Random Variables, Stochastic Processes, and Deterministic Signals

Abstract

Chapter 2 reviews the concept of random variables, stochastic processes, and deterministic signals. Many important terms and definitions, which are often used in the context of fading channel modeling, are recapitulated. Some of the terms discussed are random experiment, sample points, events, probability measure and probability space, random variables, cumulative distribution function and probability density function, joint cumulative distribution function and joint probability density function, statistically independent random variables, marginal probability density function, expected (mean) value, variance, covariance, moments, characteristic function, Chebyshev inequality, central limit theorem, addition and multiplication of random variables, functions of random variables, stochastic processes, strict-sense and wide-sense stationary stochastic processes, ergodic processes, autocorrelation and cross-correlation function, power and cross-power spectral density, Hilbert transformer, level-crossing rate and average duration of fades. Important probability density functions, such as uniform distribution, Gaussian (normal) distribution, multivariate Gaussian distribution, Rayleigh distribution, Rice distribution, lognormal distribution, Suzuki distribution, Nakagami distribution. Chapter 2 also explains the relationships between stochastic processes, random variables, sample functions, and real-valued (complex-valued) numbers. Moreover, the properties of deterministic continuous-time and discrete-time signals are introduced. Apart from that, Chapter 2 makes the reader familiar with the nomenclature used in the book.