Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 44

(1.82)

λξ

+⋅ XYor = ln() Y = exp( λξ +⋅ X )

It is clear that Y cannot be negative, because the smallest possible value of the exponential
function is 0. It can be shown that the PDF of Y can be written as

1 − [ln( ) y − λ ]

2

fy () = exp y > 0 (1.83)

2 ⋅ ξ

2

2 πξ ⋅ ⋅ y

The lognormal distribution has zero as its lower bound. The shifted lognormal distribu-
tion generalizes the lognormal distribution to account for nonzero lower bounds. If Y is
shifted lognormally, the relationship between X and Y is

X − b

*

X

= ln( Y − b ) (1.84)

Y

a

X

where a X , b * , and b Y are the parameters for the shifted lognormal distribution. The param-
eter b Y is the lower bound of Y and it is typically determined by physics. The remaining
parameters can be determined by the method of moments:

1 µ

2

⋅
−

COV

2

1

= ln 1 + b

*

= − a ln( µ − b ) (1.85)

X X Y

2 a

a

2

( µ b )

2

X Y X

It is clear that when b Y = 0, the shifted lognormal distribution reduces to lognormal dis-
tribution with a X = 1/ξ and b X =−λξ The notation in Equation 1.84 are chosen to be

different from those in Equation 1.82 to accommodate the other members of the Johnson
system of distributions.

*

/ .

1.5.2.2 Johnson system of distributions
Phoon (2008) and Phoon and Ching (2013) highlighted that the shifted lognormal distribu-
tion is a member of a more general Johnson system, which can be expressed in the following
form following the notations presented by Slifker and Shapiro (1980):

X − b Y − b

=

X Y

= κ κ() Y (1.86)

n

a a

X Y

where Y n = (Y - b Y )/ a Y is the normalized Y. The SU member is unbounded and is defined by

( )

κ() sinh Y =

−1

() ln Y = ++ Y 1 Y

2

(1.87)

n n n n

The SB member is bounded between [ b Y , a Y + b Y ] and is defined by

Y

n

κ() ln Y = (1.88)

n

1 − Y

n