The beta distribution allows the representation of a wide diversity of distributional shapes over the values of the variable between zero and unity. The beta probability density function is [1, p. 91]
| (17.1) |  |
where
| 0 ? x ? 1, ? > -1, ? > -1, | ||
| ? | = | shape parameter, |
| ? | = | shape parameter, |
and
| ?( n) | = | gamma function. |
Figure 17.1 shows the various shapes of the beta distribution for a variety of ? and ? values.
Some of the specific characteristics of the beta distribution are the following [1, p. 91]:
When both parameters ? and ? have the value of zero, it becomes the uniform distribution with f( x) = 1, for 0 ? x ? 1, and f( x) = 0 elsewhere, as shown in Fig. 17.1a.
When one parameter is zero and the other is one, it becomes a straight line with a slope of tan ? = +2 for ? = 1 and ? = 0, and a slope of tan ? = -2 for ? = 1 and ? = 0, as shown in Fig. 17.1b.
When ? > 0 and ? > 0, the distribution is single peaked with a modal value of
| (17.2) |  |
as shown in Fig. 17.1c.
When ? < 0 and ? < 0 the distribution is U-shaped, as...
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