Foundations of Markov Chains and Poisson Processes

How do we predict the unpredictable? In fields ranging from data science to operations research, understanding how random events evolve over time is essential for making accurate system predictions. This text-only course provides a clear, step-by-step pathway to mastering the mathematical frameworks used to model random systems. You will transition from basic probability concepts to constructing and analyzing sophisticated stochastic models. By reading through detailed explanations and structured written examples, you will develop the analytical skills needed to evaluate systems that change state over time. What you'll learn: - Understand the foundational terminology and core concepts of stochastic modeling. - Analyze discrete-time Markov chains, including transition matrices and state classifications. - Calculate stationary distributions to predict long-term system behavior. - Model random arrivals and event occurrences using the properties of Poisson processes. - Apply renewal theory concepts to analyze queueing systems and inventory models. - Explore modern applications of Markov processes in algorithmic decision-making and data science. The course begins with essential definitions and probability basics, ensuring you have a solid foundation before moving on to transition formulas, mathematical proofs, and practical modeling scenarios. Written exercises throughout the text allow you to practice calculations at your own pace. This course is designed for beginners, students, and professionals looking for a clear introduction to stochastic processes. No advanced mathematical background is required, though a basic understanding of algebra and introductory probability is recommended. Start reading today to build a strong foundation in probabilistic modeling.