Markov-Bernoulli Chain
Consider the transition probability matrix of two states as given by ,
with the initial distribution  and
(i) For , the transition probability matrix is: , what does it mean?

(ii) For , the transition probability matrix is: , what does it mean?

(iii) For , again use the same fundamental principle where we can write
      , where and .      Here we can easily prove that  and so for .

From this above result we easily get the following

(i) , in general the formulae would be
     , depending on the number of states, i.e., we have a multinomial      distribution.

(ii) ,
      in general the formulae would be , depending on the number of      states, i.e.,we have a multinomial distribution.

(iii)
     
     Now we already know that

, hence

If we extend this calculation for  we can easily see that

(iv)  and  for