Chapter 2: Probability, Random Variables, and Stochastic Processes

Learning Objectives

• Construct probability spaces and apply the axioms of probability, conditional probability, and Bayes' theorem
• Characterize random variables through PDF, CDF, moments, and moment-generating functions
• Derive distributions of transformed random variables and identify the key fading distributions (Rayleigh, Ricean, Nakagami)
• Analyse random vectors via joint distributions, covariance matrices, and the multivariate complex Gaussian $\mathcal{CN}(\mathbf{0}, \mathbf{R})$
• State and apply the law of large numbers, central limit theorem, and Chernoff bound
• Define stationarity, ergodicity, and power spectral density for random processes via the Wiener--Khinchin theorem
• Characterise Gaussian processes and white noise as the baseline models for communication channels
• Analyse Markov chains, compute stationary distributions, and model queuing systems (M/M/1)
• Apply Poisson process properties (memoryless, superposition, thinning) to random access and stochastic geometry