Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition

Chapter 6Stochastic Processes

6.1 Introduction

In the previous chapters we have seen the need to consider a collection or a family of random variables instead of a single random variable. A family of random variables that is indexed by a parameter such as time is known as a stochastic process (or chance or random process).

Definition (Stochastic Process)

A stochastic process is a family of random variables , defined on a given probability space, indexed by the parameter t, where t varies over an index set T.

The values assumed by the random variable are called states, and the set of all possible values forms the state space of the process. The state space will be denoted by I.

Recall that a random variable is a function defined on the sample space S of the underlying experiment. Thus the above family of random variables is a family of functions . For a fixed is a random variable [denoted by ] as s varies over the sample space S. At some other fixed instant of time , we have another ...