IE 336 Seat # Name. Closed book. One page of hand-written notes, front and back. No calculator. 60 minutes.

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1 Closed book. One page of hand-written notes, front and back. No calculator. 60 minutes. Cover page and five pages of exam. Four questions. To receive full credit, show enough work to indicate your logic. Do not spend time calculating. Unless stated otherwise, you will receive full credit if someone with no understanding of probability could simplify your answer to obtain the correct numerical solution. This exam covers through (most of) Chapter 4 of Solberg (August 2005). Definition: NEITK: "not enough information to know" Recall: The Poisson pmf with meanµis f (x )=e µ µ x / x! for x = 0, 1, 2,... Recall: The geometric pmf with probability of success p, and mean 1 / p, is f (x )=p (1 p) x 1 for x = 1, 2, 3,... Score Exam #2, Fall 2007 Schmeiser

2 Closed book. One page of hand-written notes, front and back. No calculator. 1. (Based on Solberg, Problem 7, Chapter 3) A paint manufacturer mixes colors in a large vat. There are only three colors: red, white, and blue. The color order, which is determined by external factors, can be modeled by a Markov chain with one-step transition matrix red white blue Changing colors is trivially easy when a dark color follows a dark color; that is, essentially no cleaning is needed when red follows blue or when blue follows red. Also, no cleaning is needed when red or blue follows white. Substantial cleaning is needed when white follows red or when white follows blue. (So far, this is just Solberg s problem statement.) (a) (3 points) Which states are transient? (b) (3 points) Which states are recurrent? (c) (3 points) Which states are absorbing? (d) (7 points) Determine the long-term fraction of white-paint vats. (e) (7 points) Assume that the Markov chain is in steady state. Determine the probability that the previous vat was white if the current batch is red. Exam #2, Fall 2007 Page 1 of 5 Schmeiser

3 (f) (7 points) If the vat now has white paint, determine the expected number of nonwhite vats until the vat is again filled with white paint. (g) (7 points) If the vat is now filled with white paint, what is the probability of exactly four more white vats before another color is mixed? (h) (7 points) Determine the long-term fraction of the vats that mix a dark color. Exam #2, Fall 2007 Page 2 of 5 Schmeiser

4 2. Consider again the red/white/blue paint model. Now, however, let "white" be an absorbing state, that is, the one-step probability matrix is P = red white blue Assume that the initial state distribution is red with probability, white with probability 0.6, and blue otherwise. Let e ij denote the (i,j )th element of E = (I Q ) 1, where Q is the matrix of transientto-transient one-step probabilities. Assume that these values have been computed. transient Q R (a) (7 points) Write the one-step transition matrix in the form P = recurrent 0 I. (b) (7 points) Explain, in the context of this problem, the meaning of e red,blue + e red,red. (c) (4 points) Determine the probability of eventual absorption into "white". (d) (7 points) Determine the expected number of vats before a white vat. (Include the current vat in the count.) Exam #2, Fall 2007 Page 3 of 5 Schmeiser

5 3. (Based on Solberg, Problem 6, Chapter 3) Consider the four-state Markov chain that models sequential car purchases for a particular buyer. The states are S = {GM, Ford, Chrysler,Foreign} with one-step transition matrix P = GM Ford Chrysler Foreign (a) (3 points) Is this Markov chain stationary? YES NO NEITK (b) (7 points) Explain, in the context of this problem, the meaning of the Markov Assumption. (c) (7 points) Consider the next three cars. If current car is foreign, determine the expected number of Ford cars. Exam #2, Fall 2007 Page 4 of 5 Schmeiser

6 4. (Based on Solberg, Chapter 3, Problem 13) A solar-heating device can recharge (store heat) in a single sunny day enough to provide for the next two days. In other words, the only times that the solar heat must be augmented by some other heat source is when there are three or more consecutive cloudy days. During the heating season, a period of 120 days, the weather patterns are described by the following Markov chain. sunny 1/2 1/2 P = cloudy 1/3 2/3. Let time be measured by days. Let the states be the number of days since a sunny day, with a maximum of three; that is, S = {0, 1, 2, 3}. (For example, X n = 2 means that today is cloudy, yesterday was cloudy, and the day before yesterday was sunny.) (a) (7 points) Write the one-step probability transition matrix. (b) (7 points) Sketch the corresponding one-step digraph. Exam #2, Fall 2007 Page 5 of 5 Schmeiser