Probability, Random Variables, Statistics, and Random Processes

11Introduction to Random Processes

In many practical applications, from the transmission of time‐varying signals over telecommunications networks to the observation of fluctuating stock prices over a long period of time, it is necessary to deal with functions of time. These functions are unpredictable; otherwise, their study would serve no purpose, and thus become unnecessary. As the study of random processes is important, this chapter introduces the fundamentals of random processes in a probabilistic sense. The Gaussian process and the Poisson process are also briefly discussed.

11.1 Classification of Random Processes

Suppose to every outcome (sample point) ω in the sample space Ω of a random experiment, according to some rule, a function of time t is assigned. The ensemble or collection of all such functions that result from a random experiment, denoted by X(t, ω), is a random process or a stochastic process.

The function X(t, ω) versus t, for ω fixed, is a deterministic function and is called a realization, sample path, ensemble member, or sample function of the random process. For a given ω = ω_i, a specific function of time t, i.e. X(t, ω_i), is thus produced, and denoted by x(t). For a specific time t = t_k, X(t_k, ω) is a random variable, and is called a time sample. For a specific ω_i and a specific t_k, X(t_k, ω_i) is simply a nonrandom constant. It is common to suppress ω, and simply let X(t) denote a random process.

The notion of a random process is an extension of the random ...