122. 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?