Reconsider Eddie's Bicycle Shop described in Prob. 22.4-7. Forty percent of the bicycles require only a minor repair. The repair time for these bicycles has a uniform distribution between 0 and 1 hour. Sixty percent of the bicycles require a major repair. The repair time for these bicycles has a uniform distribution between 1 hour and 2 hours. You now need to estimate the mean of the overall probability distribution of the repair times for all bicycles by using the following alternative methods.
(a) Use the uniform random numbers-0.7256, 0.0817, and 0.4392-to simulate whether each of three bicycles requires minor repair or major repair. Then use the uniform random numbers-0.2243, 0.9503, and 0.6104-to simulate the repair times of these bicycles. Calculate the average of these repair times to estimate the mean of the overall distribution of repair times.
(b) Draw the cumulative distribution function (CDF) for the overall probability distribution of the repair times for all bicycles.
(c) Use the inverse transformation method with the latter three uniform random numbers given in part (a) to generate three random observations from the overall distribution considered in part (b). Calculate the average of these observations to estimate the mean of this distribution.
(d) Repeat part (c) with the complements of the uniform random numbers used there, so the new uniform random numbers are 0.7757, 0.0497, and 0.3896.
(e) Use the method of complementary random numbers to estimate the mean of the overall distribution of repair times by combining the random observations from parts (c) and (d).
(f) The true mean of the overall probability distribution of repair times is 1.1. Compare the estimates of this mean obtained in parts (a), (c), (d), and (e). For the method that provides the closest estimate, give an intuitive explanation for why it performed so well.
(g) Formulate a spreadsheet model to apply the method of complementary random numbers. Use 300 uniform random numbers to generate 600 random observations from the distribution considered in part (b) and calculate the average of these random observations. Compare this average with the true mean of the distribution.
(h) The drawbacks of the approach described in part (a) are that (1) it does not ensure that the repair times for both minor repairs and major repairs are adequately sampled and (2) it requires two uniform random numbers to generate each random observation of a repair time. To overcome these drawbacks, combine stratified sampling and the method of complementary random numbers by using the first three uniform random numbers given in part (a) to generate six random minor repair times and the other three uniform random numbers to generate six random major repair times. Calculate the resulting estimate of the mean of the overall distribution of repair times
Prob. 22.4-7
Eddie's Bicycle Shop has a thriving business repairing bicycles. Trisha runs the reception area where customers check in their bicycles to be repaired and then later pick up their bicycles and pay their bills. She estimates that the time required to serve a customer on each visit has a uniform distribution between 3 minutes and 8 minutes. Apply the inverse transformation method as indicated below to simulate the service times for five customers by using the following five uniform random numbers: 0.6505, 0.0740, 0.8443, 0.4975, 0.8178.
(a) Apply this method graphically.
(b) Apply this method algebraically.
(c) Calculate the average of the five service times and compare it to the mean of the service-time distribution.
(d) Use Excel to generate 500 random observations and calculate the average. Compare this average to the mean of the service-time distribution.