## 1 Introduction
Also stated is the problem of the statistical distribution of the performance function. In conclusion, it is reasonable and conservative to assume that it is distributed normally for safety margin (M), and lognormally for the factor of safety (FS). [PERSON] and [PERSON] [4] conduct analyses of reliability in various geotechnical problems, using FORM, PEM, FOSM and the Monte Carlo methods. In the FOSM and PEM analyses, lognormally distributed performance functions are used. Using the example of ULS and the serviceability limit state (SLS) of piles subjected to a vertical load, they tested hypotheses that the performance functions are lognormally distributed. The soil's influence on the pile was represented by bilinear springs, and the hypothesis was tested for different values of the coefficients of variations of spring stiffnesses (COVs). The hypothesis was confirmed for the ULS and the SLS, in the case where COVs \(=0.2\). For greater COVs values--0.4 for ULS and 0.5 for SLS--the hypothesis was rejected. They conclude that a possible reason for the rejection in the sensitivity of the goodness-of-fit tests to small discrepancies in the fit, particularly in the tails, and that a lognormal distribution is a reasonable assumption. [PERSON] and [PERSON] [1] conduct reliability analyses using the stability of column-supported embankments (Case 1) and stability of embankments supported on panels under the side slope (Case 2). In their analyses, they use the Taylor Series method, PEM, Simplified FORM and the Direct Integration method. In the Taylor Series and PEM methods they choose the factor of safety (FS) for the performance function, and conduct analyses for the assumption that it is normally (Taylor Series and PEM) and lognormally (Taylor Series) distributed. The results of the conducted analyses are presented in Table 1. They choose the Direct Integration as the control method, and use it to estimate the accuracy of other methods. As expected the Taylor Series and PEM give a significantly greater error when compared to Simplified FORM. A probable cause for such an error is an incorrect estimate of the statistical distribution of FS, as well as a simplification of the aforementioned methods. Considering the error size, they recommend using the Simplified FORM method exclusively to estimate the reliability in the examined cases. Although the most precise, they do not recommend the Direct Integration method due to its complexity. [PERSON] and [PERSON] [5] analyse the relation of the reliability index for ULS and SLS of augered cast-in-place piles. They estimate the reliability indexes using FGSM, with the assumption of a lognormally distributed performance function. Due to the fact that the performance function can only assume positive values, and for mathematical convenience, such an assumption is considered reasonable. [PERSON] [6] promotes the usage of reliability methods in everyday engineering practices. He presents a procedure for determining the reliability index using the Taylor Series method. For the reliability index calculation, using the retaining wall and the slope stability examples, he assumes a lognormal distribution of the performance function, without assessing the reasonability of such an assumption. He claims there is no proof that the FS is lognormally distributed, but he believes that it is a reasonable approximation. The accuracy of the Simplified FORM, PEM and Taylor Series methods in the example of sliding and baring capacity (undrained) of retaining wall is assessed by [PERSON] and [PERSON] [2]. The results obtained were compared with the results of the Monte Carlo method, which is considered the most accurate of all methods used. PEM analyses were conducted for normal and lognormal distribution of the performance function, and the Taylor Series was used for the normal distribution. The arguments for the choice of the assumptions were not elaborated, and the results are presented in Table 2.  
Table 2 shows a good agreement between the Monte Carlo method results and the results of other methods.