Characterization of Stochastic Processes: Theory and Applications

This lecture focuses on the characterization of stochastic processes, particularly emphasizing the mathematical foundations and applications. It begins with the definition of stochastic processes and their properties, including deterministic and random functions. The instructor discusses the characterization of a stochastic process {X(s, t), t ∈ R} and introduces various types of stochastic processes, such as fixed and variable functions. The lecture covers the probability distribution functions and their significance in determining the behavior of stochastic processes. The instructor elaborates on the conditions for stochastic processes, including the relationships between different time points t₁, t₂, and tm. The mathematical formulations presented include the probability of certain outcomes and the expected values associated with these processes. The lecture concludes with practical examples illustrating the application of these concepts in real-world scenarios, reinforcing the theoretical aspects discussed throughout the session.