Question

There is a square mat, and a cat sitting on one of the corners. The cat will jump between the four corners of the mat. Every time the cat jumps, it has a probability of 1/4 of jumping to each of the two adjacent corners, probability 1/6 of jumping to the corner diagonally opposite, and probability 1/3 of jumping straight up and down and staying on the same corner. Write the transition matrix A for this Markov chain.

Answer 1

The transition matrix A for this Markov chain can be constructed by assigning the probabilities of the cat jumping between the corners of the mat to the corresponding entries of the matrix. The resulting transition matrix A is [0 1/4 1/6 0; 1/4 0 1/4 1/3; 1/6 1/4 0 1/4; 0 1/3 1/4 0].

Step-by-step explanation:

To write the transition matrix A, we need to consider the probabilities of the cat jumping between the corners of the mat. Let's number the four corners as 1, 2, 3, and 4. The transition matrix A will have dimensions 4x4, with the (i, j) entry representing the probability of the cat moving from corner i to corner j.

To fill in the entries of A, we can use the given probabilities. For example, the probability of the cat jumping from corner 1 to corner 2 is 1/4, so the entry (1, 2) of matrix A is 1/4. Similarly, the entry (1, 3) representing the probability of the cat jumping from corner 1 to corner 3 is 1/6. We can continue this process to fill in the entire matrix A.

The resulting transition matrix A for this Markov chain is:

01/41/601/401/41/31/61/401/401/31/40