About This Course

This course provides a rigorous foundation in probability theory and stochastic processes, essential for modeling uncertainty in engineering, data science, and applied mathematics. The curriculum begins with the fundamentals of probability, exploring events, conditional probability, and Bayes’ theorem. Students will gain a solid understanding of random variables, expectations, and variance, with detailed study of both discrete and continuous distributions. Emphasis is placed on understanding probability density functions (PDFs), cumulative distribution functions (CDFs), and their conditional counterparts.

Building on this foundation, the course delves into joint distributions of multiple random variables, examining the critical concepts of covariance, independence, and transformations of random variables. This prepares students to model interactions between multiple uncertain quantities.

The third module introduces random processes and their key properties, such as stationarity and ergodicity. Students explore convergence concepts of sequences of random variables and fundamental probabilistic theorems, including the Strong and Weak Laws of Large Numbers and the Central Limit Theorem, which form the backbone of statistical inference.

In the progress, attention shifts to discrete-time Markov chains. Topics include state transitions, classification of states (transient and recurrent), stopping times, and the Strong Markov property. The module also introduces counting processes, particularly the Poisson process and its real-world applications.

The final module covers renewal theory and the Renewal Reward Theorem, followed by an introduction to continuous-time Markov chains. Students explore the structure and long-term behavior of such processes, laying the groundwork for further studies in queuing theory, reliability engineering, and operations research.

This course emphasizes both theoretical understanding and practical modeling skills.