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Deterministic and Stochastic Analysis of Groundwater in Unconfined Aquifer Model
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Dineo Ramakatsa, Abdon Atangana
In the theory of probability, a log-normal distribution is a continuous statistical probability distribution of a randomly chosen parameter normally distributed by the logarithm. Therefore, if the seemingly random parameter X is dispersed log-normally, Y = ln (X) has a normal distribution. The exponential function of Y, X = exp(Y) also has a log-normal distribution if Y has a normal distribution. A randomly chosen parameter that is dispersed log-normally only takes positive real values.
Bioaerosol Particle Statistics
Published in Christopher S. Cox, Christopher M. Wathes, Bioaerosols Handbook, 2020
A straight line on the log-probability graph is convenient for analysis of the distribution. Though they have little physical significance in interpreting experimental results,10 the geometric median diameter and geometric standard deviation describe completely the lognormal distribution on that graph. The geometric median diameter is readily determined from the 50% point of the graph (Figure 5.4). The geometric standard deviation is calculated by dividing the diameter taken from the 84% probability point by the geometric median diameter, or dividing the geometric median diameter by the diameter taken from the 16% probability point. The mode, median, and mean are always unequal in a lognormal distribution, whereas they are equal in a normal distribution. However, the geometric mean diameter (Equation 5.7) is equal to the geometric median diameter in a lognormal distribution.
Quality and Warranty: Sensitivity of Warranty Cost Models to Distributional Assumptions
Published in Donald B. Owen, Subir Ghosh, William R. Schucany, William B. Smith, Statistics of Quality, 2020
Wallace R. Blischke, Sushmita Das Vij
where −∞ < η < ∞ and θ > 0. The mean and variance of the log-normal distribution are μ = eη+θ2/2 and σ2 = e2η+θ2(eθ2 - 1).
Pulling the distribution in supply chains: simulation and analysis of Dynamic Buffer Management approach
Published in International Journal of Systems Science: Operations & Logistics, 2023
Lucas Martins Ikeziri, Fernando Bernardi de Souza, Andréia da Silva Meyer, Mahesh C. Gupta
The demand and replenishment time data were generated by ProModel from log-normal distributions. The choice for this distribution is justified because it is more suitable for original data and is considered more realistic than the normal distribution (Huang, 2013). In addition, the log-normal distribution is a model that approximates the demand for products in inventory management research as it represents well environments with high variability and non-negative data skewed to the right (Cobb et al., 2013; Gholami et al., 2018; Limpert et al., 2001; van Steenbergen & Mes, 2020). Trapero et al. (2019) point out that the log-normal distribution is reasonable for products subjected to promotional periods when the observed demand is greater than the baseline demand. Lee and Rim (2019) also used the log-normal distribution as a realistic approximation to daily demand and replenishment time data. For such reasons, we also used it in the representations of these two variables.
Improved ASCE/SEI 7-10 Ground-Motion Scaling Procedure for Nonlinear Analysis of Buildings
Published in Journal of Earthquake Engineering, 2021
Juan Carlos Reyes, Catalina González, Erol Kalkan
For scaling records, various intensity measures have been evaluated to minimize the variability in the prediction of EDPs [Mazza and Labernarda, 2017]. In this study, spectral acceleration ordinates are used as the intensity measure. As it will be demonstrated later, spectral responses and EDPs may be assumed as log-normally distributed. Therefore, it is appropriate to represent the “mean” response by the geometric mean (or median), instead of the arithmetic mean [Jayaram and Baker, 2008]—the arithmetic mean is not ideal due to the skewed nature of the EDP data. For a log-normal distribution of a random variable, the geometric mean () and median () are given by the same equation: , where is the mean of a log-normal distribution. Therefore, it is not misleading to use median instead of geometric mean. Another alternative to represent “mean” response is the 50th percentile of the data, but this alternative representation may be too crude to be useful in the earthquake engineering field.
Stochastic seasonal rainfall simulation model for Irbid region, Northern Jordan
Published in Urban Water Journal, 2020
In order to obtain the conditional distribution of rainfall depth for a given duration, a histogram of the rainfall depth for each duration class is constructed as shown in Figure 6 for Irbid station. These histograms appear to fit the log-normal distribution well. Parameters of the log-normal distribution were estimated via maximum likelihood method. A Chi-square goodness-of-fit test was performed for the distribution in each duration class for all stations. The result of the tests shows that the log-normal distribution is significantly accepted at 5% level of significance. Similar distributions were found for Ajlun and Um Qeis stations. The pdf for the conditional log-normal distribution of storm depth for a given duration f(D/t) was expressed by: