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2. Consider the Markov chain X = (Xn)neN with state space I one step transition probabilities given in the following diagram: 1 2 A TT ✓ 4 72 3 4 510 E 3 2 {A, B, C, D, E, F, G, H} and F 12 H 237 (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 7 for the Markov chain Y.

2. Consider the Markov chain X = (Xn)neN with state space I one step transition probabilities given in the following diagram: 1 2 A TT ✓ 4 72 3 4 510 E 3 2 {A, B, C, D, E, F, G, H} and F 12 H 237 (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 7 for the Markov chain Y.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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(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, E;T;, for Y to first return to each state i, v. the long-term average value of f(Xn), where ƒ : I → R is a function with ƒ(A) = 1, ƒ(B) = 2, ƒ(C) = 3, ƒ (D) 4, ƒ (E) = 5, ƒ (F) = f(G) = f(H) = 6, = 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, E;T;, for Y to first return to each state i, v. the long-term average value of f(Xn), where ƒ : I → R is a function with ƒ(A) = 1, ƒ(B) = 2, ƒ(C) = 3, ƒ (D) 4, ƒ (E) = 5, ƒ (F) = f(G) = f(H) = 6, = vi. starting initially in state B, what is the average number of visits made to state C before first returning to B.
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: LH Q T|T B TT TR 34 334 1 E T|T 13 (F) H 237 (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.
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: LH Q T|T B TT TR 34 334 1 E T|T 13 (F) H 237 (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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