a. In Exercise 72, what is the median lifetime of such tubes? [Hint: Use Expression (4.1...

a. In Exercise 72, what is the median lifetime of such tubes? [Hint: Use Expression (4.12).]

b. In Exercise 74, what is the median return time?

c. If X has a Weibull distribution with the cdf from Expression (4.12), obtain a general expression for the (100p)th percentile of the distribution.

d. In Exercise 74, the company wants to refuse to accept returns after t weeks. For what value of t will only 10% of all returns be refused?

Reference exercise 72

The lifetime X (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters α and β . Compute the following:

a. E(X) and V(X)

b. P(X ≤ 6)

c.P(1.5 ≤ X ≤ 6)

(This Weibull distribution is suggested as a model for time in service in “On the Assessment of Equipment Reliability: Trading Data Collection Costs for Precision,” J. of Engr. Manuf., 1991: 105–109.)

Expression (4. 12)

Reference exercise 74

Let X = the time (in 10^-1 weeks) from shipment of a defective product until the customer returns the product. Suppose that the minimum return time is ϒ = 3.5 and that the excess X – 3.5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1.5 see “Practical Applications of the Weibull Distribution,” Industrial Quality Control, Aug. 1964: 71–78).

a. >What is the cdf of X?

b. What are the expected return time and variance of return time? [Hint: First obtain E(X – 3.5 )and V(X – 3.5).]

c. Compute P(X > 5).

d. Compute P(5 ≤ X ≤ 8).