Probability and Stochastic Processes: Fundamentals and Applications

This lecture covers the fundamentals of probability and stochastic processes, focusing on random variables and their properties. The instructor begins by reviewing the Z-transform and LTI systems, emphasizing the importance of finite difference equations for realizable systems. The discussion then shifts to random variables, defining them as quantities that take values from a specific set. The cumulative distribution function (CDF) is introduced as a key concept, illustrating how it describes the probabilistic behavior of random variables. The instructor explains the differences between discrete and continuous random variables, including their respective probability density functions. The lecture further explores moments of random variables, including expected values and variances, and introduces the concept of independence between random variables. The instructor also discusses discrete time stochastic processes, their laws, and the significance of correlation and covariance. The lecture concludes with an introduction to wide sense stationarity and the power spectral density, highlighting their relevance in statistical signal processing and pattern analysis.