In particular for problems involving more than one random variable, and/or different failure criteria which are considered simultaneously, a proper scaling of the variables is important for numerical stability of the computations. By the linear transformation

(5–9)

with and , the mean and standard deviation of X, respectively, Y has mean value zero and standard deviation one. For a random vector X of possibly correlated random variables, the covariance matrix C_XX has to be decomposed to the form C_XX = LL ^T e.g. by the Cholesky algorithm. Then

(5–10)

is a random vector with zero means, unit standard deviations and zero correlations.

Many reliability algorithms are based on standardized normal random variables U. These are produced by the algorithm and have to be transformed back to the real-world variables X in order to evaluate the simulated system with the random input. For this purpose, the marginal distributions and correlations are defined and the Nataf model, as introduced in Multivariate Distributions, is established in advance. Then by applying the inverse standardization on a realization u and then the inverse Nataf transformation, the sample x in original space is obtained.