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 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:
1
534
(0)
A
71
WIN
(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 for the Markov chain Y.
Transcribed Image Text: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: 1 534 (0) A 71 WIN (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 for the Markov chain Y.
(d) Stating any general results that you appeal to, deduce the following:
i. Y is positive recurrent,
ii. the distribution of Y after it has been running for a very long time,
iii. the long-term proportion of time spent in each of the states,
iv. the average time, ET;, for Y to first return to each state i,
v. the long-term average value of f(X₂), where ƒ : I → R is a function with f(A)
ƒ(B) = 2, ƒ(C) = 3, ƒ (D) = 4, ƒ (E) = 5, ƒ (F) = ƒ(G) = f(H) = 6,
= 1.
vi. starting initially in state B, what is the average number of visits made to state C
before first returning to B.
Transcribed Image Text:(d) Stating any general results that you appeal to, deduce the following: i. Y is positive recurrent, ii. the distribution of Y after it has been running for a very long time, iii. the long-term proportion of time spent in each of the states, iv. the average time, ET;, for Y to first return to each state i, v. the long-term average value of f(X₂), where ƒ : I → R is a function with f(A) ƒ(B) = 2, ƒ(C) = 3, ƒ (D) = 4, ƒ (E) = 5, ƒ (F) = ƒ(G) = f(H) = 6, = 1. vi. starting initially in state B, what is the average number of visits made to state C before first returning to B.
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