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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
v2/2s2
v > 0
?bathtubshaped? 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
x
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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This question was answered on: Oct 24, 2017
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