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Statistics
When we used Bayes’ theorem to update the probabilities that eKnow would be a Success, Potential, or Failure by counting the number of Definitely-Purchase responses at the National Book Conference,
A factory manager must decide whether to stock a particular spare part. The part is absolutely essential to the operation of certain machines in the plant. Stocking the part costs $10 per day in
Another useful distribution that is based on the normal is the lognormal distribution. Among other applications, this distribution is used by environmental engineers to represent the distribution of
Refer to the discussion of eKnow in the section on the beta distribution. Find the probability that the net contribution of eKnow would be greater than $600,000. What is the probability the net
Use the probability statements that follow to find the value of the random variable. a. PB (R ≤ r | n = 12; p = 0.85) = 0.024; find r. b. PP (K > k | m = 3.2) = 0.62; find k. c. PE (T ≤ t | β =
Using the alternative-parameter method, determine the parameters of the following distributions based on the given assessments. Refer to Step 5.5 if necessary.a. Find the parameter value β for the
An exponential distribution has PE (T ≥ 5 | m) = 0.24. Find m.
A Poisson distribution has PP (X = 0 | m) = 0.175. Calculate m.
It was suggested that five is the minimum number of data points for the least likely category when constructing discrete distributions. In many cases, however, we must estimate probabilities that are
As discussed in the text, it often is possible to use a theoretical distribution as an approximation of the distribution of some sample data. It is always important, however, to check to make sure
A scientist collected the following weights (in grams) of laboratory animals:Run @RISK's distribution fitting procedure on the weights of the lab-oratory animals.a. The normal distribution is often
Explain in your own words the role that data can play in the development of models of uncertainty in decision analysis.
A plant manager is interested in developing a quality-control pro-gram for an assembly line that produces light bulbs. To do so, the manager considers the quality of the products that come from the
A retail manager in a discount store wants to establish a policy of the number of cashiers to have on hand and also when to open a new cash register. The first step in this process is to determine
An ecologist studying the breeding habits of birds sent volunteers from the local chapter of the Audubon Society into the field to count nesting sites for a particular species. Each team was to
Decision analyst Sandy Baron has taken a job with an up-and-coming consulting firm in San Francisco. As part of the move, Sandy will purchase a house in the area. There are two houses that are
Ransom Global Communications (RGC) Inc., sells communications systems to companies that have worldwide operations. Over the next year, RGC will be instituting important improvements in its
Why might a decision maker be reluctant to make subjective probability judgments when historical data are available? In what sense does the use of data still involve subjective judgments?
Suppose that an analyst for an insurance company is interested in using regression analysis to model the damage caused by hurricanes when they come ashore. The response variable is Property Damage,
Estimate the 0.65 and 0.35 fractiles of the distribution of yearly operating costs of the Big Bertha solar trash compactors, based on the CDFin Figure 10.7.Figure 10.7
Choose appropriate intervals and create a relative frequency histogram based on the annual operating costs of the Big Bertha data in Table 10.4.Table 10.4
Figure 10.36 shows two empirical CDFs of the residuals of Sales for two regressions. Estimate (roughly) the 0.20 and 0.80 fractiles of both of these empirical CDFs. What do these two intervals say
Explain in your own words how Monte Carlo simulation may be useful to a decision maker.
Modify the decision-tree model in Figure 11.13 to use the extended Swanson-Megill (ES-M) approximation for demand instead of the EPT approximation.Use your ES-M model to calculate expected profit for
Use your ES-M model to calculate the standard deviation of Profit for Order-Quantity values 600, 650, 700, 750, and 800. Compare those with the corresponding standard deviations using the extended
Your boss has asked you to work up a simulation model to examine the uncertainty regarding the success or failure of five different investment projects. He provides probabilities for the success of
A decision maker is working on a problem that requires her to study the uncertainty surrounding the payoff of an investment. There are three possible levels of payoff — $1,000, $5,000, and $10,000.
In the Monty Hall Problem, the host (Monty Hall) asks a contestant to choose one of three curtains. Behind one and only one of these curtains is a fabulous prize, such as an all-expense paid
Use simulation to solve the Birthday Problem using the data file Observed BirthDays.xls at www.cengagebrain.com. The first three rows of the data are shown in the table that follows. The dataset
Explain how the simulation process works to produce results that are helpful to a decision maker.
If you were to make Leah Sanchez’s decision of how many calendars to order, what order amount would you chose? Why?
A friend of yours has just learned about Monte Carlo simulation methods and has asked you to do a simulation of a complicated decision problem to help her make a choice. She would be happy to have
Explain why a simulation model with only discrete probability distributions produces the same results as the corresponding decision tree model even though it uses very different solution methods.
Consider the statement “In simulation models, the values for the uncertain cells are chosen randomly.” Why is this statement incorrect, strictly speaking?
Why should you not model a decision variable as a random variable with a probability distribution?
Leah Sanchez is concerned that if she orders too few calendars, customers’ disappointment in not finding a calendar might drive them to shop elsewhere, resulting in more of a loss over the long
Leah Sanchez might have a different objective than maximizing expected profit. How does her optimal order quantity change if her objective is to minimize leftovers? What if her objective is to
Explain why in decision analysis we are concerned with the expected value of information.
Look again at Problem 8.11, which concerns whether you should drop your decision analysis course. Estimate EVPI for your score in the course.
In Problem 9.33, the issue is whether or not to close the plant. If your boss knew exactly how many machines would fail during your absence; he would be able to decide what to do without the fear of
Consider the Texaco – Pennzoil example from Chapter 4.a. What is EVPI to Hugh Liedtke regarding Texaco’s reaction to a counteroffer of $5 billion? Can you explain this resultintuitively?b. The
In the Texaco – Pennzoil case, what is EVPI if Liedtke can learn both Texaco ’ s reaction and the final court decision before he makes up his mind about the current $2 billion offer? Can you
In Problem 5.8, assume that the grower’s loss incurred with the burners would be $17,500 and that the loss incurred with the sprinklers would be $27,500.a. Find EVPI for the weather conditions
In Exercise 12.4, is it necessary to assume that the events are independent? What other assumption could be made?
Calculate the EVPI for the decision shown in Figure 12.11.Figure 12.11
What is the EVPI for the decision shown in Figure 12.12? Must you perform any calculations? Can you draw any conclusions regarding the relationship between value of information and deterministic
For the decision tree in Figure 12.13, assume Chance Events E and F are independent.a. Draw the appropriate decision tree and calculate the EVPI for Chance Event E only.b. Draw the appropriate
Draw the influence diagram that corresponds to the decision tree for Exercise 12.4. How would this influence diagram be changed in order to answer parts a, b, and c in Exercise 12.4?In exercisea.
The claim was made in the chapter that information always has positive value. What do you think of this? Can you imagine any situation in which you would prefer not to have some unknown information
Consider another oil-wildcatting problem. You have mineral rights on apiece of land that you believe may have oil underground. There is only a10% chance that you will strike oil if you drill, but the
Consider the preceding oil-wildcatting problem. The basic tree is shown in Figure 12.14.Based on Figure 12.14, work through these steps:a. Change P(Strike Oil) from 0.1 to 1.0 and P(No Oil) to 0.0.
In Problem 4.16, find:a. EVPI for the cost of making the processor.b. EVPI for the cost of subcontracting the processor.c. EVPI for both uncertain events.
Perhaps at some point in your life you have taken each of the actions a – e that follow. In doing so, you exhibited “options thinking.” For each, explain how the option value arises. Additional
You are acting in a consulting capacity advising a limited partner in a real-estate partnership. The partnership has negative income because of high vacancy rates and discounts on lease rates in the
A start-up venture owns the rights to new technology for a coated stent. The company is about to begin animal trials that could lead to FDA approval of the use of the stent in humans. The company
Two options are evaluated separately in Figures 13.6 and 13.7. Do the analysis for both options together, calculate the option value for the two options together, and explain your results.Figure
Acquisition: The board of directors of Quicker com Inc. is considering an acquisition of Honest Communication Inc. (HCI) for a total price of $3 billion, to be finalized January 1, 2016. After
A hybrid automobile has two motors, gasoline and electric. It switches between motors for power, sometimes using both motors. When accelerating, both electricity and gasoline are providing power to
Ruined Mound Properties is in final negotiations for the rights to mine copper on a tract of land in Indonesia. They must decide about adding an option in their agreement with the government of
The development of a drug requires Food and Drug Administration (FDA) approval at five stages (a preclinical phase; clinical phases I, II, and III; and a final approval). A pharmaceutical company
Pierre, a book broker, is about to sign a contract to buy the rights to a book about twin teenage computer-security sleuths titled Nan and Dan vs. the Monika Worm. His offer to advance $50,000 for
Shu Mei Wang is going to start a new business to provide a geographical-information system (GIS) to be used by small businesses in selecting promotions and for other marketing applications. 2 She
The lead time for delivery of the largest offshore platforms for oil and gas drilling and collecting is now 2 years. ConExoco Petroleum is thinking about putting down a $15 million nonrefundable
A biopharma project being pursued by Hopfer Pharmaceutical has a negative NPV of − $28 million Still, Hopfer may participate in this project as a “pioneer” option, giving the company an option
Why is it important for decision makers to consider their attitudes toward risk?
Consider the following game that you have been invited to play by an acquaintance who always pays his debts. Your acquaintance will flip a fair coin. If it comes up heads, you win$2. If it comes up
Assess your utility function in three different ways. a. Use the certainty-equivalent approach to assess your utility function for wealth over a range of $100 to $20,000. b. Use the
Assess your risk tolerance (R). Now rescale your exponential utility function—the one you obtain by substituting your R value into the exponential utility function—so that U($100) = 0 and
Let us return to the Texaco-Pennzoil example from Chapter 4 and think about Liedtke’s risk attitude. Suppose that Liedtke’s utility function is given by the utility function in Table 14.5.a.
Of course, Liedtke is not operating by himself in the Texaco-Pennzoil case; he must report to a board of directors. Table 14.6 gives utility functions for three different directors. Draw graphs of
How do you think Liedtke (Problem 14.14) and the directors in Problem 14.15 will be able to reconcile their differences?Problem 14.15Of course, Liedtke is not operating by himself in the
Rescale the utility function for Director A in Problem 14.15 so that it ranges between 0 and 1. That is, find constants a and b so that when you multiply the utility function by a and then add b, the
What if Hugh Liedtke were risk-averse? Based on Figure 4.2, find a critical value for Hugh Liedtke’s risk tolerance. If his risk tolerance is small enough (very risk-averse), he would accept the $2
The idea of dominance criteria and risk aversion come together in an interesting way leading to a different kind of dominance. If two risky gambles have the same expected payoff, on what basis might
We have not given a specific definition of risk. How would you define it? Give examples of lotteries that vary in riskiness in terms of your definition of risk.
This problem is related to the ideas of dominance that we discussed in Chapters 4 and 8. Investment D in the table that follows is said to show "second-order stochastic dominance" over Investment C.
Utility functions need not relate to dollar values. Here is a problem in which we know little about five abstract outcomes. What is important, however, is that a person who does know what A to E
You have considered insuring a particular item of property (such as an expensive camera, your computer, or your Stradivarius violin), but after considering the risks and the insurance premium quoted,
An investor with assets of $10,000 has an opportunity to invest$5,000 in a venture that is equally likely to pay either $15,000 or nothing. The investor’s utility function can be described by the
A bettor with utility function U(x) = ln(x), where x is total wealth, has a choice between the following two alternatives:A Win $10,000 with probability 0.2Win $1,000 with probability 0.8B Win $3,000
Repeat Problem 14.24 with U (x) = 0.0003 x – 8.48e – x = 2775. A utility function of this form is called linear-plus-exponential, because it contains both linear (0.0003 x) and exponential terms.
Show that the linear-plus-exponential utility function in Problem 14.25 has decreasing risk aversion. Repeat Problem 14.25 With U (x) = 0.0003 x – 8.48e – x = 2775. A utility function of this
Buying and selling prices for risky investments obviously are related to CEs. This problem, however, shows that the prices depend on exactly what is owned in the first place! Suppose that Peter
We discussed decreasing and constant risk aversion. Are there other possibilities? Think about this as you work through this problem. Suppose that a person ’ s utility function for total wealth
Suppose that a decision maker has the following utility function:U (x) = 0.000156 x2 + 0.028125 x 0.265625Use this utility function to calculate risk premiums for the
Explain in your own words the idea of a certainty equivalent.
The CEO of a chemicals firm must decide whether to develop a new process that has been suggested by the research division. His decision tree is shown in Figure 14.20. There are two sources of
The year is 2040, and you are in the supercomputer business. Your firm currently produces a machine that is relatively expensive (list price $6million) and relatively slow (for supercomputers in the
Show that the value of Y that yields indifference between the two alternatives in Figure 14.12 is within about 4% of the risk tolerance R.Figure 14.12
We stated that to assess the risk tolerance value of R for the exponential utility function, we should find the greatest value of Y such that the decision maker is indifferent between receiving $0 or
In the trade-off method for assessing utilities, consider the dollar amount X1 that makes you indifferent between gambles A1 and A2, and X2 that makes you indifferent between gambles B1 and B2. Show
Explain what is meant by the term risk premium.
Explain in your own words the idea of risk tolerance. How would it apply to utility functions other than the exponential utility function?
A decision maker’s assessed risk tolerance is $1,210. Assume that this individual’s preferences can be modeled with an exponential utility function. a. Find U ($1,000), U($800), U($0), and U( −
Many firms evaluate investment projects individually on the basis of expected value and at the same time maintain diversified holdings in order to reduce risk. Does this make sense in light of our
A friend of yours, who lives in Reno, has life insurance, homeowner’s insurance, and automobile insurance and also regularly plays the quarter slot machines in the casinos. What kind of a utility
In your own words, explain why the axioms that underlie expected utility are important to decision analysis.
Even without a formal assessment process, it often is possible to learn something about an individual’s utility function just through the preferences revealed by choice behavior. Two persons, A and
Assume that you are interested in purchasing a new model of a personal computer whose reliability has not yet been perfectly established. Measure reliability in terms of the number of days in the
You are in the market for a new car. An important characteristic is the life span of the car. (Define lifespan as the number of miles driven until the car breaks down, requiring such extensive
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