic Systems and Random Signals

**CATALOG DESCRIPTION: **Basic concepts of probability theory and statistics, random variables, moments; multiple random variables, conditional distributions, correlation; random signals; applications to engineering systems. **NOTE: This is the graduate level version of EECS 302; additional material and different homework will be provided for EECS 495.**

**REQUIRED TEXTS: **A. Haddad, *Probabilistic Systems and Random Signals *, Prentice Hall, 1 st edition

**REFERENCE TEXTS: **None

**COURSE COORDINATOR: **Abraham Haddad

**COURSE GOALS: **To teach students the basic concepts of probability theory and statistics, random variables, conditional distributions, and correlation as they arise in engineering signal and systems models.

**PREREQUISITE:** **Mathematics 234**

**DETAILED COURSE TOPICS**

**Week 1: **Techniques for Summarizing and Simulating Data, Sample Space.

How data is represented, models for random experiments, sample space, basic definition of probability.

**Week 2: **Basic Probability Concepts - Conditional Probability, Bayes' Rule.

Conditional probability and independence, Bayes' rule, applications to error models and reliability.

**Week 3: **Random Variables - Models of Random Variables and Distributions.

Distributions, discrete and continuous, basic engineering examples.

**Week 4: **Averages, Functions, and Various Important Distributions.

Averages, functions of random variables, conditional distributions, applications to signals, systems and networks.

**Week 5: **Joint Random Variables, Independence .

Joint densities and distributions, basic discrete and continuous examples, independent random variables.

**Week 6: **Functions of Random Variables, Correlation.

Sum of two random variables, functions of two random variables, averages and correlation, applications to the processing of signals.

**Week 7: **Gaussian Random Variables.

Joint Gaussian Random variables, applications to signals and noise, Rayleigh distribution, envelope of sinusoidal signals.

**Week 8: **Multiple Random Variables, Central Limit Theorem, Simulation.

Independent multiple random variables, sums of random variables, law of large numbers, central limit theorem, Monte Carlo simulation, applications to signals and systems.

**Week 9: **Basic statistics and applications in engineering.

Data analysis; sample mean and sample variance; histograms; convergence of sample mean.

**Week 10: **Introduction to Random Processes.

Basic definitions of discrete-time and continuous time random signals, sample paths, autocorrelation function, second-order properties, applications to signals and noise.

**COMPUTER USAGE: **Matlab Assignments

**LABORATORY PROJECTS: **None

**GRADES:**

- Homework - 20%
- Matlab Assignments - 10%
- Midterm - 30%
- Final - 40%

**COURSE OBJECTIVES: When a student completes this course, s/he should be able to:**

• Understand the basic concepts of probability and how they arise in engineering systems.

• Analyze systems involving uncertainties using random variable models.

• Understand the concepts of distributions, correlation, and other averaging properties.

• Design systems for extracting useful information from noisy data.

• Design simple reliable engineering systems in the presence of errors and failures.

• Understand the basic elements of statistics and analysis of data.

ABET CONTENT CATEGORY: 50% Math and Basic Science, 50% Engineering.