## EECS 422 - Random Processes in Communications and Control I

**CATALOG DESCRIPTION:** Fundamentals of random variables; mean-squared estimation; limit theorems and convergence; definition of random processes; autocorrelation and stationarity; Gaussian and Poisson processes; Markov chains.

**REQUIRED TEXTS:** Robert G. Gallager, *"Stochastic Processes: Theory for Applications,"* Cambridge University Press, 2014.

**REFERENCE TEXTS:** None

**COURSE DIRECTOR/INSTRUCTOR: Prof. Randall Berry**

**COURSE GOALS:** Probability and random processes are central fields of mathematics and are widely applied in many areas including risk assessment, statistics, machine learning, data networks, operations research, information theory, control theory, theoretical computer science, quantum theory, game theory, finance, and neurophysiology. This course will provide an introduction to mathematical probability and random process with a focus on techniques that are useful in studying communication and control systems as well as in many other domains. We will begin with a thorough review of basic probability theory including probability spaces, random variables, probabilistic inequalities, and laws of large numbers. We then will study a number of basic random processes including Poisson Processes, Markov Chains and Gaussian Processes. The basics of estimation and filtering of random processes will also be covered.

**PREREQUISITES BY COURSES:** One course in probability.

**PREREQUISITES BY TOPIC:**

• Probability theory.

**DETAILED COURSE TOPICS:**

- Probability review: probability spaces, axioms of probability, conditional probabilities, independence, random variables, expectation, conditional expectation, inequalities.
- Limit theorems – laws of large numbers, central limit theorems.
- Poisson Process – memoryless properties, alternative definitions, combining and splitting.
- Finite State Markov chains – first passage time analysis, steady-state analysis
- Gaussian Processes – jointly Gaussian random variables, covariance matrices, filtered processes, power spectral density.
- Bayesian Estimation – MMSE criteria, estimation and Gaussian random vectors, linear least squares estimation.

**COMPUTER USAGE:** Optional.

**LABORATORY PROJECTS:** None.

**GRADES:**

- Problem Sets – 25%
- Midterm – 35%
- Final – 40%

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

• Understand the description and behavior of random processes.

• Model and analyze systems with random signals.