Geoscience Reference
In-Depth Information
Figure 9.4
Examples of the beta distribution, all with mean 0.5. The standard deviations proceed-
ing from the tallest curve to the lowest curve at x= 0.5 are 0.05 to 0.3 in increments of 0.05.
mean of the distribution is given by
a
/(
a
+
b
) and the variance as
ab
/[(
a
+
b
1). Most
random number generation techniques for the beta distribution require you to
specify values for
)
2
(
a
+
b
+ 1)], with the mode (
a
- 1)/(
a
+
b
- 2) (mode only for
a ³
a
and
b
. For a given mean (
m
) and variance (
s
2
) or standard
deviation (
s
),
m
2
(
m
1
)
}
-
a
=
-
m
}
s
2
and
s
2
+
m
m
m
- 1)
}}}
s
[
(
- 1)](
b
=
2
However, the amount of variation possible is limited because the distribution
is bounded on the [0, 1] interval. Thus for a mean of 0.5, the maximum vari-
ance approaches 0.25 as
approach zero.
The standard deviations of the birth and death rates over time affect per-
a
and
b
