Question 2: A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (point are marked in a clockwise order). The dynamics of particle movement is as follows: Random walk 2: The particle must move clockwise or counter-clockwise at each step. •The next move is clockwise with probability 0.75 if earlier two moves were clockwise; - with probability 0.6 if immediately previous step was clockwise and the one before was counter-clockwise. •The next move is counter-clockwise - with probability 0.8 if earlier two moves were counter-clockwise; - with probability 0.7 if immediately previous ste
Question 2: A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (point are marked in a clockwise order). The dynamics of particle movement is as follows:
Random walk 2: The particle must move clockwise or counter-clockwise at each step.
•The next move is clockwise with
- with probability 0.6 if immediately previous step was clockwise and the one before was counter-clockwise.
•The next move is counter-clockwise
- with probability 0.8 if earlier two moves were counter-clockwise;
- with probability 0.7 if immediately previous step was counter-clockwise and the one before was clockwise.
Under this new model, answer the following questions:
(A) Can you model this movement as a Markov chain? Please specify the sequence of the random variable {Yn}. Also specify transition probabilities in the form of one-step transition matrix.
(B) Determine n-step transition probabilities for n = 5, 10, 20, 40, 80.
(C) Suppose last two movements were clockwise. Determine the probability of clockwise movement after 5 steps.
(D) Suppose last movement was clockwise. We don’t know the precise information about the earlier movement. We estimate that the earlier movement could be either clock-wise or counter-clockwise equally likely. Then, what is the probability of clockwise movement after 5 steps.
(E) What is the steady-state probability? How does this compare to probabilities as n grows large in part (B)?
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