Geoscience Reference
In-Depth Information
F(x)
0
.0
1
0
.
1
0
.2
0
.3 0.4
0
.5
0
.6
0.
7
0.
8
0
.
9
0
.9
9
1000
x
(m
3
s
−
1
)
9
5
2
1
4
3
100
10
−
2.5
−
2
−
1.5
−
1
−
0.5
0
0.5
1
1.5
2
2.5
y=[ln(x)
−μ
n
]/
σ
n
Fig. 13.9 Estimates of the probability distribution of the annual maxima of the rate of flow of the Cayuga Inlet
near Ithaca, NY, plotted on lognormal probability paper. The heavy straight line 1 represents the
lognormal distribution, which was calculated with the first two sample moments of the logarithms
M
=
3
.
557, and
S
=
0
.
6793. Also shown are the generalized log-gamma distribution (dashed line 2
curving upward), the first asymptote for largest values (thin solid line 3 curving downward), the
generalized extreme value distribution (dashed line 4 curving downward) and the power distribution
(solid curve 5). Both the
y
-scale and the
F
(
x
)-scale are shown. (See Example 13.6.)
13.4.3
The generalized gamma distribution
This distribution, which is often referred to as the Pearson Type III distribution, is a
generalized form of the incomplete gamma function, by the inclusion of a lower bound
c
. Its density can be written as follows
x
a
−
1
exp
−
1
−
c
(
x
−
c
)
f
(
x
)
=
(13.43)
b
(
a
)
b
b
Except for the shifted origin, this has the same form as Equation (12.41). The three
parameters can be related to the first three moments by
c
a
0
.
5
,
b
a
−
0
.
5
, and
=
μ
−
σ
=
σ
C
s
, with
a
a
c
;if
c
is known, only the first two of these
are needed (cf. Equations (12.42) and (12.43)). Once the parameters are known, the
distribution function, which is the integral of (13.43), can be obtained from tables of
the incomplete gamma function (Abramowitz and Stegun, 1964) applied to the variable
(
x
=
4
/
>
0, and
b
>
0 when
x
>
c
). An alternative method determines the probability function from its quantiles
x
p
, which in turn are obtained from the tabulated quantiles of the reduced variable
y
p
=
−
(
x
p
−
μ
)
/σ
; Table 13.2 shows these quantiles as functions of the skew coeffi-
cient, i.e.
y
p
=
y
p
(
C
s
). Note that Table 13.2 yields the values of the normal distribution
