Normal and weibull distributions

On the use of the Weibull and the normal cumulative probability models in structural design

Thanks to the performance improvement introduced by micro sized functional fillers, application of epoxy composites for electrical insulation purposes has become popular. This paper investigates the dielectric properties of epoxy micro-composites filled with alumina (Al2O3). In particular, measurements of relative permittivity, dissipation factor, and electrical breakdown are performed, and a comprehensive statistical analysis on dielectric properties was conducted. AC breakdown strength (AC-BDS) was analyzed for normal distribution using four methods (Anderson–Darling, Shapiro–Wilk, Ryan–Joiner, and Kolmogorov–Smirnov). In addition, the AC-BDS was analyzed at risk probabilities of 1%, 5%, 10%, and 50% using Weibull distribution functions. Both normal and Weibull distributions were evaluated using the Anderson–Darling (A-D) statistic and p-value. Additionally, the log-normal, gamma, and exponential distributions of AC-BDS were examined by A-D goodness-of-fit test. The hypothesis test results of AC-BDS were fit by normal and Weibull distributions, and the compliance was evaluated by p-value and each method statistics. In addition, the experimental results of AC-BDS were fit by log-normal and gamma distributions, and the goodness-of-fit was evaluated by p-value and A-D testing. On the other hand, exponential distribution was not suitable for p-value and A-D testing. The results showed that the distributions of AC-BDS were the best using log-normal distribution. Meanwhile, statistical analysis results verified the apparent effect of temperature on dielectric properties using a paired t-test. The analysis results of this paper not only contribute to better characterization of epoxy/Al2O3 micro-composites but also introduce a comprehensive approach for performing statistical analysis for electrical insulation materials.

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

Ceramic fibers, e. g., boron fibers and carbon fibers etc., are known to contain several kinds of strength-limiting defects, and in this case the statistical analysis of their strength is mainly evaluated by the theory decided due to the weakest defect among all defects, the so-called "Weibull's weakest link theory". In this paper, from the tensile-test data of two types of boron fibers, we try to analyze from the view point that tensile strength is governed by the weakest defect, which is decided by competition among the several kinds of defect groups. The resukt is that the strength distribution curves predicted by multi-modal Weibull's distribution fitted to experimental plots better than those predicted by single Weibull's distribution for every gage length tested. The application of the multi-model Weibull's distribution also showed that the analysis gives the exact expected value of the bundle strength at a certain gage length. The multi-modal Weibull's distribution function may be applied to the statistical strength anblysis for ceramic fibers accompanied by several kinds of defects.

This paper compares two commonly used functions, the Weibull and Rayleigh distribution functions, for fitting a measured wind speed probability distribution at a given location over a certain period. The monthly and annual measured wind speed data at 84 m height for the years have been statistically analyzed for the country with a large capacity - Kitka. The analysis is made in the case of the implementation of all the predicted capacity of wind turbines and by virtue of the probability of power distribution. The Weibull and Rayleigh probability distribution functions have been determined and their parameters have been identified. The average wind speed and the wind power density have been estimated using both distribution functions and compared those estimated from the measured probability distribution function. The Weibull distribution function fits the wind speed variation better than Rayleigh distribution function. The average wind speed was found to be 4.5 m/s and the average wind power density was 114.54 W/m According to results, we can conclude that such a distribution of winds in this region yields an appropriate average value of wind power.

Some Thoughts About Data Type, Distribution, and Statistical Significance

It is generally assumed that the measured strength of brittle ceramics follows a Weibull distribution. However, there seems to be few sound and direct evidences to support this assumption. Several previous studies have shown that other distributions, such as normal distribution and log-normal distribution may describe more appropriately the strength data than Weibull distribution. In this paper, the efficiency of using a normal distribution to describe the strength which follows a Weibull distribution is examined based on Monte-Carlo simulations. It was shown that there exist strong correlations between the parameters of normal distribution and those of Weibull distribution. For the designed fracture probability not lower than 0.01, analyses based on both normal distribution and Weibull distribution may give nearly identical predictions for the applicable stress levels. For lower fracture probabilities, the differences between the predictions of both distributions are not significant. It was suggested that, if there is no evidence to confirm that the measured strength follows a certain distribution, normal distribution and Weibull distribution seem to have the same efficiency in analysing the statistical variations in the measured strength of ceramics.

Hemoglobin level is a continuous variable. So, it follows some theoretical probability distribution Normal, Log-normal, Gamma and Weibull distribution having two parameters. There is low variation in observed and expected frequency of Normal distribution in bar diagram. Similarly, calculated value of chi-square test (goodness of fit) is observed which is lower in Normal distribution. Furthermore, plot of PDFof Normal distribution covers larger area of histogram than all of other distribution. Hence Normal distribution is the best fit to predict the hemoglobin level in future.

Within the fields of risk management and banking, the normality condition is one of the basic assumptions to apply value at risk, capital asset pricing or linear regression models on credit risk assessment. However, banking sector data related to loans may not be normally distributed. Hence, it needs to be put through scientific tests. For this purpose, firstly, Anderson-Darling, Jarque-Bera, Kolmogorov-Smirnov, Shapiro-Wilk, and Shapiro-Francia tests are applied to ninety-two banking sector loan variables and it is demonstrated that most of the variables are not normally distributed. Additionally, the parameters of Normal, Birnbaum-Saunders, Exponential, Extreme Value, Gamma, Generalized Extreme Value, Inverse Gaussian, Log-Logistic, Logistic, Lognormal, Nakagami, Negative Binomial, Non-Parametric, Poisson, Rayleigh, Rician, t- Location-Scale, and Weibull distributions are estimated for loan variables. Thirdly, when the data are not normally distributed, it is necessary to examine the other test results. Therefore, Kolmogorov-Smirnov, Anderson Darling, and Chi-square test results are employed for sixty-one distributions related to the variables and best fitted distribution per variable is aimed at. The results indicate that different computer codes and programs may give different outcomes in connection with the normality and best fitting distribution. Therefore, the use of different strategies may also be adopted in risk management courses along with the traditional ones since the normality assumption is an essential first step for the application of such techniques. Finally, pedagogically speaking, it should be noted that teaching the essence of mathematical background and computer codes could be strategically useful for students in internalizing these distribution concepts. Keywords: distribution, banking sector loans, risk management

Introduction and objectives The objective measurement of cough frequency has changed the standards by which we evaluate anti-tussive treatments and study the mechanisms driving cough. The distribution of individual cough events over time can vary substantially between patients with identical cough frequencies but this has received little attention. The aim of this study was to explore the differences in the temporal distribution of cough in 24 hour acoustic recordings across a range of respiratory conditions using the Weibull Distribution. Methods We studied 24 hour acoustic recordings made using the VitaloJAK™ cough monitoring system from 62 participants [healthy volunteers (n=11), chronic cough (n=15), asthma (n=10) chronic obstructive pulmonary disease (COPD) (n=16), Idiopathic Pulmonary Fibrosis (IPF) n=10)]. Using the exact location of the explosive phase of each cough sound, the lengths of inter-cough intervals were calculated in each recording and the fit to Weibull and exponential distributions assessed; the shape parameter which describes the ‘clustering’ of events was examined. Results The number of coughs per 24 recording ranged from 0 to 3133, median 158.0 coughs and was increased compared with healthy controls in all disease groups [healthy volunteers median 13.0(IQR 1.0–22.0) coughs/24 hour versus chronic cough 509.0 (296.0–916.0), asthma 35.0 (13.5–144.0) COPD 158.0 (41.0–791.5), Idiopathic Pulmonary Fibrosis 193.5 (101.3–542.8), all p Conclusions The intervals between cough events are not random and can be described by a Weibull distribution. Significant differences in the Weibull shape parameter suggest differences in the temporal clustering of coughs between different respiratory conditions and therefore may be an important novel parameter from objective cough monitoring.

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

In North America and Europe, there is a significant number of centenary metallic bridges that require maintenance and strengthening operations. These structures need to be adapted to increasing traffic intensities, and their structural integrity should be constantly evaluated. Fatigue damages were not considered in the original design of old metallic bridges, and riveted connections are one of the most frequent sources of fatigue damages on these structures. This paper intends to be a contribution for the reliable assessment of the fatigue behavior of riveted connections by proposing S-N curves for these structural details. Experimental fatigue data is analyzed from multiple bridges across Europe, and different statistical methods were implemented. A normal distribution function was implemented following the ISO 12107 standard, and the results were compared with the implementation of the two-parameter Weibull distribution function. Different estimation methodologies were implemented to determine the parameters of the Weibull distribution. S-N curves obtained by the statistical analysis were then compared with design recommendations from North American and European standards.

<div class="htmlview paragraph">The failure time of automobile and its components is typical random variable. In studying reliability automobile and its components some simple theoretical statistical models, such as Normal distribution, Lognormal distribution and Weibull's distribution, index distribution etc., are used to describe automobile reliability. These distributions can not often represent actual datum well, then more complex theoretical statistical model is needed. If using complex statistical theoretical model the more parameters must be determined. Methods of moment, Maximum likelihood method can not estimate the parameters of complex statistical models yet. With graph method though the type of random variable distribution can be recognized, the precision of estimated parameters is not high. Meanwhile there is not an objectively criterion to judge statistical model reasonability.</div> <div class="htmlview paragraph">Based on the adaptability of Weibull's distribution, multiple-mixed Weibull's distribution suits to describe the complex statistical distribution of random variable. To estimate the parameters of multiple Weibull's distribution correctly, optimization model for estimating parameters of multiple-mixed Weibull's distribution is presented, according to the principle of least-square.</div> <div class="htmlview paragraph">In order to obtain the optimal estimation, the Levebberg-Marquardt method with line-search and Gauss-Newton method are alternatively used. A safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation are chosen as algorithm of line-search. The example shows that multiple Weibull's distribution has quite good adaptability in describing the complex statistical distribution of random variable. The model and algorithm of optimization for estimating multiple Weibull's distribution parameters have steady convergence.</div>

In general, distribution theory plays a crucial role in modeling various real-life phenomena, making it a fundamental tool in statistical analysis and decision-making. Over the years, extensive research has been conducted on different statistical distributions and estimation techniques. While the literature abounds with information regarding well-known distributions, there is always room for exploring new variants that can better capture the characteristics of complex phenomena. In this paper, we contribute to the field of distribution theory by introducing novel probability distribution called the Weibull-Weibull distribution. The Weibull-Weibull distribution is derived by compounding two Weibull distributions, and it offers a flexible framework for modeling phenomena that exhibits a complex interplay of factors. By combining the strength of the Weibull distribution with itself, we are able to capture a wider range of shapes and behaviour, providing more accurate representations of real-world occurrences. To facilitate the practical application of the Weibull-Weibull distribution, we employ the maximum likelihood estimation (MLE) approach to estimate its shape and scale parameters. The MLE method is a widely used statistical technique that allows us to determine the most likely values of the parameters based on observed data. By applying this estimation method to the Weibull-Weibull distribution, we enable researchers and practitioners to effectively utilize this new distribution in their analyses and modeling efforts. Furthermore, we delved into a comprehensive study of statistical theory and properties of the Weibull-Weibull distribution. We investigate its moments, cumulative distribution function, probability density function, and other key measures. Through rigorous analysis, we establish the theoretical foundations of the Weibull-Weibull distribution and provide insights into its behaviour and characteristics. This comprehensive examination equips researchers with a solid understanding of the distribution, enabling them make informed decisions and interpretations when working with real-life data. In conclusion, our research introduces the Weibull-Weibull distribution as a valuable addition to the existing repertoire of statistical distributions. By leveraging the power of compounding two Weibull distributions, we provide a flexible and robust framework of modeling complex phenomena. With the use of maximum likelihood estimation and the given Chernoff bound, practitioners are able to estimate the distribution's parameters as well as analyze its tails accurately. Our extensive statistical analysis further enhances the understanding of the Weibull-Weibull distribution, facilitating its practical application in a wide range of fields, including reliability analysis, survival analysis, and risk assessment.

Quantifying inequality in the leaf area distribution within a single module is critical for elucidating plant resource allocation strategies, but the accuracy of theoretical Gini coefficients derived from statistical distributions remains poorly validated against observed values. To resolve this gap, we analyzed 9,242 leaves from 121 culms of the bamboo Semiarundinaria densiflora, a model system with minimal ontogenetic noise and moderate leaf counts (36–187 leaves per culm) that enables robust Lorenz curve construction. Four candidate distributions were tested: the normal, log-normal, two-parameter Gamma, and two-parameter Weibull distributions. The parameters of the normal and log-normal distributions were estimated directly from sample statistics, whereas the parameters of the Gamma and Weibull distributions were estimated using the maximum likelihood method. Goodness of fit was assessed using the Kolmogorov-Smirnov (K-S) test for distributional validity, and the Akaike’s information criterion (AIC) for model selection. Although the Gamma distribution passed the K-S test for a slightly higher percentage of culms (99%) than the Weibull distribution (97.5%), the Weibull distribution was selected as the superior model because it yielded significantly lower AIC values. Crucially, the theoretical Gini coefficients of the Gamma and Weibull distributions (denoted as GG and GW, respectively) were tested against the observed Gini coefficients (GP) calculated nonparametrically using the polygon method. Linear regression demonstrated that GW​ predicted GP with near isometric accuracy: the intercept’s 95% confidence interval included zero (−0.006 to 0.017) and the slope’s 95% confidence interval included unity (0.929 to 1.039). In contrast, GG exhibited significant bias. Notably, pooling leaves across culms violated all distributions due to microhabitat driven multimodality, confirming that intra-culm inequality assessments require organism level analysis. This work provides an empirical validation that the Weibull shape parameter reliably quantifies intra-culm leaf area inequality. By bridging theoretical distribution models with field-derived inequality metrics, our approach provides insights into canopy efficiency, photosynthetic optimization, and hydraulic trade-offs. Future work should test this approach in other grass species and assess its generalizability in plants with contrasting canopy architectures.