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.
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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 18EQ
Related questions
Question
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.
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Write the given information
VIEWStep 2: Identify the communicating classes and state the period of each class
VIEWStep 3: Write down the one-step transition matrix P for the Markov chain Y with state space R
VIEWStep 4: State the conditions for the invariant probability mass function π and find π for the Markov chain Y
VIEWSolution
VIEW
Step by step
Solved in 5 steps with 47 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning