Reliability: Modeling, Prediction, and Optimization

APPENDIX B

Introduction to Stochastic Processes

B.1 BASIC CONCEPTS

In this Appendix, we give a brief introduction to stochastic processes and discuss some of the processes that are used in the book. Our presentation will be intuitive and nonrigorous and will highlight the important concepts. Readers interested in a deeper understanding of the underlying theory should consult the references given at the end of the book.

1.1 Stochastic Processes

In Appendix A.1 we defined a random variable, X(ω), as function that maps outcomes from the sample space to real numbers. A stochastic process X(t, ω), t ∈ T, where T is a set of nonnegative numbers, can be viewed as an extension of X(ω) in the following sense: t represents a time instant in the set T, which may be either finite or infinite. For a fixed t ∈ T, X(t, ω) is a random variable in the usual sense. For a fixed ω (outcome), X(t, ω) can be viewed as a function of t. X(t, ω) denotes the state of the process at time t. If T is countable, then X(t, ω) is called a discrete time stochastic process and if T is a continuum, then it is called a continuous time stochastic process. Henceforth, we omit ω and represent X(t, ω) as simply X(t).

Let t_i(i = 1, 2, …, n) denote n different time instants. The probabilistic characterization of the process X(t) at these n points can be done through the joint probability distribution

As n increases, this ...