Question: Question 1 [12 marks] Let X, t 2 0, be a homogenous Markov chain taking states X, C {1,2, ... ,5} with transition matrix governing

Question 1 [12 marks] Let X, t 2 0, be a homogenous Markov chain taking states X, C {1,2, ... ,5} with transition matrix governing jumps and initial distribution 0 1/4 0 1/4 1/2 1/3 1/ 0 1/6 1/6 1/3 pjump = B 1/3 0 0 O 1/5 3/5 0 0 1/ , P(0) = | 0 5 1/6 respectively. Suppose that the waiting times: var(T) = 1/16 var(T,) = 1/36 . var(T,) = 1/25 . var(T5) = 1/9. (a) Compute P(X, 3, X, = 3) [3 marks]. (b) Compute the conditional CF E[exp(iuX,X,)|X = 1] for u = 1 [3 marks]. (c) Find all stationary distributions [3 marks]

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