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
Using Equation 1.95 and the probability model parameters ( a X , b X , a Y , b Y ) given in Table
1.18 , the values in standard normal space can be computed as X 3 = 1.174, X 4 = −0.737,
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
σ X are the updated mean and standard deviation of X′. Comparing Equations
1.86 and 1.117, it is clear that Y′ remains a Johnson random variable with the distribution
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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