NON-NORMAL VARIABLES

Decoding UPC Code In Visual Studio .NETUsing Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.

uncorrelated variables. The remainder of the calculation proceeds as in the uncorrelated case, using the new variables. When the iterations have converged, the x variables are back-calculated.

GS1 - 12 Printer In .NETUsing Barcode encoder for .NET Control to generate, create UCC - 12 image in Visual Studio .NET applications.

16.5.3 Low and Tang s Approach

Scan UCC - 12 In .NET FrameworkUsing Barcode decoder for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.

Low and Tang (1997a) proposed a very ef cient procedure that takes advantage of the optimization and solution techniques that are available in modern spreadsheets and mathematical software packages. They work directly with the correlated variables themselves without any rotations or transformations. If x is the vector of uncertain variables and C is their covariance matrix (not the correlation matrix) and is the vector of their means, they show that the Hasofer Lind reliability index is = min (x )T C 1 (x ) (16.56)

Barcode Generator In .NET FrameworkUsing Barcode generation for .NET Control to generate, create bar code image in .NET framework applications.

subject to the constraint that x must satisfy the failure criterion that M = 0. The procedure is as follows: 1. Values of the means and covariance matrix are de ned and C 1 is calculated. 2. The functional form of the following equation is entered 2 = (x )T C 1 (x ) (16.57)

Decoding Barcode In VS .NETUsing Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications.

3. The failure criterion (M = 0) is expressed as a constraint in terms of the variables in x. 4. The minimize or solve command (depending the spreadsheet or mathematical software used) is invoked to minimize 2 by changing the values of x subject to the constraint that the failure criterion is satis ed. 5. The result consists of values of the terms in x at the failure point and a corresponding value of 2 . Computing and the probability of failure follows easily. This technique is extremely easy to use. It can deal with correlated as well as uncorrelated variables. Low and Tang also show how distributions other than Normal can be employed.

Drawing Universal Product Code Version A In C#.NETUsing Barcode drawer for .NET Control to generate, create UPC-A image in .NET framework applications.

16.6 Non-normal Variables

UPC-A Supplement 5 Drawer In .NETUsing Barcode drawer for ASP.NET Control to generate, create UPC-A Supplement 5 image in ASP.NET applications.

The Hasofer Lind approach is based on independent, Normal variables. We have shown how the procedure can be modi ed when the variables are correlated. When some of the variables are logNormally distributed, the problem can be reformulated in terms of the logarithms of those variables, which are Normally distributed. The mathematical manipulations will become more complicated, but the basic procedures will be those described in this chapter. For other distributions there is a procedure known as the Rosenblatt transformation (Rosenblatt 1952), which is also described by Ang and Tang (1990). This presentation follows Ang and Tang s. The transformation considers a set of n random variables

UPC-A Supplement 5 Drawer In VB.NETUsing Barcode encoder for Visual Studio .NET Control to generate, create GTIN - 12 image in VS .NET applications.

THE HASOFER LIND APPROACH (FORM)

Data Matrix Printer In VS .NETUsing Barcode printer for .NET framework Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications.

(X1 , X2 , . . . , Xn ). If their joint CDF is F (x1 , x2 , . . . , xn ) and CDF, then we can develop a set of relations: (u1 ) = F1 (x1 ) (u2 ) = F2 (x2 |x1 ) . . . (un ) = Fn (xn |x1 , . . . , xn 1 )

Make EAN13 In .NET FrameworkUsing Barcode maker for .NET framework Control to generate, create EAN 13 image in VS .NET applications.

(u) is the standard normal

Code 39 Full ASCII Creator In Visual Studio .NETUsing Barcode encoder for Visual Studio .NET Control to generate, create Code-39 image in .NET applications.

(16.58)

Making International Standard Book Number In .NET FrameworkUsing Barcode printer for Visual Studio .NET Control to generate, create International Standard Book Number image in .NET framework applications.

In words, this means that the u s are scaled so that their standard Normal CDFs correspond to the conditional CDFs of the x s. Equation (16.58) implies that u1 = u2 = . . . un =

Bar Code Maker In .NET FrameworkUsing Barcode drawer for ASP.NET Control to generate, create bar code image in ASP.NET applications.

1 1 1

UCC.EAN - 128 Drawer In JavaUsing Barcode generator for Java Control to generate, create UCC.EAN - 128 image in Java applications.

[F1 (x1 )] [F2 (x2 |x1 )]

GTIN - 13 Reader In VS .NETUsing Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications.

[Fn (xn |x1 , . . . , xn )]

Making EAN 13 In .NETUsing Barcode printer for ASP.NET Control to generate, create EAN-13 image in ASP.NET applications.

(16.59)

Drawing Code 128B In Visual Basic .NETUsing Barcode encoder for .NET Control to generate, create Code 128 image in .NET applications.

Figure 16.8 shows how this works in the case of one variable. On the right is the CDF of a function x that has an arbitrary distribution. For each value of x, there corresponds a value of the CDF. On the left is the CDF of the standard Normal variable, and for the particular value of the CDF there corresponds a value u. In most cases the failure functions and its derivatives are expressed in terms of the x s, but the analysis is driven by the u s. During the analysis one moves from a set of the u s to corresponding values of the x s by the following procedure: 1. 2. 3. 4. 5. For all the ui and nd (ui ). 1 Find x1 = F1 [ (u1 )]. 1 Find F2 [ (u2 )] and from the result and the value of x1 calculate x2 . 1 Find F3 [ (u3 )] and from the result and the values of x1 and x2 calculate x3 . Repeat for the rest of the variables.

Code 128 Code Set B Encoder In VS .NETUsing Barcode drawer for ASP.NET Control to generate, create Code 128B image in ASP.NET applications.

Painting ANSI/AIM Code 39 In JavaUsing Barcode creator for Java Control to generate, create Code 3/9 image in Java applications.