Abstract
In Chap. 1, we introduced the concept of a random event: a collection of one or more outcomes resulting from a random experiment (e.g., a randomly selected device works for 1,000 h, or a randomly selected person has brown hair). In Chaps. 2–4, we studied random variables: numerical values resulting from random experiments (the number of flaws on a randomly selected wafer, the number of wins in 5 games of chance, the mass of a randomly selected object). In this chapter, we look at random processes, also called stochastic processes (“stochastic” is a synonym for “random”): time-dependent functions resulting from random phenomena.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Loosely speaking, a random process is ergodic if its time and ensemble properties “match.” We will define ergodicity more carefully later in this section; for now, you may assume the processes referenced in this section are ergodic unless noted otherwise.
- 2.
Readers not familiar with the triangular or “tri” function should consult Appendix B.
- 3.
A rv Y equals a constant c in the mean square sense if E[(Y − c)2] = 0. This is equivalent to requiring that E(Y) = c and Var(Y) = 0. In the definition above, Y = 〈X(t)〉 is the rv and c = E[X(t)] is the constant.
- 4.
Readers not familiar with o(h) notation should consult Appendix B.
- 5.
Albert Einstein showed in 1905 from physical considerations that the conditional pdf of B(t 0 + t) given B(t 0) = x must satisfy the partial differential equation \( \partial f/\partial t=\frac{1}{2}\alpha \cdot {\partial}^2 f/\partial {x}^2 \), where the “diffusion constant” α involves a gas constant, temperature, a coefficient of friction, and Avogadro’s number. He also showed that the unique solution to this PDE is the normal pdf.
- 6.
When the state space is infinite, it can sometimes happen that q i = ∞. This will not occur for the situations considered in this section.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Carlton, M.A., Devore, J.L. (2017). Random Processes. In: Probability with Applications in Engineering, Science, and Technology. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-52401-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-52401-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-52400-9
Online ISBN: 978-3-319-52401-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)