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(Solution Download) 149. Let X denote the lifetime of a component, with f ( x ) and F ( x ) the pdf and cdf of X ....

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149.   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 probabil-

ity 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        f 1x 2    

1  2       1 - F1x 2

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



f 1v 2 = s2     e


v > 0

?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.    This pdf is a member of what family introduced in this chapter?

b.   If s = 20 mm (close to the value suggested in the paper), what is the probability that a dart will land within 25 mm (roughly 1 in.) of the target?

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

c.    Since r(x) = -(d/dx) ln[1 -F(x)], ln[1 - F(x)]

= -J r(x) dx. Suppose




r1x 2 = ? a a 1 - b b       0 x b

0             otherwise

so that if a component lasts b hours, it will last for- ever (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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