2. Consider the Markov chain X = (Xn)neN with state space I = {A, B, C, D, E, F, G, H} and one step transition probabilities given in the following diagram: B TH HIN TH 3 4 E 11 13 1 2137 (a) Decompose the state space into its communicating classes and state the period of each class. Hence, identify the set of transient states T and a communicating class of recurrent states R. (b) Write down the one-step transition matrix P for the discrete parameter Markov chain Y with state space R, that is, the restriction of the Markov chain X to the recurrent class RCI. (c) What conditions does an invariant probability mass function for a discrete time Markov chain satisfy? Find π for the Markov chain Y.

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2. Consider a Markov chain {Xn}n≥o having the following transition diagram: 1/2 1 2 5 1/4 3 4 1/2 1/4 6 7 1/2 For this chain, there are two recurrent classes R₁ = {6,7} and R₂ = {1,2,5}, and one transient class R3 = {3,4}. (a) Find the period of state 3. (b) Find f33 and f22. (c) Starting at state 3, find the probability that the chain is absorbed into R₁. (d) Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R₁ or R₂. Note: there are missing transition probabilities for this chain, but no impact for your solution.

Refer to the sample data for pre-employment drug screening shown below. If one of the subjects is randomly selected, what is the probability that the test result is a false positive? Who would suffer from a false positive result? Why? Positive test result Negative test result Drug Use Is Indicated Drug Use Is Not Indicated Subject Uses Drugs 39 14 Subject Is Not a Drug User 2 36 Pre-Employment Drug Screening Results The probability of a false positive test result is (Round to three decimal places as needed.)

A state vector X for a four-state Markov chain is such that the system is three times as likely to be in state 4 as in 3, is not in state 2, and is in state 1 with probability 0.2. Find the state vector X.

Let {Xn: n = 0, 1, 2, ...} be a Markov chain with the four states 1,2,3,4. If p44 = 1, what is state 4 called? %3D an absorbing state O a certain state a recurring state O a cyclic state • Previous Simpfun