Markov chain on state {1, 2, 3, 4, 5, 6 , 7}

AI Thread Summary
The discussion centers on using a Markov chain with states {1, 2, 3, 4, 5, 6, 7} for a programming project. Participants emphasize the importance of the transition matrix, specifically a 6 x 6 sparse matrix, to analyze long-term behavior through the limit of the matrix raised to the power of n. The geometric distribution of returns to state 1, with a parameter of 1/4, is noted, raising questions about its relevance to the overall problem. There is a call for clarification on how to determine the long-term fraction of time spent in state 3. The conversation encourages sharing attempts at solving the problem and simplifying the state diagram for better understanding.
Janji
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Homework Statement
Let's suppose the chain starts at state 1. The distribution of the number of times that the chain returns to state 1 is geometric and the parameter would be 1/4.
Relevant Equations
In the long run, what fraction of the time does the chain spend in state 3?
I need this for a programming project. Could you help?
7_reducible.png
 
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What have you tried? What do you mean formally with "in the long run"?
 
Math_QED said:
What have you tried? What do you mean formally with "in the long run"?
In the long run (n→∞):
 
Janji said:
Homework Statement:: Let's suppose the chain starts at state 1. The distribution of the number of times that the chain returns to state 1 is geometric and the parameter would be 1/4.
Relevant Equations:: In the long run, what fraction of the time does the chain spend in state 3?

I need this for a programming project. Could you help?View attachment 260957
The diagram can be represented by a transition matrix. For this problem it is a 6 x 6 sparse matrix; i.e., most of the entries are 0 since many transitions aren't defined. To find the long-term behavior, you look at ##\lim_{n \to \infty}A^n##, where A is the transition matrix.

Janji said:
The distribution of the number of times that the chain returns to state 1 is geometric and the parameter would be 1/4.
It's been many years since I've done problems like this -- I don't know how this information fits into the problem.
 
Janji said:
The distribution of the number of times that the chain returns to state 1 is geometric and the parameter would be 1/4.
Is that given or something to be proved? If given it would seem redundant - all the info is in the initial state and the diagram.
Janji said:
In the long run, what fraction of the time does the chain spend in state 3?
You must show some attempt.
Can you see how simplify the state diagram in respect of this question?
 

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