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4.138 The random variable xcan be adequately approximated by an exponential probability distribution with$\mathit{胃}\mathbf{=}\mathbf{2}$ . Find the probability that xassumes a value

a. More than 3 standard deviations from$\mathit{渭}$

b. Less than 2 standard deviations from$\mathit{渭}$

c. Within half a standard deviation of$\mathit{渭}$

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.

Preventative maintenance tests. The optimal schedulingofpreventative maintenance tests of some (but not all) ofnindependently operating components was developed in Reliability Engineering and System Safety(January$\mathbf{2006}$).The time (in hours) between failures of a component wasapproximated by an exponential distribution with mean$\mathit{胃}$.

a. Suppose$\mathit{胃}\mathbf{=}\mathbf{1000}$ hours. Find the probability that the time between component failures ranges between $\mathbf{1200}$and$\mathbf{1500}$hours.

b. Again, assume$\mathit{胃}\mathbf{=}\mathbf{1000}$hours. Find the probability that the time between component failures is at least$\mathbf{1200}$hours.

c. Given that the time between failures is at leastrole="math" localid="1658214710824" $\mathbf{1200}$ hours, what is the probability that the time between failures is less than$\mathbf{1500}$hours?

Maintaining pipe wall temperature. Maintaining a constant pipe wall temperature in some hot-process applications is critical. A technique that utilizes bolt-on trace elements to maintain temperature was presented in the Journal of Heat Transfer (November 2000). Without bolt-on trace elements, the pipe wall temperature of a switch condenser used to produce plastic has a uniform distribution ranging from 260掳 to 290掳F. When several bolt-on trace elements are attached to the piping, the wall temperature is uniform from 278掳 to 285掳F.

a. Ideally, the pipe wall temperature should range between 280掳 and 284掳F. What is the probability that the temperature will fall in this ideal range when no bolt-on trace elements are used? When bolt-on trace elements are attached to the pipe?

b. When the temperature is 268掳F or lower, the hot liquid plastic hardens (or plates), causing a buildup in the piping. What is the probability of plastic plating when no bolt-on trace elements are used? When bolt-on trace elements are attachedto the pipe?

Lead in metal shredder residue. On the basis of data collectedfrom metal shredders across the nation, the amount xof extractable lead in metal shredder residue has an approximateexponential distribution with mean$\mathit{胃}$= 2.5 milligramsper liter (Florida Shredder鈥檚 Association).

a. Find the probability that xis greater than 2 milligramsper liter.

b. Find the probability that xis less than 5 milligrams perliter.

Critical-part failures in NASCAR vehicles. In NASCAR races such as the Daytona 500, 43 drivers start the race; however, about 10% of the cars do not finish due to the failure of critical parts. University of Portland professors conducted a study of critical-part failures from 36 NASCAR races (The Sport Journal, Winter 2007). The researchers discovered that the time (in hours) until the first critical-part failure is exponentially distributed with a mean of .10 hours.

a. Find the probability that the time until the first critical part failure is 1 hour or more.

b. Find the probability that the time until the first critical part failure is less than 30 minutes.

Social network densities. Social networking sites for business professionals are used to promote one鈥檚 business. Each social network involves interactions (connections) between members of the network. Researchers define network density as the ratio of actual network connections to the number of possible one-to-one connections.

For example, a network with 10 members has a$$\begin{gathered} \left(10 \\ 2\right)\mathbf{=}\mathbf{45} \end{gathered}$$ total possible connections. If that network has only 5 connections, the network density is $\frac{\mathbf{5}}{\mathbf{45}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{111}$. Sociologists at the University of Michigan assumed that the density x of a social network would follow a uniform distribution between 0 and 1 (Social Networks, 2010).

a. On average, what is the density of a randomly selected social network?

b. What is the probability that the randomly selected network has a density higher than .7?

c. Consider a social network with only 2 members. Explain why the uniform model would not be a good approximation for the distribution of network density.

Boeing 787 Dreamliner. An assessment of the new Boeing 787 鈥淒reamliner鈥� airplane was published in the Journal of Case Research in Business and Economics (December 2014). Boeing planned to build and deliver 3,500 of these aircraft within the first 20 years of the 787 program. The researchers used simulation to estimate the number of annual deliveries of 787 airplanes (expressed as a percentage of total Boeing aircraft deliveries during the year). In the fifth year of the program, deliveries were uniformly distributed over the range 2.5% to 5.5%. Assuming the simulated results are accurate, what is the probability that the number of Boeing 787s built and delivered in the fifth year of the program exceeds 4% of the total Boeing aircraft deliveries during the year?

Soft-drink dispenser. The manager of a local soft-drink bottling company believes that when a new beverage dispensing machine is set to dispense 7 ounces, it in fact dispenses an amount at random anywhere between 6.5and 7.5 ounces inclusive. Suppose has a uniform probability

distribution.

a.Is the amount dispensed by the beverage machine a discreteor a continuous random variable? Explain.

b. Graph the frequency function for$\mathit{X}$ , the amount of beverage the manager believes is dispensed by the new machine when it is set to dispense 7 ounces.

c. Find the mean and standard deviation for the distribution graphed in part b, and locate the mean and theinterval $\mathbf{渭}\mathbf{卤}\mathbf{2}\mathbf{蟽}$on the graph.

d. Find $\mathbf{P}(\mathbf{x}鈮�\mathbf{7})$.

e. Find$\mathbf{P}(\mathbf{x}\mathbf{<}\mathbf{6})$ .

f. Find$\mathbf{P}(\mathbf{6}\mathbf{.}\mathbf{5}鈮�\mathbf{x}鈮�\mathbf{7}\mathbf{.}\mathbf{25})$ .

g. What is the probability that each of the next six bottles filled by the new machine will contain more than7.25 ounces of beverage? Assume that the amount of beverage dispensed in one bottle is independent of the amount dispensed in another bottle.

Explain why each of the following is or is not a valid probability distribution for a discrete random variable x:

a.

b.

c.

d.