## 5 Numerical and Engineering Examples
### Numerical Example 2  
The mathematical example presented in Equation (20) was used to demonstrate the effectiveness of the proposed algorithm in the uncertainty propagation analysis problem:  
\[y=x_{1}+x_{2}+x_{3}, \tag{20}\]  
where \(x_{1}\in[2,5]\) is a single interval variable, \(x_{2}\) is a triangular fuzzy variable that is represented within \([6,10,11]\), and \(x_{3}\) is represented using evidence theory, where the subintervals are \([10,15]\), \([13,18]\), and \([17,20]\), and the corresponding BPAs are 0.3, 0.5, and 0.2, respectively.  
The bounds of four statistical moments \(M_{1}\sim M_{4}\) for \(x_{1}\sim x_{3}\) were calculated and are listed in Table 3. The probabilistic CDF bounds of \(x_{1}\sim x_{3}\) were calculated using the proposed algorithm in Section 4.2, and the results are shown in Figure 8.  
The performance function \(y\) in Equation (20) monotonically increases according to the increases of \(x_{1}\), \(x_{2}\), and \(x_{3}\), respectively. Therefore, the upper bound of the CDF of \(y\) was calculated directly using the probabilistic uncertainty propagation analysis method with the upper bound of the CDFs of \(x_{1}\), \(x_{2}\), and \(x_{3}\). The lower bound of the CDF of \(y\) was calculated using the probabilistic uncertainty propagation analysis method with the lower bounds of the CDFs of \(x_{1}\), \(x_{2}\), and \(x_{3}\). The response CDF bounds of \(y\) calculated using the proposed unified probabilistic representation approach are shown by dashed lines in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(I\)-th Order & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** \\ \hline \(l=1\) & \([10,20]\) & \([14.33,16.33]\) & \([11.50,16.50]\) & \([14.51,15.59]\) & \([14.35,15.98]\) \\ \(l=2\) & \([0,25]\) & \([6.48,16.53]\) & \([0.75,8.78]\) & \([4.03,9.04]\) & \([3.83,9.25]\) \\ \(l=3\) & \([-96.70,96.10]\) & \([-18.22,19.28]\) & \([-4.82,21.19]\) & \([-11.94,6.88]\) & \([-17.75,10.22]\) \\ \(l=4\) & \([0,835.87]\) & \([63.06,409.98]\) & \([1.01,156.05]\) & \([43.29,161.35]\) & \([33.60,158.52]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Four statistical moments of epistemic uncertainties in example 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(I\)**-th Order & \(x_{1}\) & \(x_{2}\) & \(x_{3}\) \\ \hline \(l=1\) & \([2,5]\) & \([10.50,12.50]\) & \([14.35,15.98]\) \\ \(l=2\) & \([0,2.25]\) & \([2.08,6.75]\) & \([3.83,9.25]\) \\ \(l=3\) & \([-2.60,2.60]\) & \([-7.14,0]\) & \([-17.75,10.22]\) \\ \(l=4\) & \([0,6.76]\) & \([7.81,82.01]\) & \([33.60,158.52]\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Four statistical moments of epistemic uncertainties in example 2.  
Figure 7: Cumulative probability density function comparison of epistemic uncertainties with different types of uncertainty information.  
To verify the effectiveness of the proposed algorithm, the MCS method was also used to solve the uncertainty propagation problem. The results are shown by a series of solid lines in Figure 9. It can be observed that the CDF bounds calculated using the proposed algorithm cover the CDFs calculated with the MCS methods, which demonstrates the effectiveness of the proposed algorithm. The evaluation number of the calculating performance function \(y\) using the MCS method was \(10^{5}\)  
Figure 8: **(a)** Probabilistic representation of epistemic uncertain variables \(x_{1}\): **(b)** probabilistic representation of epistemic uncertain variables \(x_{2}\); **(c)** probabilistic representation of epistemic uncertain variables \(x_{3}\).  
Figure 9: Cumulative probability density function of response \(y\).  
while the evaluation number of the proposed algorithm was 202, which shows an improvement in the calculation effectiveness. The time taken to evaluate the calculation of the uncertainty of the performance function using the proposed algorithm and MCS method was 4.75 and 14.67 s, respectively, and the computational efficiency was improved by 67.6%.