Example 2.4

Consider a communications system which has states 0 and 1 only, but through many channels of communication, say  in number. It is like you are calling your friend in Mumbai and you stay in Kanpur, and say the communications can be routed through many different cities phone lines. So as there are two states let us denote the following

 = P[digit which is entered is transmitted unaltered]
 = P[digit which is entered is transmitted altered]

Also assume that  and . Then the transition matrix is given as , where
; ; ;

Now

Utilizing this we can deduce , ,….,

Also we know that , i.e., , i.e., . One can easily verify that the row sum is 1 as .
Furthermore we can show that
Thus we can find



 and so on

Now as  and as we always have , hence

By now we should be aware that the following sets of equations given below can be utilized for solving different transition matrices values and the set of equations are given as:

These are known as the Chapman Kolmorogrov equations and they are widely used in the study of stochastic processes.