Titel_SS06

*

()*^1 ( i)

ii

ii

iX

Xi

XX

x

fx



-



 @



@

(6.32)

where X@i and Xi are the unknown mean value and standard deviation of the approximating

normal distribution.

Solving Equations (6.31) and (6.32) with respect to @Xi and Xi there is:

1*

*

*1 *

((()))

()

(())





,

@ 

@@,

i

i

i

ii

Xi

X

Xi

Xi Xi

Fx

fx

xFx

-



 Xi

(6.33)

This transformation may easily be introduced in the iterative evaluation of the reliability index
 as a final step before the basic random variables are normalised.

The Rosenblatt Transformation

If the joint probability distribution function of the random vector can be obtained in terms
of a sequence of conditional probability distribution functions e.g.:

X

Fx F xxx x F x xx xXXn()nn( 12 , ,nXn  1 )  1 ( 112 , , nX 2 )F x 1 ( ) 1 (6.34)

the transformation from the X-space to the U-space may be performed using the so-called
Rosenblatt transformation:

1

2

11

221

12 1

() ()

() ( )

() n( ,, )

X

X

nXn n

uFx

uFxx

uFxxxx

,

,

,





(6.35)

where is the number of random variables, n FxxxXi(,,,i 12  xi 1 ) is the conditional

probability distribution function for the i’th random variable given the realisations of
x 12 ,,xx i 1 and is the standard Normal probability distribution function. From the

transformation given by Equation

,()

(6.35) the basic random variables X may be expressed in
terms of standardised Normal distributed random variables U by

1

2

1

11

1

221

1

12 1

(())

(( )

n(( , , )

X

X

nX n n

xF u

xF ux

x Fuxxx









,

,

,





(6.36)

In some cases the Rosenblatt transformation cannot be applied because the required
conditional probability distribution functions cannot be provided. In such cases other
transformations may be useful such as e.g. the Nataf transformation see.g. Madsen et al.