Suppose you work for an automotive engineering group that is responsible for modeling the length of life of brake components. While consumers know that brake pads have to be replaced every so often, repeated replacements in a short length of time will drive away customers. Your boss has asked you to help calculate the probability of replacing brake pads four times in fewer than five years (for example).
The life of a component X is often assumed to be Exp(Beta). We're interested in the life of four components, replaced one after the other.
Your group doesn't have an established Beta value for this component type, so you'll need to estimate it. Let Y =Sigma(i=1 to 4) Xi.
(a) Find the maximum likelihood estimator for Beta.
(b) You're also asked for a possible condence interval for . Find an 80% condence interval for Beta using the pivotal quantity Y/Beta.
(c) Suppose you gather a sample of size n = 20, and find that Sigma(i=1 to 20) yi = 85:6. Estimate .
(d) Use collected customer data to estimate the probability of four replacements in fewer than five years.