1.45. Consider a general chain with state space S = {1, 2} and write the transition probability as P(Xn+1 = 1)- Use the Markov property to show that b a+b and then conclude P(X₂ = 1) = 1 2 b a+b 1 1-a b 2 a 1-b = (1 − a { -b) P(Xn = 1) - ; + (1 − a −b)" { P(Xo = 1) - b a+b b a+b This shows that if 0
1.45. Consider a general chain with state space S = {1, 2} and write the transition probability as P(Xn+1 = 1)- Use the Markov property to show that b a+b and then conclude P(X₂ = 1) = 1 2 b a+b 1 1-a b 2 a 1-b = (1 − a { -b) P(Xn = 1) - ; + (1 − a −b)" { P(Xo = 1) - b a+b b a+b This shows that if 0
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 47E: Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.
Related questions
Question
S
Transcribed Image Text:1.45. Consider a general chain with state space S = {1, 2} and write the transition
probability as
P(Xn+1 = 1) -
Use the Markov property to show that
b
a+b
and then conclude
P(Xn = 1) =
b
1
2
1
1-a
b
a+b
2
a
1-b
=
= (1 − a −b)} {P{Xn \
P(X₂ 1) -
¡ + (1 − a − b)" { P(Xo = 1) —
b
a+b)
b
a+b
This shows that if 0 < a + b < 2, then P(Xn = 1) converges exponentially fast to
its limiting value b/(a + b).
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