follow the exponential distribution, and vice versa. Figure 24-6 shows the shape of two different

exponential distributions.

The Microsoft

makes exponentially distributed random

numbers with mean

.

© John Wiley & Sons, Inc.

FIGURE 24-6: The exponential distribution.

The Weibull Distribution

This distribution describes failure times for devices (such as light bulbs), where the failure rate can be

constant, or can change over time depending on the shape parameter, k. It is also used in human

survival analysis, where failure is an outcome (such as death). In the Weibull distribution, the failure

rate is proportional to time raised to the

power, as shown in Figure 24-7a.

If

, the failure rate has a lot of early failures, but these are reduced over time.

If

, the failure rate is constant over time, following an exponential distribution.

If

, the failure rate increases over time as items wear out.

Figure 24-7b shows the corresponding cumulative survival curves.

The Weibull distribution shown in Figure 24-7 leads to survival curves of the form Survival

, which are widely used in industrial statistics. But survival methods that don’t assume a

distribution for the survival curve are more common in biostatistics (we cover examples in Chapters

21, 22, and 23).