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University of Wisconsin Milwaukee Markov Chain M Odel P Redictions Questions

University of Wisconsin Milwaukee Markov Chain M Odel P Redictions Questions

Question Description



Exercise 2.6(b) in the Givens & Hoeting book (instead of Newton-Raphson you can use the quasi-Newton method BFGS in the R function “optim”, or the Nelder-Mead method).



Assume the state of the weather in Sydney on any particular day can be modelled using a state space S = {1 ? sunny, 2 ? overcast, 3 ? wet}. Also assume that the likelihood of any state depends only on the state on the previous day. A sunny day is followed by a sunny day 60% of the time and otherwise it is overcast. An overcast day is equally likely to be followed by either a sunny or a wet day. 20% of days after a wet day are also wet, but 20% are sunny. (a). Write down the transition probability matrix for this Markov chain. (b). Draw the state transition diagram for this chain and check if it is irreducible or reducible. If reducible, identify the equivalence classes. (c). If it is sunny all weekend, calculate the probability it is also sunny on Monday. (d). Calculate the probability that a Monday is wet if the previous Monday was wet. (e). Calculate the probability that Wednesday through Friday are all sunny, if the previous Sunday was sunny. (f). Write down the equilibrium equations and, if possible, solve for the equilibrium distribution both algebraically and using R.



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