Environmental Engineering Reference
In-Depth Information
the sample l-moments to the theoretical l-moments of
a distribution and then solving for the distribution
parameters constitutes the method of l-moments.
sists of
N
measurements, such that each measurement
is given by the random variable
X
i
,
i
∈ [1,
N
], then any
parameter estimated from the sample set will depend
on the values of the measurements,
x
i
,
i
∈ [1,
N
]. There-
fore, any derived statistic will necessarily be a random
variable with a probability distribution dependent on
the number of observed outcomes and the probability
distribution of each of the outcomes,
X
i
,
i
∈ [1,
N
]. The
probability distribution of statistics derived from
random samples are called
sampling distributions
. In
most cases of practical interest, the calculated statistics
are estimates of population parameters, such as the
mean and variance.
EXAMPLE 10.12
It is estimated that the concentration data in Example
10.10 can be represented by a log-normal distribution.
use the method of l-moments to estimate the mean
and standard deviation of the log-normal distribution.
Solution
From the given data,
N
= 30 and the l-moments can be
calculated using Equations (10.69-10.76) as
10.7.1 Mean
The mean,
X
, of a sample of
N
realizations is defined
by
N
1
1
30
∑
b
=
c
i
=
(
149 1
. )
=
4 97
.
0
N
i
=
1
N
1
∑
X
=
X
i
(10.77)
N
1
1
30 30 1
∑
N
b
=
(
i
−
1
)
c
i
=
)
(
3310 6
. )
=
3 81
.
1
i
=
1
N N
(
−
1
)
(
−
i
=
2
If the probability distributions of
X
i
are identical,
with a (population) mean,
μ
x
, and variance
,
σ
2
, t
he
in it
can be shown that the expected value of
X
,
E X
L b
1
=
= .
4 97
0
=
(
)
−
L
=
2
b
−
b
2 3 81
.
4 97
.
=
2 64
.
(
)
, is
2
1
0
equal to the population mean,
μ
x
, that is
using the relationships between the distribution
parameters and the l-moments given in Table 10.5
requires that
) = µ
(10.78)
E X
(
x
and the standard deviation of
X
,
σ
X
, is related to the
standard deviation of
X
i
,
σ
x
, by
ˆ
σ
y
L
L
2 64
4 97
.
.
=
2
erf
=
=
0 531
.
2
1
σ
x
N
(10.79)
σ
=
X
which yields
ˆ
σ
y
= 1 03 mg/L
, and the parameter rela-
tionships in Table 10.5 also require that
.
The standard deviation of a sample statistic is com-
monly called the
standard error
. The results given in
Equations (10.78) an
d
(10.79) relate the properties of
the random function
X
to the properties of the random
variables
X
i
. If the population distribution
of
X
is
normal, then the sampling distribution of
X
will be
normal. If the p
op
ulation distribution is nonnormal, the
distribution of
X
will be more nearly normal than the
population distribution (Berthouex and Brown, 2002).
ˆ
σ
2
1 03
2
.
2
ˆ
y
µ
=
ln
L
−
=
ln( .
4 97
)
−
=
1 08 mg/L
.
y
1
2
Hence, based on these results, the parameters of the
log-normal distribution using the method of l-moments
are estimated as
ˆ
µ
y
= 1 08 mg/L
and
ˆ
.
σ
y
= 1 03 mg/L
.
.
10.7 PROBABILITY DISTRIBUTIONS OF
SAMPLE STATISTICS
10.7.2 Variance
The sample variance,
S
2
, of
N
realizations of the random
variables,
X
i
,
i
∈ [1,
N
] is defined by the equation
A
sample
consists of observed outcomes, while a
popu-
lation
consists of the entire set of possible outcomes.
Constants in population probability distributions are
called
parameters
, and estimates of population param-
eters that are derived from samples are called
sample
statistics
, or simply
statistics
. Suppose a sample set con-
N
1
∑
(
S
2
=
X X
−
)
2
(10.80)
x
i
N
−
1
i
=
1
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