Stochastic Modeling: Analysis and Simulation (Dover Books on Mathematics) by Nelson Barry L

Author:Nelson, Barry L. [Nelson, Barry L.]
Language: eng
Format: epub, mobi
Publisher: Dover Publications
Published: 2012-10-11T00:00:00+00:00

What is the long-term market share for Box O’ Spuds?

6.12.Todd and Diane plan to take out a mortgage from Humorous Money Company to purchase a home. The current fixed-interest rate on a 30-year loan is 8%. For no additional service charge Humorous Money will allow Todd and Diane to “lock in” the current rate of 8% (option 1). For a service charge of $50 they can take whatever the current rate happens to be in 4 weeks (option 2). For a service charge of $100 Humorous Money will let the rate drift for 4 weeks but lock it in if it hits 8.2% or 7.8% (option 3). The sales agent says that option 3 protects Todd and Diane from getting stuck with a very high rate while giving them a chance for a lower rate.

Interest rates can change weekly, and Diane believes that interest rates are more likely to go down than up. For the next 4 weeks she believes that the following model is a good representation of how the interest rate will change each week: The rate will go up 0.1% with probability 0.3, go down 0.1% with probability 0.4, and stay the same with probability 0.3. Under this model, help Todd and Diane decide which option is best for them.

6.13.Find a building with at least five floors that has an elevator with a floor indicator so that you can observe the elevator’s movement from floor to floor. Define the state of your system to be the floor on which the elevator is currently stopped, and the time index to be the number of stops. Observe this system for a substantial period of time, and record the sample path (sequence of floors on which it stops).

(a)Suppose that the movement from floor to floor is well modeled as a Markov chain, so that pij is the probability that an elevator currently stopped on floor i next stops on floor j. Estimate the one-step transition matrix, and provide a standard error for each estimate. Do you think that the Markov property is appropriate? Why or why not?

(b)To test the Markov property, select two intermediate floors j and k (for example floors j = 2 and k = 3 in a five-story building). Let pijk ≡ Pr{Sn + 1 = k | Sn = j, Sn – 1 = i}. Let nijk be the number of times you observed a stop on i followed by a stop on j followed by a stop on floor k, for all possible floors i. Estimate pijk by

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