Example 1.2

Consider a PBX/EPBAX (Electronic Public Branch Automatic Exchange) where we denote , , as the number of incoming calls arriving during a particular time . Individually  are the realized values of a random variable (r.v). In case one is interested to know how  changes as time, , changes then we look into a dynamic process, which can be termed as a stochastic process.

In both these examples one would also be interested to know about the joint distribution of the members of this family of random variables (r.v), and how many random variables (r.v's) are there, that depend on . One should remember that this parameter space  may be discrete/denumerably infinite or non denumerable infinite. Few examples for discrete parameter space are , . When  is discrete we usually use  to denote the parameter, e.g.,  and call this a stochastic sequence. When  is continuous, say for example, , , we use  to denote the parameter, e.g., .

Measure theoretic approach

Before going into more detail about stochastic process we give here a very brief preview of measure theoretical approach which is relevant for probability and statistics.Let X be a random variable (r.v) on , where, (i)  is the sample space, (ii)  is the sigma field, i.e., very simply stated it is the subsets of  which defines a set of events having certain properties which is predefined and common, (iii)  be the probability which is defined on , or in other words it is a function mapping from  to .