18-751   Applied Stochastic Processes

Course description

This course provides a fundamental understanding of probability theory and its applications to stochastic systems. The course covers the basic concepts of random variables, random vectors, stochastic processes, and random fields. In addition, common random processes such as Markov chains, Wiener and Gaussian processes are introduced. The course touches on elements of estimation theory and optimal filtering including Wiener and Kalman filtering. Advanced topics in modern statistical signal processing such as linear prediction, linear models, and spectrum estimation are also discussed. Throughout the course, students will gain hands-on experience in applying the concepts to solve a wide range of complex and relevant problems.

Learning objectives

Students in this class will learn:

• The fundamental concepts of probability theory and how to apply them in modeling stochastic systems, including random variables, vectors, processes, and fields.
• The common random processes, such as Markov chains, Wiener and Gaussian processes, and their applications in modeling real-world systems.
• The basic inference methodologies (for both estimation and hypothesis testing) and how to apply them.
• How to process random signals with specific applications in estimation theory.

Outcomes

At the end of the course, students will be able to:

• Demonstrate an understanding of the fundamental concepts of probability theory and its applications to stochastic systems.
• Apply the concepts of random variables, random vectors, stochastic processes, and random fields to model real-world systems.
• Analyze common random processes such as Markov chains, Wiener and Gaussian processes, and use them to model complex systems.
• Apply the principles of estimation theory and optimal filtering to solve real-world problems, including Wiener and Kalman filtering.
• Apply advanced topics in modern statistical signal processing such as linear prediction, linear models, and spectrum estimation to solve complex problems.

Content details

Module 1: Probability and random variables

• Discrete random variables
• Continuous random variables
• Multivariate random variables
• Sum of random variables

Module 3: Statistical inference

• Estimation
• Hypothesis

Module 4: Stochastic processes

• Introduction to Stochastic Processes
• Markov Chain
• Random Walks
• Gaussian Process
• Weiner Process
• Poisson Process

Module 5: Signal processing and filtering

• Basics of signal processing
• Power spectrum density
• Introduction to linear filtering
• Kalman Filter

Prerequisites

None

Faculty

Moise Busogi