Two Probability Plots of the Three-Parameter Lognormal Distribution

The two-parameter lognormal distribution is a variant of the normal distribution and the three-parameter lognormal distribution is an extension of the two-parameter lognormal distribution by introducing a location parameter. The Q-Q plot of the three-parameter lognormal distribution is widely used. To obtain the Q-Q plot one needs to iteratively try different values of the shape parameter and subjectively judge the linearity of the Q-Q plot. In this paper,a mathematical method was proposed to determine the value of the shape parameter so as to simplify the generation of the Q-Q plot. Then a new probability plot was proposed,which was more easily obtained and provided more accurate parameter estimates than the Q-Q plot. These are illustrated by three realworld examples.

Applying the Different Statistical Tests in Analysis of Electrical Breakdown Mechanisms in Nitrogen Filled Gas Diode

This paper presents the results of our investigations of breakdown mechanisms, as well as a description of their influence on the distributions of time delay distributions, for a gas tube filled with nitrogen at 4 mbar. The values of the time delay are measured for different voitages, and the values of the relaxation times and their distributions and probability plots are analyzed. The obtained density distributions have Gaussian distributions and exponential distributions for different values of relaxation times （Gaussian for small values and exponential for large values of relaxation time）. It is shown that for middle values of relaxation time the delay distributions have a shape between Caussian and exponential distributions, which is a result of the different influences of electrical breakdown.

Test of Normality of Waist Measurement Data of Young Male and Female Adults based on the Quantile - Quantile Plot

This paper investigates the normality of some real data set obtained from waist measurements of a group of 49 young adults. The quantile - quantile （Q-Q） plot and the analysis of correlation coefficients for the Q-Q plot is used to determine the normality or otherwise of the data set. In this regards, the probabilities of the quantiles were computed, modified and plotted. Thereafter the correlation coefficients for the quantile - quantile plots were obtained. Results indicate that at 0.1 level of significance, the data for young adult males of the sample were not normally distributed, and had a mean value that is within the range of low risk, healthwise, whereas the distribution of the data for young female adults showed reasonable normality, but also with a mean value that is within the range of low risk in terms of health condition.

Probability distributions of arsenic in soil from brownfield sites in Beijing （China）： statistical characterization of the background populations and implications for site assessment studies

A probabilistic analysis was performed on soil arsenic concentration data from 4 brownfield sites at Beijing （Chaoyang and Haidian Districts）, involved in environmental assessment studies. The available data sets were processed to provide a statistical characterization of the background populations and differentiate ＂anomalous data＂ from the natural range of variation of arsenic concentrations in soil. The site-specific background distributions and the existing wide-scale background values defined for the Beijing area were compared, discussing related implications for the definition of metal contamination soil screening levels （SSLs） in site assess- ment studies. The statistical analysis of As data sets discriminated site-specific background populations, encompassing 88% to 94% of the sample data, from oufliers values, associated with either subsoil natural enrichments or possible anthropogenic releases. Upper Baseline Concentration （UBC） limits （＋ 2σ level）, including most of the site-specific metal background variability, were derived based on the statistical characterization of the background populations. Sites in the Chaoyang South District area had UBC values in the range 10.4-12.6mg·kg^-1. These ranges provide meaningful SSL values to be adopted for As in local site assessment studies. Using the wide-scale background value for the Beijing area would have erroneously classified most of the areas in the subject sites as potentially contaminated.

Contribution to sample failure probability plot and its solution by Kriging method

To analyze the effect of the region of the model inputs on the model output,a novel concept about contribution to the sample failure probability plot(CSFP) is proposed based on the contribution to the sample mean plot(CSM) and the contribution to the sample variance plot(CSV).The CSFP can be used to analyze the effect of the region of the model inputs on the failure probability.After the definition of CSFP,its property and the differences between CSFP and CSV/CSM are discussed.The proposed CSFP can not only provide the information about which input affects the failure probability mostly,but also identify the contribution of the regions of the input to the failure probability mostly.By employing the Kriging model method on optimized sample points,a solution for CSFP is obtained.The computational cost for solving CSFP is greatly decreased because of the efficiency of Kriging surrogate model.Some examples are used to illustrate the validity of the proposed CSFP and the applicability and feasibility of the Kriging surrogate method based solution for CSFP.

Study on Probability Plot Correlation Coefficient of the Log-Weibull Distribution

The log-Weibull distribution is a variant of the three-parameter Weibull distribution. The probability plot of a distribution model is desired since it can help to decide on whether the model is appropriate for fitting a given dataset and can provide the initial estimate of the model parameters. The decision on the appropriateness of a distribution is somehow subjective. This paper presents a probability plot of the log-Weibull distribution(LWPP). The distribution of the probability plot correlation coefficient is studied. From this distribution, a lower confidence limit is determined for determining whether the probability plot correlation coefficient derived from a given data set is large enough. The appropriateness and usefulness of this study are illustrated by two real-world examples.