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 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 α and β, 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 r(x) = - (d/dx)ln[1 2 F(x)], ln[1 - F(x)] = - ∫r(x)dx. Suppose

so that if a component lasts β 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?

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r(x) =-F(x) flr) 1 - F(r) r(x) 0 otherwise