Suppose that whether or not it rains today depends on the weather conditions in the last three days. In particular, if it rained for the past three days, then it will rain today with probability 0.8; if it did not rain for any of the past three days, then it will rain today with probability 0.2; and in any other case, the weather today will, with probability 0.6, be the same as the weather yesterday.
(a) Show that this system can be modeled by using a Markov chain. How would you define the states of the process? How many states are needed?
(b) Write the transition probability matrix for this Markov Chain.