^{1,a}, Yan-Gang Zhao

^{2,3,b}and Zhao-Hui Lu

^{1,3,c}

^{1}School of Civil Engineering, Central South University, Changsha, China

^{a}zhangxuanyi@csu.edu.cn

^{2}Department of Architecture, Kanagawa University, Kanagawa-ku, Yokohama, Japan

^{b}Zhao@kanagawa-u.ac.jp

^{3}Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China

^{c}luzhaohui@csu.edu.cn

Probability distributions of random variables are necessary in engineering practices. However, the distributions of the random variables are usually unknown. It is in this regard that efficient approximation of the distributions is important to accurately quantify practical engineering problems. Generally, probability distributions are determined based on the mean and standard deviation of statistical data, which are not flexible enough to represent the skewness and kurtosis of statistical data. In this paper, we investigate various distribution types of the cubic normal distribution based on the first four moments of the random variables. The explicit expressions of the probability density function and cumulative distribution function of each distribution type are formulated, and the boundaries among different distribution types are presented. Practical examples are applied to demonstrate the efficiency and accuracy of the proposed distribution types in fitting probability distributions of statistical data and structural reliability assessment. We anticipate our assay to be a starting point for precise simulation of the distributions of random variables and thus fulfill more precise evaluation of practical problems.