When calculating the long-run probability that a markov process is in a particular state, for example state 2 of 3 different states. Do we have to write the pi equations in terms of the long-run state that we are trying to find. i.e. do we have to write pi1, pi2 and pi3 in terms of pi2. Thanks in advance for any help.
I don't think it matters how you write it, if the question says "find long run probability of being in state 2" the only way to do it is to find the stationary distribution. Doing this by solving the system interms of either pi1 pi2 or pi3 will all lead to the same answer because of the constraint that the sum of pi_j always totals to 1.
Devon is right. It doesn't matter which one you expressed in terms of because the answer will be the same anyway. However, it does pay to know which pi to express it as, especially when you have more than 4 pi's to make it easier/quicker to solve all those simultaneous equations. 4 pi's means you need 3 different equations, because one will be redundant.
