## 1 Introduction
The reliability design problem with aleatory and epistemic uncertainties is simultaneously a multiple layer nesting optimization problem. Common gradient optimization methods cannot be implemented because of the discontinuity of epistemic uncertainties. To improve the computational efficiency and reduce the complexity of reliability design, some probabilistic representation methods for epistemic uncertainties have been proposed, and the unified uncertainty calculations of aleatory and epistemic uncertainties have been implemented [27; 28; 29]. [PERSON] [30; 31; 32] presented a likelihood-based probabilistic representation method of epistemic uncertainty for which only sparse point and/or interval data are available, and quantified the individual contributions of variability and uncertain distribution parameters using Bayesian model averaging and Bayesian hypothesis testing approaches. [PERSON] [33; 34; 35] proposed a probabilistic framework for the representation and propagation of uncertain interval variables, which were fitted to the Johnson distribution using the moment matching approach. Then, the proposed framework was applied to the robust optimization design of engineering problems. [PERSON] [36] presented the Bayesian approach and Monte Carlo method to take insufficient input data into consideration in reliability optimization design, and then determined the probability of failure probability and reliability of optimum results. [PERSON] [37] proposed equivalent evidence-based fuzzy variables to represent the interval variables, fuzzy variables, and evidence theory variables simultaneously. [PERSON] [38] employed the multi-model inference method to quantify and propagate the uncertainty created by a lack of sufficient statistical data, and applied the proposed algorithm into uncertainty analysis of the plate buckling strength. [PERSON] [39; 40] derived a complete probabilistic expression of epistemic uncertainties according to the mean, variance, skewness, and kurtosis of available statistical data, and applied the proposed cubic normal distribution in a variety of areas in reliability engineering. [PERSON] [41] and [PERSON] [42] constructed Johnson p-boxes to equivalently represent evidence variables, and converted the uncertainty analysis problem of evidence theory into two rounds of probabilistic uncertainty analysis problem. Though a series of research studies about the probabilistic representation of epistemic uncertainties have been published, there are still some limitations in this area:  
(1) In the actual engineering product, there are multiple types of epistemic uncertainties with different representation functions according to the available uncertainty information. Multiple different epistemic uncertainties cannot be represented in a unified probabilistic analysis framework. [PERSON] [37] used the evidence-based fuzzy variable to represent multiple types of epistemic uncertainties, but the variables were calculated using the non-probabilistic uncertain analysis method. (2) The representation function of epistemic uncertainties is discontinuous, so repeated extreme analysis of epistemic uncertainties at discrete subintervals should be implemented, which reduces the computational efficiency in the reliability design of engineering products. To resolve these issues effectively, a unified probabilistic uncertainty representation approach for multiple types of epistemic uncertainties is proposed based on the cubic normal distribution method. The cubic normal distribution method has been applied to the probabilistic uncertainty representation of interval variables [43] and probabilistic variables with unknown distribution types [44], which is effective in the calculation of the reliability index and reliability design [45,46]. Therefore, this article extends the application range of the cubic normal transformation method from the representation of interval variables to probabilistic representation for multiple types of epistemic uncertainties.