Answer: Let X denote the lifetime of a component, with

Chapter 4, Problem 4.130

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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: r(x) 1 f(x F ) (x) 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 r(x) (d/dx)ln[1F(x)], ln[1F(x)] r(x) dx. Suppose r(x) 1 x 0 x 0 otherwise 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?