The integers from 1 to 7 are written in order clockwise around a circle. I start with a counter on number 1 and repeatedly roll a standard six-sided fair die. If the die shows 1 then I move the counter to the next number clockwise around the circle; if the die shows 2 or 3 then I move the counter to the next number anticlockwise around the circle; if the die shows 4, 5 or 6 then I do not move the counter. (i) Describe how to model this process as a discrete-time Markov chain. Write the transition matrix (ii) Explain why the Markov property is satisfied and give an example of how you could make a small modification after which the process would be a discrete-time Markov chain which is not homogeneous
The integers from 1 to 7 are written in order clockwise around a circle. I start with a counter on number 1 and repeatedly roll a standard six-sided fair die. If the die shows 1 then I move the counter to the next number clockwise around the circle; if the die shows 2 or 3 then I move the counter to the next number anticlockwise around the circle; if the die shows 4, 5 or 6 then I do not move the counter.
(i) Describe how to model this process as a discrete-time Markov chain. Write the transition matrix
(ii) Explain why the Markov property is satisfied and give an example of how you could make a small modification after which the process would be a discrete-time Markov chain which is not homogeneous.
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