A.1 Introduction

As noted in the introduction to Chapter 12, the material in this appendix is not required for a discussion of system reliability at the level presented in this text. However, some discussion of random processes is useful in connection with the material covered elsewhere in this text, and its treatment logically follows from that already presented. We therefore include the required discussion in this appendix to avoid interrupting the continuity of the material on reliability analysis. As always, we omit derivations that can be found in standard texts, seeking instead to provide the reader with an understanding of the key ideas and results.

A.2 Concept of a Random Process

If a random variable X is a function of time, i.e., X = X( t), then X( t) is said to be a random process or stochastic process. Unlike simple random variables, random processes are characterized both by their properties at a given time and by their behavior as it evolves across time.

The value of X( t) at any particular time, for example X( t ₀) = x ₀, is a random variable characterized by a probability density function f( x, t ₀) and having a mean, variance, etc., just as for any random variable. For example, if the density function is Gaussian, we have by analogy to Eq. (12.30),

then the process is said to be a Gaussian random process.

A...