Probability & Stochastic Proc

Q2

[7 marks] A mouse moves around the following (4 x 4)-checkerboard: At each step, the mouse moves from its current square to a randomly selected neigh- bouring square either vertically or horizontally (but not diagonally). So, if the mouse is on one of the corner squares, 2 moves are allowed, each occurring with probability 1/2; if it is on a side (but not corner) square, 3 moves are allowed, each occurring with probability 1/3; and if it is on a central square, 4 moves are allowed, each occurring with probability 1/4. a) [3 marks] In the long-term, what is the fraction of time that the mouse spends on each square? b) [1 mark] The mouse is currently on a corner square. After how many steps, on average, will it be back on the same square? c) [1 mark] The mouse is currently on a corner square. After how many steps, on average, will it be back on a corner square (not necessarily the same)? d) [2 marks] The mouse is currently on a corner square. What is the probability that, two steps earlier, it was on a central square? Assume that the mouse has been on that checkerboard for a very long time