Show transcribed image text Q.1 A Simple Markov Chain Consider a road for three cars that is monitored at discrete instants of time. The dynamics of the system are summarized below: More than one car can be present in the same location (the road has more than one lane) In the next monitoring interval, the car at the end of the road will occupy the position one unit in front of either of the other two cars with equal probability Assume that the cars start off exactly in line one behind another, with a uniform spacing of one unit between them. Using only 3 states, model the above scenario as a Discrete Time Markov Chain. Also write the one-step probability transition matrix. For simplicity, assume that all the cars are indistinguishable. (The one-step probability matrix is a square matrix with the same dimensions as the number of states i.e. it is an n Ã— n matrix where n is the number of states. The entry in the ith row and the jth column of this matrix gives us the probability of a process in state i transitioning to state j in the next instant. The rows of this matrix must sum to 1 which stems from the fact that the chain must transition to any of the states with probability 1.)

Q.1 A Simple Markov Chain Consider a road for three cars that is monitored at discrete instants of time. The dynamics of the system are summarized below: More than one car can be present in the same location (the road has more than one lane) In the next monitoring interval, the car at the end of the road will occupy the position one unit in front of either of the other two cars with equal probability Assume that the cars start off exactly in line one behind another, with a uniform spacing of one unit between them. Using only 3 states, model the above scenario as a Discrete Time Markov Chain. Also write the one-step probability transition matrix. For simplicity, assume that all the cars are indistinguishable. (The one-step probability matrix is a square matrix with the same dimensions as the number of states i.e. it is an n Ã— n matrix where n is the number of states. The entry in the ith row and the jth column of this matrix gives us the probability of a process in state i transitioning to state j in the next instant. The rows of this matrix must sum to 1 which stems from the fact that the chain must transition to any of the states with probability 1.)