Page 169 - Probability, Random Variables and Random Processes

Chapter 5

Random Processes

5.1 INTRODUCTION
In this chapter, we introduce the concept of a random (or stochastic) process. The theory of
random processes was first developed in connection with the study of fluctuations and noise in physi-
cal systems. A random process is the mathematical model of an empirical process whose development
is governed by probability laws. Random processes provides useful models for the studies of such
diverse fields as statistical physics, communication and control, time series analysis, population
growth, and management sciences.

5.2 RANDOM PROCESSES
A1. Defintion:
A random process is a family of r.v.'s (X(t), t E T) defined on a given probability space, indexed
by the parameter t, where t varies over an index set T.
Recall that a random variable is a function defined on the sample space S (Sec. 2.2). Thus, a
random process (X(t), t E T) is really a function of two arguments {X(t, c), t E T, 5 E S}. For a fixed
t(=tk), X(tk, 5) = Xk(c) is a r.v. denoted by X(tk), as 5 varies over the sample space S. On the other
hand, for a fixed sample point ci E S, X(t, ci) = Xi(t) is a single function of time t, called a sample
function or a realization of the process. The totality of all sample functions is called an ensemble.
Of course if both 5 and t are fixed, X(t,, ci) is simply a real number. In the following we use the
notation X(t) to represent X(t, c)..

B. Description of a Random Process:
In a random process (X(t), t E T}, the index set T is called the parameter set of the random
process. The values assumed by X(t) are called states, and the set of all possible values forms the state
space E of the random process. If the index set T of a random process is discrete, then the process is
called a discrete-parampter (or discrete-time) process. A discrete-parameter process is also called a
random sequence and is denoted by {X,, n = 1, 2, . . .). If T is continuous, then we have a continuous-
parameter (or continuous-time) process. If the state space E of a random process is discrete, then the
process is called a discrete-state process, often referred to as a chain. In this case, the state space E is
often assumed to be (0, 1, 2, . . .). If the state space E is continuous, then we have a continuous-state
process.
A complex random process X(t) is defined by

where Xl(t) and X,(t) are (real) random processes and j = ,fq. Throughout this book, all random
processes are real random processes unless specified otherwise.

5.3 CHARACTERIZATION OF RANDOM PROCESSES
A. Probabilistic Descriptions:

Consider a random process X(t). For a fixed time t,, X(t,) = X, is a r.v., and its cdf F,(xl; t,) is
defined as