A flexible distribution and its application in reliability engineering

Probability distributions of random variables are necessary for reliability evaluation. Generally, probability distributions are determined using one or two parameters evaluated from the mean and standard deviation of statistical data. However, these distributions are not sufficiently flexible to represent the skewness and kurtosis of data. This study therefore proposes a probability distribution based on the cubic normal transformation, whose parameters are determined using the skewness and kurtosis, as well as the mean and standard deviation of available data. This distribution is categorized into six different types based on different combinations of skewness and kurtosis. The boundaries of each type are identified, and the completeness of each type is proved. The cubic normal distribution is demonstrated to provide significant flexibility, and its applicable range covers a large area in the skewness kurtosis plane, thus enabling it to approximate well-known distributions. The distribution is then applied in reliability engineering: simulating distributions of statistical data, calculating fourth-moment reliability index, finding optimal inspection intervals for condition-based maintenance system, and assessing the influence of input uncertainties on the whole output of a system. Several examples are presented to demonstrate the accuracy and efficacy of the distribution in the above-mentioned reliability engineering practices.