MATH 410 Homework #6 (Due Wednesday, 1 April 2015) Read: Review Dobrow, Chapter 3. 1. A particle randomly moves around a circle of 5 nodes labeled 1 to 5. At each node, the particle is equally likely to move to the two neighboring nodes. Apply our gambler’s ruin formulas to find the following: (a) The expected number of steps for the particle to move from node 1 to node 3. (b) The chance that the particle hits node 3 before hitting node 4, starting at node 1. 2. Consider a 6 × 6 chessboard. The diagram shows that a king can move to any adjoining square, horizontally, vertically or diagonally. Informally, knight moves look like the letter L in some orientation (more precisely, if it sits at one corner of a 3-by-2 rectangle, then it can move to the opposite corner). King Knight (b) Your computations are based on the stationary distributions. Does either walk havethe a limiting (a) Compare mean distribution? return time Explain. to a corner square for the King’s and the Knight’s random walk. This means that at each step you choose at random among the available legal moves. Hint: Consider a graph whose vertices correspond to squares on the board. (b) Does either walk have a limiting distribution? Explain. Hint: Colors. 3. Consider M balls distributed between two urns, A and B. Let Xn be the number of balls in Urn A at time n. At each time period, we select one ball at random, and remove it from its urn. We then replace the ball in Urn A with probability p (where 0 < p < 1), or in Urn B with probability q = 1 − p, independent of any previous step. (a) Describe the transition probabilities for {Xn }. (b) Make a conjecture about the stationary distribution, and then verify that your conjecture is correct. (c) Is your answer the limiting distribution? 4. Refer to class notes on the “success runs” chain. We showed that π = (q, qp, qp2 , . . .) gave a probability vector solution to π = πP , and then verified that µ0 = 1/π0 gave the mean return time to state 0. Verify that the other 1/πj values make sense as mean return times.