Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

From Gaussian Distribution to Weibull Distribution Xu Jiajin α & Gao Zhentong σ Abstract- The Gaussian distribution is one of the most widely used statistical distributions, but there are a lot of data that do not conform to Gaussian distribution. For example, structural fatigue life is mostly in accordance with the Weibull distribution rather than the Gaussian distribution, and the Weibull distribution is in a sense a more general full state distribution than the Gaussian distribution. However, the biggest obstacle affecting the application of the Weibull distribution is the complexity of the Weibull distribution, especially the estimation of its three parameters is relatively difficult. In order to avoid this difficulty, people used to solve this problem by taking the logarithm to make the data appear to be more consistent with the Gaussian distribution. But in fact, this approach is problematic, because from the physical point of view, the structure of the data has changed and the physical meaning has changed, so it is not appropriate to use logarithmic Gaussian distribution to fit the original data after logarithm. The author thinks that Z.T. Gao method can give the estimation of three parameters of Weibull distribution conveniently, which lays a solid mathematical foundation for Weibull distribution to directly fit the original data. I. I ntroduction he Gaussian distribution is also commonly known as the Gaussian distribution, and it is generally known that the height, weight, and even IQ of a group of people are relatively consistent with the Gaussian distribution. However, like fatigue life of structures is often far from the Gaussian distribution and more in line with the Weibull distribution. In [1] it was pointed out that the Weibull distribution is a full state distribution, i.e., it can depict not only left-skewed and right-skewed data, but to some extent also symmetric as well as data satisfying a power law. In this sense it is more versatile than the Gaussian distribution [2], [3] and plays a very important role especially in fitting the fatigue life of structures. However, because of the difficulties encountered in determining the three parameters of the Weibull distribution, the problem was solved by taking the logarithm to make the data appear to be more in line with the Gaussian distribution. In fact, this approach is problematic. This paper points out that logging the original data is only a spatial transformation from a Author α : L&Z international leasing Co. Ltd, Canton. e-mail: xujiajin666@163.com Author σ : School of Aeromoutical Science and Engineering, Beihang University, Beijing. mathematical point of view, but from a physical point of view, it changes the structure of the data, and the physical meaning is changed, so it is not appropriate to use logarithmic Gaussian distribution to fit the original data after logarithm. To determine the three parameters of the Weibull distribution, the graphical and analytical methods [4] were previously adopted, the former being inconvenient to use and with relatively large errors; the latter involves solving a system of three joint transcendental equations, which, despite the availability of computers to do so, still has the problem of being inconsistent. This problem can now be solved relatively well by using T.Z. Gao method proposed by [1]. II. T he C haracteristics of the G aussian D istribution It is well known [4] that the so-called Gaussian distribution is a distribution in which the random variable is a PDF of X with the form, f( x )=[1/(2π) 1/2 σ]exp[-( x -μ) 2 /2σ 2 ] (1) where μ and σ 2 are the mean and variance of the Gaussian distribution, respectively. And when the mean μ = 0 and the standard deviation σ = 1 is called the standard Gaussian distribution as follows, [1/(2π) 1/2 ]exp(- x 2 /2) (2) From the definition of Gaussian distribution it is easy to see that Gaussian distribution has the following characteristics [5] : 1. Single-peaked, a distribution that is unimodal. And symmetry, with its Mode and median and mean are the same. 2. Universality, a significant proportion of random variables encountered in real life are or approximately conform to the Gaussian distribution. Even in an arbitrary distribution, in the case of a large sample, the distribution of the mean will approximate the Gaussian distribution. 3. Simplicity, i.e., only two parameters ( μ , σ 2 ) are needed to determine the shape of the entire distribution. Because the normal distribution has so many good characteristics, it has become the most studied and applied distribution. However, it is obvious that not all data conform to Gaussian distribution, and in most cases the data conform to Gaussian distribution is only T © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 1 Year 2023 ( ) I Keywords: gaussian distribution; three-parameter weibull distribution; full state distribution; safe life; Z.T. Gao (or GZT) method.