Our study of probability refers to an experiment consisting of a procedure and observations. When we study random variables, each observation corresponds to one or more numbers. When we study stochastic processes, each observation corresponds to a function of time. The word stochastic means random. The word process in this context means function of time. Therefore, when we study stochastic processes, we study random functions of time. Almost all practical applications of probability involve multiple observations taken over a period of time. For example, our earliest discussion of probability in this book refers to the notion of the relative frequency of an outcome when an experiment is performed a large number of times. In that discussion and subsequent analyses of random variables, we have been concerned only with how frequently an event occurs. When we study stochastic processes, we also pay attention to the time sequence of the events.
In this chapter, we apply and extend the tools we have developed for random variables to introduce stochastic processes. We present a model for the randomness of a stochastic process that is analogous to the model of a random variable, and we describe some families of stochastic processes (Poisson, Brownian, Gaussian) that arise in practical applications. We then define the autocorrelation function and autocovariance function of a stochastic process. These time functions are useful summaries of the time structure of a ...