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16.2-1. Assume that the probability of rain tomorrow is 0£ if it israining today, and assume that the probability of its being clear (norain) tomorrow is 0.9 if it is clear today. Also assume that theseprobabilities do not change if information is also provided aboutthe weather before today.(a) Explain why the stated assumptions imply that the Markovian

property holds for the evolution of the weather.(b) Formulate the evolution of the weather as a Markov chain by

defining its states and giving its (one-step) transition matrix.

16.2-2. Consider the second version of the stock market modelpresented as an example in Sec. 16.2. Whether the stock goes uptomorrow depends upon whether it increased today and yesterday.If the stock increased today and yesterday, it will increase tomor-row with probability a i . If the stock increased today and decreasedyesterday, it will increase tomorrow with probability «2. If the stockdecreased today and increased yesterday, it will increase tomorrow

with probability «3. Finally, if the stock decreased today and yes-terday, it will increase tomorrow with probability «4.(a) Construct the (one-step) transition matrix of the Markov chain.(b) Explain why the states used for this Markov chain cause the

mathematical definition of the Markovian property to hold eventhough what happens in the future (tomorrow) depends uponwhat happened in the past (yesterday) as well as the present(today).

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16.3-3. A particle moves on a circle through points that have beenmarked 0, 1, 2, 3, 4 (in a clockwise order). The particle starts atpoint 0. At each step it has probability QS of moving one pointclockwise (0 follows 4) and 0£ of moving one point counter-clockwise. Let X,, (n 5: 0) denote its location on the circle afterstep n. [Xn] is a Markov chain,(a) Construct the (one-step) transition matrix.

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16.4-3. Given the following (one-step) transition matrix of aMarkov chain, determine the classes of the Markov chain andwhether they are recurrent.

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C 16.5-4. The leading brewery on the West Coast (labeled /O hashired an OR analyst to analyze its market position. It is particu-larly concerned about its major competitor (labeled 5). The ana-lyst believes that brand switching can be modeled as a Markovchain using three states, with states A and B representing customersdrinking beer produced from the aforementioned breweries andstate C representing all other brands. Data are taken monthly, andthe analyst has constructed the following (one-step) transition ma-trix from past data.

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16.5-7." Consider the inventory example introduced in Sec. 16.1,but with the following change in the ordering policy. If the num- — ^ 1her of cameras on hand at the end of each week is 0 or 1, two ad- lo • -? " ^~/ditional cameras will be ordered. Otherwise, no ordering will take ^ w / - .. -,place. Assume that the storage costs are the same as given in the & / /) — . Q \ ^ ff &S-f & {jsecond subsection of Sec. 16.5. * * +• "C (a) Find the steady-state probabilities of the state of this Markov ^ ••* £. ? X" &

chdn. *(b) Find the long-run expected average storage cost per week.

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