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Chapter 4: Q10. (page 217)
Stock market. Give an example of a continuous random variable that would be of interest to a stockbroker.
Example: The time taken by a stockbroker for the completion of the transactions of the stocks.
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4.131 Chemical composition of gold artifacts. The Journal of Open Archaeology Data(Vol. 1, 2012) provided data onthe chemical composition of more than 200 pre-Columbiangold and gold-alloy artifacts recovered in the archaeologicalregion inhabited by the Muisca of Colombia (a.d.600–1800). One of many variables measured was the percentageof copper present in the gold artifacts. Summary statisticsfor this variable follow: mean = 29.94%, median = 19.75%,standard deviation = 28.37%. Demonstrate why the probabilitydistribution for the percentage of copper in thesegold artifacts cannot be normally distributed.
Elevator passenger arrivals. A study of the arrival process of people using elevators at a multilevel office building was conducted and the results reported in Building Services Engineering Research and Technology (October 2012). Suppose that at one particular time of day, elevator passengers arrive in batches of size 1 or 2 (i.e., either 1 or 2 people arrive at the same time to use the elevator). The researchers assumed that the number of batches, n, arriving over a specific time period follows a Poisson process with mean . Now let xn represent the number of passengers (either 1 or 2) in batch n and assume the batch size has probabilities . Then, the total number of passengers arriving over a specific time period is . The researchers showed that if are independent and identically distributed random variables and also independent of n, then y follows a compound Poisson distribution.
a. Find , i.e., the probability of no arrivals during the time period. [Hint: y = 0 only when n = 0.]
b. Find , i.e., the probability of only 1 arrival during the time period. [Hint: y = 1 only when n = 1 and .]
Shopping vehicle and judgment. Refer to the Journal of Marketing Research (December 2011) study of whether you are more likely to choose a vice product (e.g., a candy bar) when your arm is flexed (as when carrying a shopping basket) than when your arm is extended (as when pushing a shopping cart), Exercise 2.85 (p. 112). The study measured choice scores (on a scale of 0 to 100, where higher scores indicate a greater preference for vice options) for consumers shopping under each of the two conditions. Recall that the average choice score for consumers with a flexed arm was 59, while the average for consumers with an extended arm was 43. For both conditions, assume that the standard deviation of the choice scores is 5. Also, assume that both distributions are approximately normally distributed.
a. In the flexed arm condition, what is the probability that a consumer has a choice score of 60 or greater?
b. In the extended arm condition, what is the probability that a consumer has a choice score of 60 or greater?
4.139 Load on timber beams. Timber beams are widely used inhome construction. When the load (measured in pounds) perunit length has a constant value over part of a beam, the loadis said to be uniformly distributed over that part of the beam.Uniformly distributed beam loads were used to derive thestiffness distribution of the beam in the American Institute of
Aeronautics and Astronautics Journal(May 2013). Considera cantilever beam with a uniformly distributed load between100 and 115 pounds per linear foot.
a. What is the probability that a beam load exceeds110 pounds per linearfoot?
b. What is the probability that a beam load is less than102 pounds per linear foot?
c. Find a value Lsuch that the probability that the beamload exceeds Lis only .1.
Variable speed limit control for freeways. A common transportation problem in large cities is congestion on the freeways. In the Canadian Journal of Civil Engineering (January 2013), civil engineers investigated the use of variable speed limits (VSL) to control the congestion problem. A portion of an urban freeway was divided into three sections of equal length, and variable speed limits were posted (independently) in each section. Probability distributions of the optimal speed limits for the three sections were determined. For example, one possible set of distributions is as follows (probabilities in parentheses). Section 1: 30 mph (.05), 40 mph (.25), 50 mph (.25), 60 mph (.45); Section 2: 30 mph (.10), 40 mph (.25), 50 mph (.35), 60 mph (.30); Section 3: 30 mph (.15), 40 mph (.20), 50 mph (.30), 60 mph (.35).
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