MHB How Do You Calculate Long-Term Proportions in a Markov Chain?

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To calculate long-term proportions in a Markov chain with the transition matrix T = |0.7 0.4| |0.3 0.6|, the nth state can be determined using the formula s_n = T^n × s_0. For convergence, it's recommended to try n = 50 and n = 100; if the results stabilize, those values represent the long-term proportions. The initial state vector s_0 is crucial for this calculation; if it's not provided, testing scenarios like a = b = 0.5 can be useful. This approach allows for the determination of the long-term behavior of the states A and B. Understanding these calculations is essential for analyzing Markov chains effectively.
musad
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Not really sure how to get started on this one:Find the long-term proportions, a and b, of the two states, A and B, corresponding to the transition matrix T=|0.7 0.4|
| 0.3 0.6|


Note, the matric is a 2x2 matrix

Thanks
 
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Consider the nth state of a,b to be given by $s_n = T^n \times s_0$

For long term convergence try n = 50 and n=100, if they do not vary then you have your answer.

The only bit we are missing is $s_0$ were you given that? If not try some scenarios i.e. $a=b=0.5$
 
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