Question

# Question 2 In a Markov chain model for the progression of a disease, Xn denotes the level of severity in year n,

for n = 0, 1, 2, 3, . . .. The state space is {1, 2, 3, 4} with the following interpretations: in state 1 the symptoms are under control, state 2 represents moderate symptoms, state 3 represents severe symptoms and state 4 represents a permanent disability. The transition matrix is: P = 1 4 1 2 0 1 4 0 1 4 1 2 1 4 0 0 1 2 1 2 0 0 0 1 (a) Classify the four states as transient or recurrent giving reasons. What does this tell you about the long-run fate of someone with this disease? (b) Calculate the 2-step transition matrix. (c) Determine (i) the probability that a patient whose symptoms are moderate will be permanently disabled two years later and (ii) the probability that a patient whose symptoms are under control will have severe symptoms one year later. (d) Calculate the probability that a patient whose symptoms are moderate will have severe symptoms four years later. A new treatment becomes available but only to permanently disabled patients, all of whom receive the treatment. This has a 75% success rate in which case a patient returns to the "symptoms under control" state and is subject to the same transition probabilities as before. A patient whose treatment is unsuccessful remains in state 4 receiving a further round of treatment the following year. (e) Write out the transition matrix for this new Markov chain and classify the states as transient or recurrent. (f) Calculate the stationary distribution of the new chain.

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