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 (x, x + ?x) is approximately f(x) ? ?x. The conditional probability that it fails in (x, x + ?x) given that it has lasted at least x is f(x) ? ?x/[1  F(x)]. Dividing this by ?x produces the

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 a and b, what is r(x)? For what parameter values will r(x) be increasing? For what parameter values will r(x) decrease with x?

so that if a component lasts b 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?