Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the component fails in the interval is approximately

. The conditional probability that it fails in given that it has lasted at least x is

. Dividing this by produces the failure rate function:

An increasing failure rate function indicates that older components are increasingly likely to wear out, whereas a decreasing failure rate is evidence of increasing reliability with age. In practice, a “bathtub-shaped” failure is often assumed.

a. If X is exponentially distributed, what is r(x)?

b. If X has a Weibull distribution with parameters and , what is r(x)? For what parameter values will r(x) be increasing? For what parameter values will r(x) decrease with x?

c. Since

. Suppose so that if a component lasts hours, it will last forever

(while seemingly unreasonable, this model can be used to study just “initial wearout”). What are the cdf and pdf of X?