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 step was counter-clockwise and the one before was clockwise.

Under this new model, answer the following questions:

(D) 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.

(E) Determine n-step transition probabilities for n = 5, 10, 20, 40, 80.

(F) Suppose last two movements were clockwise. Determine the probability of clockwise movement after 5 steps.

(G) 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.

(H) What is the steady-state probability? How does this compare to probabilities as n grows large in part (E)?

(I have posted the entire question below, but please only help with answering the Random Walk 2. I just wanted to add the attachment in case you may need it for reference, however I am sure that the questions listed above from Question 2 do not need information given in question 1. Thank you!)

Question 1. A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (points are marked in a clockwise order). The dynamics of particle movement is as follows: Random walk 1: The particle must move either clockwise or counterclockwise at each step. It has probability 0.6 of moving one point clockwise (0 follows 4) if earlier move was clockwise. Similarly, it has the probability 0.7 of moving one point counter-clockwise (4 follows 0) if the earlier move was counter-clockwise. (a) Can you model this movement as a Markov chain? Please specify the sequence of the random variable {Xn}. Also specify transition probabilities in the form of onestep transition matrix. (b) Determine n-step transition probabilities for n = 5, 10, 20, 40, 80. (c) What is the steadystate probability? How does this compare to probabilities as n grows large in part (b)? After some analysis, it was realized that the earlier model for particle movement was wrong. Question 2. A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (points are marked in a clockwise order). The dynamics of particle movement is as follows: Consider the newly proposed dynamics: Random walk 2: The particle must move clockwise or counterclockwise at each step. 0 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 counterclockwise. o The next move is counter-clockwise - with probability 0.8 if earlier two moves were cou nter-clockwise; - with probability 0.7 if immediately previous step was counterclockwise and the one before was clockwise. Under this new new model, answer the following questions: (D) Can you model this movement as a Markov chain? Please specify the sequence of random the variable {Yn}. Also specify transition probabilities in the form of one-step transition matrix. (E) Determine n-step transition probabilities for n = 5, 10, 20, 40, 80. (F) Suppose last two movements were clockwise. Determine the probability of clockwise movement after 5 steps. (G) 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. (H) What is the steadystate probability? How does this compare to probabilities as n grows large in part (E)? How does this compare to steady-state probabilities in part (B)? Bonus. Consider the first Random Walk model for particle movement. You are given that the currently the particle is in position 0 and could have landed in position 0 from either position 4 or position 1 equally likely. What is the probability that after 5 steps, particle is in position 3