Section Summary

In this section we have expanded on and formalized our study of expected failure times in the preceding section. In particular, we have seen that the probability of failure of a component between times `t = a` and `t = b` can be expressed as

where `f text[(] t text[)] = re^(-rt)` is the exponential density function and `F text[(] t text[)] = 1-e^(-rt)` is the exponential distribution function. Since every component fails eventually, we are led to the concept of improper integral and, in particular, the observation that

We saw that the expected or average time for failure of a randomly selected component of a given type could be expressed as

and we interpreted this result geometrically as both a moment and a center-of-mass coordinate of an infinitely long region in the plane.

In preparation for the next section, we introduced left-hand and two-sided improper integrals to complement the right-hand integrals involved in the study of failure times. Because of the frequent appearance here of limiting values as an independent variable becomes large, we introduced the standard notation for such limiting values. At the end of this chapter we will take up variations of this notation for other limiting values, some of which will play an important role in the next chapter.