## 4 Fitting Cubic Normal Distribution of Epistemic Uncertainties
### Fit Continuous Probability Density Function  
Several approaches of constructing continuous Johnson distribution functions have been proposed to fit interval variables according to four statistical moments or percentiles. However, additional parameters \(\gamma\) and \(\delta\) must be optimized in the construction of every Johnson distribution under every sampling point of four statistical moments, which increases the calculation complexity. Therefore, a new probabilistic approach for simultaneous representation of interval variables, triangular fuzzy variables, and evidence theory variables is developed using cubic normal distributions, which do not require the calculation of additional parameters. The detailed procedure is shown as follows. **Step 1:** The uncertain design interval of epistemic uncertainty \(x\) is determined according to its non-probabilistic characterization method. (1) \(x\) is represented using the interval approach. If the available information is single interval \([\underline{x},\overline{x}]\), the uncertain design interval \([x_{lower},x_{upper}]\) of \(x\) is also a single interval \([\underline{x},\overline{x}]\). While the available information is presented in multiple intervals \([a_{i},b_{i}],i=1,\cdots,m\), the lower bound \(x_{lower}=\min[a_{1},\cdots,a_{m}]\) is the minimum value of multiple lower bounds of the available intervals, and the upper bound \(x_{upper}=\max\{b_{1},\cdots,b_{m}\}\) is the maximum value of multiple upper bounds of the available intervals. (2) \(x\) is represented using the triangular fuzzy approach. The lower bound \(x_{\alpha=0}^{L}\) and upper bound \(x_{\alpha=0}^{L}\) at \(\alpha=0\) construct the uncertain design interval of \(x\). (3) \(x\) is represented using evidence theory. The lower bound \(x_{lower}\) is the minimum value of the lower bounds of multiple subintervals, and the upper bound \(x_{upper}\) is the maximum value of the upper bounds of multiple subintervals. **Step 2:** The sampling points of epistemic uncertainty \(x\) are available randomly in its uncertain design interval \([x_{lower},x_{upper}]\) according to its uncertainty type. (1) \(x\) is represented by using the interval approach. If \(x\) is represented with a single interval, random sampling points are available in the single interval. While \(x\) is represented with multiple intervals \([a_{i},b_{i}](i=1,\cdots,m)\), the same numbers of sampling points are obtained using the uniform sampling method in every subinterval \([a_{i},b_{i}]\). (2) \(x\) is represented by using the triangular fuzzy approach. For the given \(\alpha\) value, the uncertain design interval is decomposed into two subintervals \(\left[\underline{x}_{\alpha}^{L},b\right]\) and \(\left[b,x_{\alpha}^{L}\right]\), and the same number of sampling points are collected randomly in the two uncertainty subintervals. (3) \(x\) is represented by using evidence theory. The same number of sampling points is obtained in every subinterval, and then the total probability of every subinterval is set to be its basic probability assignment. **Step 3:** The four statistical moments \(M_{1}\sim M_{4}\) are calculated based on the sampling points in Step 2 using the methods in Section 3 for all types of epistemic uncertainties. Figure 4: Representation functions of epistemic variables (**a**) probability density function (PDF) (**b**) cumulative CDF. **Step 4:** The continuous cubic normal distribution variable \(u\) is calculated using the transformation method presented in Section 4.1. For every sampling value of \(M_{1}\sim M_{4}\), the standardized variable \(x_{s}\) for \(x\) is calculated to be \(x_{s}=(x-M_{1})/\sqrt{M_{2}}\); the values of \(p\), \(a_{4}\), and \(M_{3}\) are calculated; and then the uncertainty type of \(u\) is chosen from type I to VI. The PDF of \(u\) is calculated, and the PDF of \(x\) is calculated using Equation (19).