2. Consider a Markov chain {Xn}nzo 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.
2. Consider a Markov chain {Xn}nzo 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.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 49E: Consider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show...
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Transcribed Image Text: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.
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