Environmental Engineering Reference
In-Depth Information
To illustrate the practical usefulness of a conditional distribution, the following values
measured at a depth = 3.16 m are extracted from a field study in Berthierville (Canada)
reported by Rochelle et al. (1988):
Y
3
= LI = 1.793
Yln/Pln42
=
(
σ
va
′
)
=
(.
00
)
= −
0
.
912
4
Y
8
= B
q
= 0.504
Yln
=
[(
qu
t
−
)
/
σ
′ =
]
ln 3 541
(.
)
=
1 265
.
1
0
2
v
X
8
= −0.411, and X
10
= −0.024. Hence, the conditional mean vector is
1 174
0 737
0 411
0 024
.
−
0 228
.
0 691
.
−
0 035
.
0 230
.
−
−
−
.
.
.
0 769
0 0189
−
.
[]
1
µ
update
=
=
(1.116)
−
0 152
.
−
0 419
.
0 233
.
0 634
.
.
In other words, the updated version of X
5
is a normal random variable with mean = −0.769
and standard deviation = (0.359)
1/2
= 0.599. The updated version of X
6
is a normal ran-
dom variable with mean = 0.0189 and standard deviation = (0.451)
1/2
= 0.671. Note that the
updated versions of X
5
and X
6
are no longer standard normal random variables. For brevity,
we drop the subscript and denote the updated
nonstandard
normal random variable pro-
duced by conditioning as X′. The symbol X denotes a standard normal random variable. The
updated physical random variable corresponding to X′ is Y′ and its probability distribution
can be deduced from
Equation 1.86
:
Y
′ −
b
X
′ −
b
(
σ
′
X
+ ′ −
µ
)
b
=
−−′
X
(
b
a
µσ
)
/
′
=
Y
X
X
X
X
X
X
X
κ
=
(1.117)
a
a
a
/
σ
′
Y
X
X
X
X
type unchanged, whereas the parameters are updated into
(
where ′
µ
X
and
′
σ µ σ
In
general, the distribution function for Y′ (conditioned) is not the same as the distribution
function for Y (unconditioned). Additional efforts are needed to obtain the distribution
function for Y′. For a Johnson distribution, both Y and Y′ follow the same κ(⋅) function. The
only difference is the numerical values of the model parameters and these updated model
parameters for Y′ can be calculated in closed form. This is a significant practical advantage
and explains why a Johnson distribution is recommended in Section 1.5.
On the basis of the above observation, it is clear that the updated version of Y
5
is an
SU distribution with
a
X
= 4.600/0.599 = 7. 682 ,
b
X
= (21.548 + 0.763)/0.599 = 37. 2 6 7,
a
Y
= 576.785, and
b
Y
= −4.793. The unconditioned mean and standard deviation of Y
5
are
0.61 and 1.17, respectively. The conditioned mean and standard deviation of Y
5
are −0.28
a
/
,
(
b
−
′
)/
,
ab
,
).
′
′
XX X
X
XYY
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