The Imperial Electric Co. makes high-quality portable compact disc players for sale in international and domestic markets. The company operates two plants in the United States, where one plant is located in the Pacific Northwest and one is located in the South. At either plant, once a disc player is assembled, it is subjected to a stringent quality-control inspection, at which time the disc player is either approved for shipment or else sent back for adjustment before it is shipped. On any given day, the proportion of the units produced at each plant that require adjustment before shipping, and the total production of disc players at the company's two plants, are outcomes of a trivariate random variable, with the following joint PDF:

\(f(x, y, z)=\frac{2}{3}(x+y) e^{-x} I_{(0, \infty)}(x) I_{(0,1)}(y) I_{(0,1)}(z)\), where

\(x=\) total production of disc players at the two plants, measured in thousands of units,

\(y=\) proportion of the units produced at the Pacific Northwest plant that are shipped without adjustment, and \(z=\) proportion of the units produced in the southern plant that are shipped without adjustment.

a. In this application, the use of a continuous trivariate random variable to represent proportions and total production values must be viewed as only an approximation to the underlying real-world situation. Why? In the remaining parts, assume the approximation is acceptably accurate, and use the approximation to answer questions where appropriate.

b. What is the probability that less than 50 percent of the disc players produced in each plant will be shipped without adjustment and that production will be less than 1,000 units on a given day?

c. Derive the marginal PDF for the total production of disc players at the two plants. What is the probability that less than 1,000 units will be produced on a given day?

d. Derive the marginal PDF for the bivariate random variable \((Y, Z)\). What is the probability that more than 75 percent of the disc players will be shipped without adjustment from each plant?

e. Derive the conditional density function for \(X\), given that 50 percent of the disc players are shipped from the Pacific Northwest plant without adjustment. What is the probability that 1,500 disc players will be produced by the Imperial Electric Co. on a day for which 50 percent of the disc players are shipped from the Pacific Northwest plant without adjustment?

f. Answer

(e) for the case where 90 percent of the disc players are shipped from the Pacific Northwest plant without adjustment.
g. Are the random variables \((X, Y, Z)\) independent random variables?
h. Are the random variables \((Y, Z)\) independent random variables?