Geoscience Reference
In-Depth Information
2) for
g
, rather than
n
and
n
2
, are introduced
to reduce the bias in these estimators (Weatherburn, 1961).
The factors (
n
−
1) for
S
, and (
n
−
1)(
n
−
13.2.2
Quantiles
By definition, the quantiles are the (
n
1) values of the random variable, which par-
tition the probability function domain (normally between 0 and 1), into
n
equal parts.
Accordingly, the
m
th quantile, say
x
, is obtained by solving the following integral for
x
−
x
x
m
n
=
f
(
y
)
dy
or
dF
(
y
)
(13.14)
−∞
−∞
The median is the most widely used quantile; it is the value of the variable where the
probability distribution equals 1
/
2, and it is often used as another measure of the central
tendency of the population, beside the mean. Thus, for a sample of
n
items, the sample
median is the value of the item, which is found in the middle after all items have been
ranked. The lower quartile of the sample is the value of
x
for which
m
/
n
=
1
/
4in
Equation (13.14), and the upper quartile that for which
m
/
n
=
3
/
4. When
n
=
100,
quantiles (times 100) are also called percentiles.
13.2.3
Return period
The reciprocal of the probability that a certain value
x
will be exceeded is referred to as
the
return period
, also called the recurrence interval or the exceedance interval, or
1
T
r
(
x
)
=
(13.15)
−
1
F
(
x
)
This return period is the expected number of observations required until
x
is exceeded
once.
This can readily be shown as follows. Let
p
(
F
(
x
)) denote the probability that, at
any trial in an experiment, the magnitude of the event will not be larger than
x
; then, the
probability, that this magnitude will not be exceeded in the first (
k
=
−
1) trials and finally
will be exceeded in the last trial, is given by
p
k
−
1
(1
P
{
k
trials until
X
>
x
}=
−
p
)
(13.16)
This probability function is known as the geometric distribution (see Section 13.3.1
below). The average number of trials is the first moment, or from Equation (13.7),
∞
m
1
=
kp
k
−
1
(1
k
=
−
p
)
(13.17)
k
=
1
p
2
Except in case of certainty, one
h
as 0
<
p
<
1, so that (1
+
p
+
+···
)
=
(1
−
p
)
−
1
; hence (13.17) reduces to
k
p
)
−
1
, which proves the statement below
=
(1
−
Equation (13.15).
Equation (13.15) shows how the return period is uniquely related to
F
(
x
). It can there-
fore be considered as an alternative to, and equivalent with, the probability distribution
