Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the compo- nent fails in the interval (x, x + Ax) is approximately f(x) Ax The conditional probability that it fails in (x,xAx) given that it has lasted at least x is f(x) Ax/[1 F(x)] Dividing this by Ax produces the failure rate function: r(x) = f(x) 1 - 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 a and B. what is r(x)? For what parameter values will r(x) be increasing? For what parameter values will r(x) de- crease with x?

c. Since r(x) = -(d/dx)In[1 - F(x)], In[1 - F(x)] = - fr(x)dx. Suppose r(x) = 0 x B 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?