Definition of Stochastic Process

What is a stochastic process?

It is a process in which a group of random variables, distributed over a period of time t, and is measured on a same sample space is indexed.

During this process we see that the random variable X1 changes over a period of time t i.e. from t1 to tn. But in case of identical random variable the distribution of random variable remains same over time t. Another important point to remember while dealing with stochastic process is that it is important to mention the period in which the random variable is taking place. The spacing between different periods of time may be equal or may vary according to the environment on which this process is taking place.

Examples:

These examples will make the concept clear:

1. It is seen that section 2 of chapter 8 represents a poison distribution and a stochastic process as well. The condition in which this process is occurring is infinitely defined i.e. if it starts at time 0 then the random variable X can take place at any point of time from time 0 to t. thus the condition at which it is occurring may be given as X= 0,1,2,……… .
2. Another example of stochastic process is seen by a dice. The sample occurs from 1 to 6 which shows equal sample spacing. The time at which each throw of dice is taking place is also the same and thus this gives you the example of identical random variables.

Thus, both the example proves that distribution of random variable may differ with time or may stay same over the same period.