normal distributions
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2022 ◽  
Vol 40 (1) ◽  
Author(s):  
José Rodriguez-Avi

A macroeconomic indicator of productivity and economic development, used to obtain information on the economic and social conditions of a country, is the GDP per capita, which is also used as an indicator of social welfare. By construction it can be used directly to compare areas of interest. It is an indicator of great variability to which it is difficult to assign a probabilistic model to describe its distribution. In fact, it usually appears as a strongly asymmetric and frequently multimodal variable, which directly indicates a strong non-normality. In this work we propose to deal with the problem of finding a probabilistic model for this variable through the estimation of a model of finite mixtures of normal distributions. As an application example, we present the model obtained through the finite mixture for GDP per capita data from the NUTS 3 zones in the nomenclature of the European Union, EU countries and neighbouring countries. Thus, the model is estimated, its validity is checked and the results obtained are analysed, both for the GDP per capita variable and as a function of the countries to which the studied areas belong.


2022 ◽  
Vol 15 (1) ◽  
pp. 149-164
Author(s):  
Alberto Sorrentino ◽  
Alessia Sannino ◽  
Nicola Spinelli ◽  
Michele Piana ◽  
Antonella Boselli ◽  
...  

Abstract. We consider the problem of reconstructing the number size distribution (or particle size distribution) in the atmosphere from lidar measurements of the extinction and backscattering coefficients. We assume that the number size distribution can be modeled as a superposition of log-normal distributions, each one defined by three parameters: mode, width and height. We use a Bayesian model and a Monte Carlo algorithm to estimate these parameters. We test the developed method on synthetic data generated by distributions containing one or two modes and perturbed by Gaussian noise as well as on three datasets obtained from AERONET. We show that the proposed algorithm provides good results when the right number of modes is selected. In general, an overestimate of the number of modes provides better results than an underestimate. In all cases, the PM1, PM2.5 and PM10 concentrations are reconstructed with tolerable deviations.


2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Shovan Biswas ◽  
Sudhansu S. Maiti

Abstract This article develops multiple dependent state (MDS) sampling inspection plans based on the mean of lifetime quality characteristic that follows non-normal distributions viz., exponential and Lindley distribution. In this plan, the lot quality is measured by the lot mean (𝜇). We have estimated the optimal plan parameters of the proposed technique by non-linear optimization approaches considering acceptable quality level and rejection quality level. We have compared the sample size between the MDS sampling inspection plan and the single sampling inspection plan for the variable. Finally, we have taken two examples to illustrate the proposed technique.


Author(s):  
FAUSTO CORRADIN ◽  
DOMENICO SARTORE

This paper computes the Non-central Moments of the Truncated Normal variable, i.e. a Normal constrained to assume values in the interval with bounds that may be finite or infinite. We define two recursive expressions where one can be expressed in closed form. Another closed form is defined using the Lower Incomplete Gamma Function. Moreover, an upper bound for the absolute value of the Non-central Moments is determined. The numerical results of the expressions are compared and the different behavior for high value of the order of the moments is shown. The limitations to the use of Truncated Normal distributions with a lower negative limit regarding financial products are considered. Limitations in the application of Truncated Normal distributions also arise when considering a CRRA utility function.


Materials ◽  
2021 ◽  
Vol 15 (1) ◽  
pp. 152
Author(s):  
Jan Stefan Bihałowicz ◽  
Wioletta Rogula-Kozłowska ◽  
Adam Krasuski ◽  
Małgorzata Majder-Łopatka ◽  
Agata Walczak ◽  
...  

This study aimed to determine the relative densities of populations of particles emitted in fire experiments of selected materials through direct measurement and parametrization of size distribution as number (NSD), volume (VSD), and mass (MSD). As objects of investigation, four typical materials used in construction and furniture were chosen: pinewood (PINE), laminated particle board (LPB), polyurethane (PUR), and poly(methyl methacrylate) (PMMA). The NSD and VSD were measured using an electric low-pressure impactor, while MSD was measured by weighing filters from the impactor using a microbalance. The parametrization of distributions was made assuming that each distribution can be expressed as the sum of an arbitrary number of log-normal distributions. In all materials, except PINE, the distributions of the particles emitted in fire experiments were the sum of two log-normal distributions; in PINE, the distribution was accounted for by only one log-normal distribution. The parametrization facilitated the determination of volume and mass abundances, and therefore, the relative density. The VSDs of particles generated in PINE, LPB, and PUR fires have similar location parameters, with a median volume diameter of 0.2–0.3 µm, whereas that of particles generated during PMMA burning is 0.7 µm. To validate the presented method, we burned samples made of the four materials in similar proportions and compared the measured VSD with the VSD predicted based on the weighted sum of VSD of raw materials. The measured VSD shifted toward smaller diameters than the predicted ones due to thermal decomposition at higher temperatures.


Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 609
Author(s):  
Arthur G. Rattew ◽  
Yue Sun ◽  
Pierre Minssen ◽  
Marco Pistoia

The efficient preparation of input distributions is an important problem in obtaining quantum advantage in a wide range of domains. We propose a novel quantum algorithm for the efficient preparation of arbitrary normal distributions in quantum registers. To the best of our knowledge, our work is the first to leverage the power of Mid-Circuit Measurement and Reuse (MCMR), in a way that is broadly applicable to a range of state-preparation problems. Specifically, our algorithm employs a repeat-until-success scheme, and only requires a constant-bounded number of repetitions in expectation. In the experiments presented, the use of MCMR enables up to a 862.6x reduction in required qubits. Furthermore, the algorithm is provably resistant to both phase-flip and bit-flip errors, leading to a first-of-its-kind empirical demonstration on real quantum hardware, the MCMR-enabled Honeywell System Models H0 and H1-2.


2021 ◽  
pp. 096228022110327
Author(s):  
Anita Brobbey ◽  
Samuel Wiebe ◽  
Alberto Nettel-Aguirre ◽  
Colin Bruce Josephson ◽  
Tyler Williamson ◽  
...  

Discriminant analysis procedures that assume parsimonious covariance and/or means structures have been proposed for distinguishing between two or more populations in multivariate repeated measures designs. However, these procedures rely on the assumptions of multivariate normality which is not tenable in multivariate repeated measures designs which are characterized by binary, ordinal, or mixed types of response distributions. This study investigates the accuracy of repeated measures discriminant analysis (RMDA) based on the multivariate generalized estimating equations (GEE) framework for classification in multivariate repeated measures designs with the same or different types of responses repeatedly measured over time. Monte Carlo methods were used to compare the accuracy of RMDA procedures based on GEE, and RMDA based on maximum likelihood estimators (MLE) under diverse simulation conditions, which included number of repeated measure occasions, number of responses, sample size, correlation structures, and type of response distribution. RMDA based on GEE exhibited higher average classification accuracy than RMDA based on MLE especially in multivariate non-normal distributions. Three repeatedly measured responses namely severity of epilepsy, current number of anti-epileptic drugs, and parent-reported quality of life in children with epilepsy were used to demonstrate the application of these procedures.


Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7335
Author(s):  
Łukasz Blacha

A non-linear modification to Miner’s rule for damage accumulation is proposed to reduce the scatter between experimental fatigue life and fatigue life predicted according to the original Miner’s sum. Based on P-s-n probability distribution and design s-n curves, the modification satisfies the assumption of equality between the mean damage degree (at the critical level) and fatigue life random variables, which is not covered in the original formulation. The adopted formulation shows the discrepancies between the fatigue lives predicted according to the design s-n curves and the estimated probability distribution. It also proves that it is inappropriate to apply a normal distribution to fatigue life analysis and that the model becomes non-linear only for non-normal distributions. The predictions according to the established model were compared to the predictions obtained with Miner’s rule.


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