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16.2-1. Assume that the probability of rain tomorrow is 0£ if it is
raining today, and assume that the probability of its being clear (no
rain) tomorrow is 0.9 if it is clear today. Also assume that these
probabilities do not change if information is also provided about
the 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.
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16.2-2. Consider the second version of the stock market model
presented as an example in Sec. 16.2. Whether the stock goes up
tomorrow depends upon whether it increased today and yesterday.
If the stock increased today and yesterday, it will increase tomorrow with probability a i. If the stock increased today and decreased
yesterday, it will increase tomorrow with probability « 2 . If the stock
decreased today and increased yesterday, it will increase tomorrow
with probability «3. Finally, if the stock decreased today and yesterday, 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 even
though what happens in the future (tomorrow) depends upon
what 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 been
marked 0, 1, 2, 3, 4 (in a clockwise order). The particle starts at
point 0. At each step it has probability QS of moving one point
clockwise (0 follows 4) and 0£ of moving one point counterclockwise. Let X,, (n 5: 0) denote its location on the circle after
step 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 a
Markov chain, determine the classes of the Markov chain and
whether they are recurrent.
State
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chain that has the following (one-step) transition matrix.
State
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hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled 5). The analyst believes that brand switching can be modeled as a Markov
chain using three states, with states A and B representing customers
drinking beer produced from the aforementioned breweries and
state C representing all other brands. Data are taken monthly, and
the analyst has constructed the following (one-step) transition matrix from past data.
B
0.7
0.2
0.1
0.2
0.1
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What arc the steady-state market shares for the two major breweries?
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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 numher of cameras on hand at the end of each week is 0 or 1, two additional cameras will be ordered. Otherwise, no ordering will take
place. Assume that the storage costs are the same as given in the
second subsection of Sec. 16.5.
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