Final answer:
The question addresses constructing the transition probability matrix for a random walk with reflecting boundaries and determining the limiting distribution of the Markov chain's time spent at each state.
Step-by-step explanation:
The question deals with constructing the transition probability matrix for a reflecting random walk on a line and finding the limiting distribution of the Markov chain that describes the movement of a particle along the points marked 1, 2, 3, and 4 with a certain probability of moving right or left.
For part (a), we need to consider the probabilities of moving from each state to another, while taking into account the reflecting boundaries at points 1 and 4. The transition probabilities can be summarized in a matrix where the element Pij represents the probability of transitioning from state i to state j. This matrix is as follows:
- From point 1, the chain can either stay at 1 with probability 1/3 (attempting to move left but reflected) or move to 2 with probability 2/3.
- From point 2, the chain can move to 1 or 3 each with probability 2/3 and 1/3 respectively.
- From point 3, the chain can move to 2 or stay at 3 with probabilities 2/3 and 1/3 (attempting to move right but reflected).
- From point 4, the chain can only move to 3 with a probability 1/3 or stay at 4 with probability 2/3.
For part (b), the limiting distribution is the steady-state probability vector that the chain spends at each state in the long run. It can be calculated by finding the eigenvector corresponding to the eigenvalue 1 of the transition matrix and normalizing it so that the sum of its components equals 1.