The Gracious Living Senior Care Facility has an Alzheimer's unit. All residents live in the regular care ward or the Alzheimer's care ward. Assume no new residents come for a full year and that the only way residents leave is if they die. In any month, 85% of residents in the regular care ward stay there, 10% move to the Alzheimer's care ward and 5% die. In any month, 75% of residents in the Alzheimer's care ward stay there and 25% die. Required:
a. Construct a Markov Chain to describe this situation. Remember, a Markov Chain must consist of well-defined states and a transition matrix. So, be sure to say what your variables represent and be sure to label all rows and columns of your transition matrix. And, since this is clearly an absorbing Markov chain, be sure your matrix is in standard form.
b. Find the fundamental matrix, F.
c. On average, how many months does a Gracious Living resident stay in a regular care ward? Round to the nearest tenth of a month? d) At the end of 1 year, will all of the patients have died?