魅影直播

Stock market. Give an example of a continuous random variable that would be of interest to a stockbroker.

Buy-side vs. sell-side analysts鈥� earnings forecasts. Financial analysts who make forecasts of stock prices are categorized as either 鈥渂uy-side鈥� analysts or 鈥渟ell-side鈥� analysts. Refer to the Financial Analysts Journal (July/August 2008) comparison of earnings forecasts of buy-side and sell-side analysts, Exercise 2.86 (p. 112). The mean and standard deviation of forecast errors for both types of analysts are reproduced in the table. Assume that the distribution of forecast errors are approximately normally distributed.

a. Find the probability that a buy-side analyst has a forecast error of +2.00 or higher.

b. Find the probability that a sell-side analyst has a forecast error of +2.00 or higher

Box plots and the standard normal distribution. What relationship exists between the standard normal distribution and the box-plot methodology (Section 2.8) for describing distributions of data using quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal.

a. Calculate the values of the standard normal random variable z, call them z_L and z_U, that correspond to the hinges of the box plot鈥攖hat is, the lower and upper quartiles, Q_L and Q_U鈥攐f the probability distribution.

b. Calculate the zvalues that correspond to the inner fences of the box plot for a normal probability distribution.

c. Calculate the zvalues that correspond to the outer fences of the box plot for a normal probability distribution.

d. What is the probability that observation lies beyond the inner fences of a normal probability distribution? The outer fences?

e. Can you better understand why the inner and outer fences of a box plot are used to detect outliers in a distribution? Explain.

If a population data set is normally distributed, what isthe proportion of measurements you would expect to fallwithin the following intervals?

$$\begin{gathered} \mathbf{a}\mathbf{.}\mathbf{}\mathbf{渭}\mathbf{卤}\mathbf{蟽} \\ \mathbf{b}\mathbf{.}\mathbf{}\mathbf{渭}\mathbf{卤}\mathbf{2}\mathbf{蟽} \\ \mathbf{c}\mathbf{.}\mathbf{}\mathbf{渭}\mathbf{卤}\mathbf{3}\mathbf{蟽} \end{gathered}$$

Normal probability plots for three data sets are shown on the next page. Which plot indicates that the data are approximately normally distributed?

Shear strength of rock fractures. Understanding the characteristics

of rock masses, especially the nature of the fracturesis essential when building dams and power plants.The shear strength of rock fractures was investigated inEngineering Geology(May 12, 2010). The Joint RoughnessCoefficient (JRC) was used to measure shear strength.Civil engineers collected JRC data for over 750 rock fractures.The results (simulated from information provided in the article) are summarized in the accompanying SPSShistogram. Should the engineers use the normal probabilitydistribution to model the behavior of shear strength forrock fractures? Explain

4.133 Suppose xis a random variable best described by a uniform

probability distribution with c= 20 and d= 45.

a. Find f(x)

b. Find the mean and standard deviation of x.

c. Graph f (x) and locate $\mathit{渭}$and the interval $\mathit{渭}\mathbf{卤}\mathbf{2}\mathit{蟽}$onthe graph. Note that the probability that xassumes avalue within the interval $\mathit{渭}\mathbf{卤}\mathbf{2}\mathit{蟽}$is equal to 1.

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.

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?

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.