A Signal Theoretic Introduction to Random Processes

5INTRODUCTION TO RANDOM PROCESSES

5.1 RANDOM PROCESSES

The random variation with time, of measures associated with phenomena in the physical, technical, and social worlds, underpins significant interest in random processes. This chapter defines a random process and establishes notation that is used in subsequent chapters. Underpinning a random process is an experiment, and the two prototypical experiments, the Bernoulli and the Poisson, are defined along with a discussion related to the specification of experimental outcomes. Finally, a preliminary classification of random processes is given.

5.2 DEFINITION OF A RANDOM PROCESS

Definition: random process

When a function, whose range is a subset of the real line or complex plane, is associated with each experimental outcome of an experiment, a random process X is defined. Formally,

(5.1)

where S is the sample space of experimental outcomes and the set S_X is a set of functions, denoted the signal sample space, defined by the random process.

Definition: Signal Sample Space

The signal sample space, S_X, is the set of functions defined by the random process:

(5.2)

5.2.1 Notation

A random process is a mapping:

(5.3)

and the set of signals ...