Question

please help urgently solve number 1

#1. For a markov chain (X(n): n = 0, 1, ..} with state space {0, 1, 2, ...} and transition probability matrix P = Pial, let po be the probability mass function of X(0); that is, pli) = P(x(0) =i}. Give an expression for the probabilty mass function of X(n):

Answer 1

Given the probability mass function of X(0) is p(). The transition probability matrix is given to be P=[p_i,j].

Then probability mass function after the first transition is given by p*P, i.e. the initial distribution is multiplied to the the transition probability matrix.

Now for the successive transition, we multiply the transition probability matrix to the obtained probability mass function of previous state i.e.

$\small p(n)=p(n-1)*P$, where p(n) denotes the mass function of X(n). Thus using the transition successively, we have:

$\small p(n)=p(n-1)*P=p(n-2)*P^2=.....=p()P^n$. Thus the answer is:

$\small p()P^n$