scholarly journals Modeling Particle Size Distribution in Lunar Regolith via a Central Limit Theorem for Random Sums

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1409
Author(s):  
Andrey Gorshenin ◽  
Victor Korolev ◽  
Alexander Zeifman
Keyword(s):  
Particle Size ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Empirical Distribution ◽  
Lunar Regolith ◽  
Real Data ◽  
Theoretical Background ◽  
Normal Mixtures ◽  
Promising Tool

A version of the central limit theorem is proved for sums with a random number of independent and not necessarily identically distributed random variables in the double array limit scheme. It is demonstrated that arbitrary normal mixtures appear as the limit distribution. This result is used to substantiate the log-normal finite mixture approximations for the particle size distributions of the lunar regolith. This model is used as the theoretical background of the two different statistical procedures for processing real data based on bootstrap and minimum χ2 estimates. It is shown that the cluster analysis of the parameters of the proposed models can be a promising tool for revealing the structure of such real data, taking into account the physico-chemical interpretation of the results. Similar methods can be successfully used for solving problems from other subject fields with grouped observations, and only some characteristic points of the empirical distribution function are given.

Unimodality of Wave Amplitude in the Northern Hemisphere

2008 ◽  
Vol 65 (3) ◽  
pp. 1077-1086 ◽  
Author(s):  
Maarten H. P. Ambaum
Keyword(s):  
Distribution Function ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Wave Amplitude ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Local Wave

Abstract A novel statistic for local wave amplitude of the 500-hPa geopotential height field is introduced. The statistic uses a Hilbert transform to define a longitudinal wave envelope and dynamical latitude weighting to define the latitudes of interest. Here it is used to detect the existence, or otherwise, of multimodality in its distribution function. The empirical distribution function for the 1960–2000 period is close to a Weibull distribution with shape parameters between 2 and 3. There is substantial interdecadal variability but no apparent local multimodality or bimodality. The zonally averaged wave amplitude, akin to the more usual wave amplitude index, is close to being normally distributed. This is consistent with the central limit theorem, which applies to the construction of the wave amplitude index. For the period 1960–70 it is found that there is apparent bimodality in this index. However, the different amplitudes are realized at different longitudes, so there is no bimodality at any single longitude. As a corollary, it is found that many commonly used statistics to detect multimodality in atmospheric fields potentially satisfy the assumptions underlying the central limit theorem and therefore can only show approximately normal distributions. The author concludes that these techniques may therefore be suboptimal to detect any multimodality.

Statistical Inference and Simulation with StatKey

2016 ◽  
Vol 109 (9) ◽  
pp. 708-711 ◽  
Author(s):  
Anne Quinn
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Statistical Inference ◽  
Confidence Intervals ◽  
Central Limit ◽  
Real Data ◽  
Web Based ◽  
The Central Limit Theorem

StatKey, a free Web-based app, supplies real data to help with the central limit theorem, confidence intervals, and much more.

scholarly journals HOW TO EXPLAIN THAT CHANGES IN ELDERLIES DEPRESSION LEVEL ARE UNIFORMLY DISTRIBUTED

2020 ◽  
pp. 104-107
Author(s):  
Griselda Acosta ◽  
Eric D. Smith ◽  
Vladik Kreinovich
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Uniform Distribution ◽  
Central Limit ◽  
Empirical Distribution ◽  
The Central Limit Theorem ◽  
Depression Level

Changes in the elderlies depression level result from a large number of small independent factors. Such situations are ubiquitous in applications. In most such cases, due to the Central Limit Theorem, the corresponding distribution is close to Gaussian. For the changes in the elderlies depression level, however, the empirical distribution is far from Gaussian: it is uniform. In this paper, we provide a possible explanation for the emergence of the uniform distribution.

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