356 PART 7 The Part of Tens

The Exponential Distribution

If a set of events follows the Poisson distribution, the time intervals between con-

secutive events follow the exponential distribution, and vice versa. Figure 24-6

shows the shape of two different exponential distributions.

The Microsoft Excel statement

LN RAND

(

()) makes exponentially distributed

random numbers with mean 1.

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 out-

come (such as death). In the Weibull distribution, the failure rate is proportional

to time raised to the k – 1 power, as shown in Figure 24-7a.»

» ^{If k}

1, the failure rate has a lot of early failures, but these are reduced

over time.»

» ^{If k}

1, the failure rate is constant over time, following an exponential

distribution.»

» ^{If k}

1, 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

l

e ^Timek

, 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).

FIGURE 24-6:

The exponential

distribution.

© John Wiley & Sons, Inc.