**6. Final remarks**

The application of the method of moments is carried out in a similar way as in the first case study. Considering that the top event is also log-normally distributed, its 5th percentile, median, and 95th percentile are estimated. As can be seen in **Table 5**, the median values of the method of moments show a good agreement with Wilks method and are 25.8% and 20.4% greater than the results of Monte Carlo simulation and FW method, respectively. This is also illustrated in **Figure 7**, where the cumulative distribution function obtained by method of moments is compared with the data in the mentioned literature [26]. As can be seen, the results of the method of moments agree reasonably with the Wilks method, being slightly lower, moving toward the analyses of uncertainty propagation using Monte Carlo simulation, which is considered for

Overall, uncertainty propagation using the method of moments in fault trees, as shown in the two case studies, or in event trees, is quite simple in small systems and does not require the specification of probability density functions of basic events but only their means and standard deviations. For more complex systems and large fault and event trees, computer implementation of the described bottom-up approach can be performed, for instance, using specialized computer software for

**Median 95th**

**percentile**

**% difference of median of method of moments**

many purposes to be close to the exact solution for simple models.

*Reliability and Maintenance - An Overview of Cases*

obtaining the minimal cut sets and quantitatively assessing the top event

Monte Carlo simulation<sup>1</sup> 8.80 <sup>10</sup><sup>11</sup> 1.55 <sup>10</sup><sup>9</sup> 2.10 <sup>10</sup><sup>8</sup> 25.8% Fenton-Wilkinson<sup>1</sup> 1.26 <sup>10</sup><sup>10</sup> 1.62 <sup>10</sup><sup>9</sup> 2.08 <sup>10</sup><sup>8</sup> 20.4% Wilks1 1.85 <sup>10</sup><sup>10</sup> 1.95 <sup>10</sup><sup>9</sup> 2.46 <sup>10</sup><sup>8</sup> 0.0% Moments<sup>2</sup> 1.65 <sup>10</sup><sup>10</sup> 1.95 <sup>10</sup><sup>9</sup> 2.31 <sup>10</sup><sup>8</sup> —

*Comparison of core melt frequency obtained by the method of moments with data from literature.*

*Comparison of cumulative distribution function for core melt frequency obtained by the method of moments*

**percentile**

**Method 5th**

*1 Ref. [26]. 2*

*Current work.*

**Table 5.**

**Figure 7.**

**138**

*with data from literature [26].*

This work addresses the uncertainty propagation in fault and event trees in the scope of probabilistic risk assessment (PRA) of industrial facilities. Given the uncertainties of the primary input data (component reliability, system failure probabilities, or human error rates), the method of moments is proposed for the evaluation of the confidence bounds of top event probabilities of fault trees or event sequence frequencies of event trees. These types of analyses are helpful in performing a systematic PRA uncertainty treatment of risks and system reliabilities associated with complex industrial facilities, mainly in risk-based decision-making.

Two illustrative examples using the method of moments for carrying out the uncertainty propagation in fault trees are presented, and their results are compared with available analyses in literature using different uncertainty assessment approaches. The method of moments proved to be conceptually simple to be used. It confirmed findings postulated in literature, when dealing with simple and small systems. More complex systems will require the support of specialized reliability and risk assessment software, in order to implement the proposed approach.
