Probability Random Variables And Stochastic Processes 4th Ed Athanasios Papoulis

Athanasios Papoulis specialized in engineering mathematics, his work covers probability, statistics, and estimation in the application of these fields to modern engineering problems. Papoulis also taught and developed subjects such as stochastic simulation, mean square estimation, likelihood tests, maximum entropy methods, Monte Carlo method, spectral representations and estimation, sampling theory, bispectrum and system identification, cyclostationary processes, deterministic signals in noise (part of deterministic systems and dynamical system studies), wave optics and the Wiener and Kalman filters.

The course covers the basics of random processes, which is a foundation technology for many engineering domains. The course begins with a review of probability theory and functions of a random variable, joint probability distributions, and the multivariate Gaussian distribution. The course then covers the fundamentals of random processes including autocorrelation and power spectra, estimation, filtering, and prediction of random processes, and Poisson and Markov Processes.

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Random variables and random vectors: definition of random variable, discrete and continuous random variables, mixed random variables, distribution function and its properties, probability density and its properties, mean value and variance of a random variable, expectation theorem, moments of a random variable, transformation of a random variable, notable random variables (Gaussian, exponential, Rayleigh, Rice, Poisson, binomial) Definition of a random vector, joint probability density of a pair of random variables, probability density marginal, conditional probability density, independent random variables, concept of correlation of a pair of random variables, transformation of a pair of random variables, jointly Gaussian random variables, Gaussian random vectors and their properties.

Stochastic processes: definition of stochastic process, parametric processes, distribution function and probability density of order N, stationarity of the first and second order, stationary processes in the broad and narrow sense, independent and uncorrelated processes, autocorrelation function and its properties , covariance function of a process, Gaussian processes, power spectral density and its properties, relationship between power spectral density and autocorrelation function, mutual power spectral density between two processes, linear filtering of a process, cross-correlation between entry and exit processes from an LTI system, white processes.

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