When you pick up a package of light bulbs in a store, you might notice a statement about the average life of a typical bulb — say, 750 hours for an incandescent bulb or 10,000 hours for a fluorescent bulb. In this chapter we investigate what this means and how we could check the average lifetime by accumulating data on the actual lifetimes of a number of bulbs. This leads us to another application of integration: We find a formula for the average lifetime in terms of an integral over all possible lifetimes.

But any length of time is a possible lifetime for a bulb. (The so-called Centennial Light has been glowing continuously for over 100 years.) This means we have to integrate from 0 to infinity! That involves finding limiting values of a process in which the upper limit of integration is a variable approaching `oo`. Limiting values play an increasingly important role in this chapter and the next, so we introduce the standard notation for them and study them for their own sake.

Having introduced a continuous probability distribution to study failure data (bulb burnouts), we move on the study several important distributions for modeling various other kinds of data. The most important of these is the normal distribution — the famous bell-shaped curve. Here the issue is not so much what the average value is — that's usually obvious from symmetry of the data — but rather how spread out the data are. We are led to measures of spread, including standard deviation. We will find that we can study all normal distributions on a standardized scale by a two-step process of centering the data at the average value and then scaling to the standard deviation as the unit of distance.

The standard normal distribution is defined by an integral that cannot be evaluated by simple formulas. This leads into Chapter 10, in which we will study approximation of functions by polynomials — for which integration is always easy.