Hypergeometric Probability Distribution Research Articles

On the Convergence of Hypergeometric to Binomial Distributions

This study presents a measure-theoretic approach to estimate the upper bound on the total variation of the difference between hypergeometric and binomial distributions using the Kullback-Leibler information divergence. The binomial distribution can be used to find the probabilities associated with the binomial experiments. But if the sample size is large relative to the population size, the experiment may not be binomial, and a binomial distribution is not a good choice to find the probabilities associated with the experiment. The hypergeometric probability distribution is the appropriate probability model to be used when the sample size is large compared to the population size. An upper bound for the total variation in the distance between the hypergeometric and binomial distributions is derived using only the sample and population sizes. This upper bound is used to demonstrate how the hypergeometric distribution uniformly converges to the binomial distribution when the population size increases relative to the sample size.

Due date quoting and rescheduling in a fixed production sequence

In this article, the due date setting problem consisting of due date quoting, order insertion and rescheduling with relevant sequence-dependent setups and a frozen zone is considered. A differential equation model is introduced. The differential equation can be solved analytically and requires low computing effort. Explicit expressions are gained concerning the earliest possible due date (due date quoting problem) and the latest possible insertion position (rescheduling problem) in order to meet the customer requested due date. The differential equation model is compared with an approach based on a recently introduced hybrid binomial inverse hypergeometric probability distribution for the due date quoting problem. The analytical results of the proposed model are very close to the expectation values (these cannot be explicitly calculated, and high computation effort is required) of the recently introduced approach.

Using Probability to Reduce the Size of a Voting Body

Suppose a large group of people, such as a university faculty, traditionally makes decisions by meeting together and voting. Because of the COVID-19 pandemic, however, large group meetings have become impractical. Having a relatively small subset of the group meet and take votes would be safer and easier to organize. The purpose of this research is to use probability theory to choose a size for this subset which guarantees that the subset has a high probability of voting the same way as the entire group. Our research involves the study of the hypergeometric probability distribution and leads to a computer algorithm for determining the subset size.

Flexible models for overdispersed and underdispersed count data

Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

Li-rich K giants, dust excess, and binarity

The origin of the Li-rich K giants is still highly debated. Here, we investigate the incidence of binarity among this family from a nine-year radial-velocity monitoring of a sample of 11 Li-rich K giants using the HERMES spectrograph attached to the 1.2 mMercatorTelescope. A sample of 13 non-Li-rich giants (8 of them being surrounded by dust according to IRAS, WISE, and ISO data) was monitored alongside. When compared to the binary frequency in a reference sample of 190 K giants (containing 17.4% of definite spectroscopic binaries – SB – and 6.3% of possible spectroscopic binaries – SB?), the binary frequency appears normal among the Li-rich giants (2/11 definite binaries plus 2 possible binaries, or 18.2% SB + 18.2% SB?), after taking account of the small sample size through the hypergeometric probability distribution. Therefore, there appears to be no causal relationship between Li enrichment and binarity. Moreover, there is no correlation between Li enrichment and the presence of circumstellar dust, and the only correlation that could be found between Li enrichment and rapid rotation is that the most Li-enriched K giants appear to be fast-rotating stars. However, among the dusty K giants, the binary frequency is much higher (4/8 definite binaries plus 1 possible binary). The remaining 3 dusty K giants suffer from a radial-velocity jitter, as is expected for the most luminous K giants, which these are.

Концептуальные особенности французского военного песенного дискурса (корпусное исследование)

This paper presents a corpus-based empirical research, which uncovers the key concepts structuring the French military song discourse. We view this type of discourse as an integral element of military discourse containing institutional and non-institutional segments. The actualization of statutory relationships is the most striking example of institutionalization. Therefore, we have started our research from the hypothesis that the officially regulated (statutory) and less regulated parts of the military song discourse may manifest some conceptual differences. The selection of this subject is determined by its contribution to understanding the French military's cultural peculiarities, as well as by the lack of exploitations regarding the French military song discourse. Within the contemporary cognitive paradigm, the designatum is regarded as a pointer to the concept, which allows grasping a concept globally, without reconstructing its elements. Thus, the analysis of lexemes frequency may serve to identify the inherent concepts structuring some kind of discourse. Using a freely distributed textometric platform TXM (version 0.7.9), we have created a linguistic corpus of 462 texts of French service songs totaling 88305 tokens. Then, we have divided our corpus into two parts, depending on the statutory belonging of the texts. The statistical scores were calculated by applying the hypergeometric probability distribution law. The analysis of statistically significant lemmas has permitted to draw the lexical profile of each part and to identify thereby the most specific concepts. Moreover, we have complemented our research by routine frequency and contextual analysis in order to reveal the additional basic background concepts. Our study leads us to the conclusion that the French military song discourse foregrounds 8 main concepts, but the conceptual structure of each part reveals some differences due to the specifics of the institutional requirements and the military ethos.

An elementary heuristic for Hardy–Littlewood extended Goldbach’s conjecture

The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood’s extended Goldbach’s conjecture. We examine common features of other heuristics in additive prime number theory, such as Cramér’s model and density-type arguments, both of which our heuristic draws from. Apart from the prime number theorem, our argument is entirely elementary, in the sense of not involving complex analysis. The idea is to model sums of two primes by a hypergeometric probability distribution, and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.

Economic aspects of decision making in production processes with uncertain component quality

Production and service processes always introduce discrepancies between real and expected parameters. Accurate modelling of uncertainties in manufacturing processes has great impact on logistics related factors. In this representation, instead of a deterministic analysis of manufacturing processes with uncertain quality of components, decision making is described with stochastic decision trees. The objective of this work is to show how to include the uncertain quality of components into the optimization of manufacturing processes. This paper proposes a new decision tree model with stochastic parameters that can determine the optimal operations depending on the quality of components to be assembled. After a careful literature review, this paper introduces a model to formulate the problem of decision making in manufacturing and assembly processes depending on the quality of produced batches of components. The model seeks the optimal pre-assembly operation, like testing or repairing while taking into account specific costs, prior and posterior probabilities and likelihoods. Next, we demonstrate an enhanced stochastic decision tree with cumulative hypergeometric probability distributions to find the minimum of expected cost based objective function. Numerical results demonstrate how the proposed model supports the decision making process and increase the efficiency, flexibility and reliability of the manufacturing process. DOI: DOI: http://dx.doi.org/10.5755/j01.ee.30.1.19324

A hybrid binomial inverse hypergeometric probability distribution: Theory and applications

In this paper, we define a novel probability distribution triggered by a due date setting problem from a multi-item single level production system. As a result, we derive a recurrence relation for the underlying distribution function and, finally, we also infer the novel discrete distribution function. We underpin our analytical findings by numerical results when computing the distribution function, the expected value as well as the variance. In order to apply and translate our apparatus to other problems, we come up with an application when determining due dates for a production system with sequence-dependent setup costs.

Wallenius Bayes

This paper introduces a new event model appropriate for classifying (binary) data generated by a “destructive choice” process, such as certain human behavior. In such a process, making a choice removes that choice from future consideration yet does not influence the relative probability of other choices in the choice set. The proposed Wallenius event model is based on a somewhat forgotten non-central hypergeometric distribution introduced by Wallenius (Biased sampling: the non-central hypergeometric probability distribution. Ph.D. thesis, Stanford University, 1963). We discuss its relationship with models of how human choice behavior is generated, highlighting a key (simple) mathematical property. We use this background to describe specifically why traditional multivariate Bernoulli naive Bayes and multinomial naive Bayes each are suboptimal for such data. We then present an implementation of naive Bayes based on the Wallenius event model, and show experimentally that for data where we would expect the features to be generated via destructive choice behavior Wallenius Bayes indeed outperforms the traditional versions of naive Bayes for prediction based on these features. Furthermore, we also show that it is competitive with non-naive methods (in particular, support-vector machines). In contrast, we also show that Wallenius Bayes underperforms when the data generating process is not based on destructive choice.

Design and Analysis of an Optimal Inventory Model for Perishable Goods with Fixed Life Time

Background/Objectives: The main purpose of this paper is to plan and implement an optimum inventory model for the deteriorating things that minimizes the entire expected cost over a finite horizon. Methods/Statistical Analysis: The model presents a projected price operates for the inventory system with fixed life time. This paper accords with the hassle of ciphering optimum ordering policies for one item for successive intervals. Hyper geometric probability distribution is taken into consideration to examine the expected outdated cost. Findings: The optimum inventory model is solved analytically to review the impact of modification within the worth of the parameters. The solution methodology provided within the model helps the choice maker to enforce the first in first out policy by exposing customers to merchandise of a similar age. The results have been explained with numerical examples and graphical illustration. Applications/Improvements: First in First out issuing policy is designed to reveal the fact that it minimizes the expected outdates.

Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

Gene expression changes in human mesenchymal stem cells from patients with osteoporosis.

The aim of the present study was to investigate the underlying molecular mechanisms of osteoporosis and to identify novel candidate genes involved in this disease. The gene expression profile of GSE35958 was downloaded from Gene Expression Omnibus, including five samples of human mesenchymal stem cells from patients with osteoporosis and four control samples. Differentially expressed genes (DEGs) were initially identified following an analysis using Student’s t-test. Subsequently, a protein-protein interaction (PPI) network of the significant pathways was constructed, based on the Human Protein Reference Database. In the significant pathways, DEGs were screened using cut-off criteria of FDR<0.1 and |log2FC|>1.5. A co-change network for pathways was also constructed using the method of cumulative hypergeometric probability distribution. Finally, the transcriptional regulatory network for DEGs was constructed based on the TRANSFAC database. In total, 1,127 DEGs, including 554 upregulated and 573 downregulated DEGs, were screened. The constructed PPI network for the DEGs involved in the two significant pathways, including focal adhesion and lysosome, demonstrated that the five DEGs with a high degree (>60) were β-catenin, SHC-transforming protein 1, RAC-α serine/threonine-protein kinase, caveolin 1 and filamin A, with degrees of 135, 117, 117, 73 and 63, respectively. The pathway with the degree of 22 in the constructed co-change network was neuroactive ligand receptor interaction. The nine genes with a high (≥9) degree in the constructed transcriptional regulatory network were REL-associated protein, upstream stimulatory factor 1, specificity protein 1, Fos-related antigen 1, cyclin-dependent kinase inhibitor 1A, upstream stimulatory factor 2, ETS domain-containing protein Elk-1, JUND and retinoic acid receptor α, with degrees of 29, 27, 19, 18, 17, 13, 11, 11 and 9, respectively. The DEGs with high degree in the PPI and transcriptional regulatory networks may be candidate target molecules, which may be used to monitor, diagnose and treat osteoporosis.

ВИКОРИСТАННЯ WEB-СЕРВІСУ WOLFRAM|ALPHA ДЛЯ РОЗВ’ЯЗУВАННЯ ЗАДАЧ З ТЕОРІЇ ЙМОВІРНОСТЕЙ

The trend towards the use of remote network resources on the Internet clearly delineated.Traditional training combined with increasingly networked, remote technologies become popularcloud computing. Research methods of probability theory are used in various fields. Of particular note is the use of methods of probability theory in psychological and educational research in statistical analysis of experimental data. Conducting such research is impossible without the use of modern information technology. Given the advantages of web-based software, the article describes web-service Wolfram|Alpha. Detailed analysis of the possibilities of using web-service Wolfram|Alpha for solving problems of probability theory. In the case studies described the results of queries for solving of probability theory, in particular the sections random events and random variables. Considered and analyzed the problem of the number of occurrences of event A in n independent trials using Wolfram|Alpha, detailed analysis of the possibilities of using the service Wolfram|Alpha for the study of continuous random variable that has a normal and uniform probability distribution, including calculating the probability of getting the value of a random variable in a given interval. The problem in applying the binomial and hypergeometric probability distribution of a discrete random variable and demonstrates the possibility of using the service Wolfram|Alpha for solving it.

TRANSIT AND RADIAL VELOCITY SURVEY EFFICIENCY COMPARISON FOR A HABITABLE ZONE EARTH

Transit and radial velocity searches are two techniques for identifying nearby extrasolar planets to Earth that transit bright stars. Identifying a robust sample of these exoplanets around bright stars for detailed atmospheric characterization is a major observational undertaking. In this study we describe a framework that answers the question of whether a transit or radial velocity survey is more efficient at finding transiting exoplanets given the same amount of observing time. Within the framework we show that a transit survey's window function can be approximated using the hypergeometric probability distribution. We estimate the observing time required for a transit survey to find a transiting Earth-sized exoplanet in the HZ with an emphasis on late type stars. We also estimate the radial velocity precision necessary to detect the equivalent HZ Earth-mass exoplanet that also transits when using an equal amount of observing time as the transit survey. We find that a radial velocity survey with sig_rv~0.6 m/s precision has comparable efficiency in terms of observing time to a transit survey with the requisite photometric precision sig_phot~300 ppm to find a transiting Earth-sized exoplanet in the HZ of late M~dwarfs. For Super-Earths, a sig_rv~2.0 m/s precision radial velocity survey has comparable efficiency to a transit survey with sig_phot~2300 ppm.

Response of human skin to esthetic scarification

This study was undertaken to investigate changes in RNA expression in previously healthy adult human skin following thermal injury induced by contact with hot metal that was undertaken as part of esthetic scarification, a body modification practice. Subjects were recruited to have pre-injury skin and serial wound biopsies performed. 4mm punch biopsies were taken prior to branding and 1h, 1 week, and 1, 2 and 3 months after injury. RNA was extracted and quality assured prior to the use of a whole-genome based bead array platform to describe expression changes in the samples using the pre-injury skin as a comparator. Analysis of the array data was performed using k-means clustering and a hypergeometric probability distribution without replacement and corrections for multiple comparisons were done. Confirmatory q-PCR was performed. Using a k of 10, several clusters of genes were shown to co-cluster together based on Gene Ontology classification with probabilities unlikely to occur by chance alone. OF particular interest were clusters relating to cell cycle, proteinaceous extracellular matrix and keratinization. Given the consistent expression changes at 1 week following injury in the cell cycle cluster, there is an opportunity to intervene early following burn injury to influence scar development.

Oxygen carrier dispersion in inert packed beds to improve performance in chemical looping combustion

Various packed beds of copper-based oxygen carriers (CuO on Al2O3) were tested over 100 cycles of low temperature (673K) Chemical Looping Combustion (CLC) with H2 as the fuel gas. The oxygen carriers were uniformly mixed with alumina (Al2O3) in order to investigate the level of separation necessary to prevent agglomeration. It was found that a mass ratio of 1:6 oxygen carrier to alumina gave the best performance in terms of stable, repeating hydrogen breakthrough curves over 100 cycles. In order to quantify the average separation achieved in the mixed packed beds, two sphere-packing models were developed. The hexagonal close-packing model assumed a uniform spherical packing structure, and based the separation calculations on a hypergeometric probability distribution. The more computationally intensive full-scale model used discrete element modelling to simulate random packing arrangements governed by gravity and contact dynamics. Both models predicted that average ‘nearest neighbour’ particle separation drops to near zero for oxygen carrier mass fractions of x⩾0.25. For the packed bed systems studied, agglomeration was observed when the mass fraction of oxygen carrier was above this threshold.

Learning small gallery size for prediction of recognition performance on large populations

This paper addresses the estimation of a small gallery size that can generate the optimal error estimate and its confidence on a large population (relative to the size of the gallery) which is one of the fundamental problems encountered in performance prediction for object recognition. It uses a generalized two-dimensional prediction model that combines a hypergeometric probability distribution model with a binomial model and also considers the data distortion problem in large populations. Learning is incorporated in the prediction process in order to find the optimal small gallery size and to improve the prediction. The Chernoff and Chebychev inequalities are used as a guide to obtain the small gallery size. During the prediction, the expectation–maximization (EM) algorithm is used to learn the match score and the non-match score distributions that are represented as a mixture of Gaussians. The optimal size of the small gallery is learned by comparing it with the sizes obtained by the statistical approaches and at the same time the upper and lower bounds for the prediction on large populations are obtained. Results for the prediction are presented for the NIST-4 fingerprint database.

A New Approach for Assessment of Mental Architecture: Repeated Tagging

A new approach to the study of a relatively neglected property of mental architecture—whether and when the already-processed elements are separated from the to-be-processed elements—is proposed. The process of numerical proportion discrimination between two sets of elements defined either by color or by orientation can be described as sampling with or without replacement (characterized by binomial or hypergeometric probability distributions respectively) depending on the possibility to tag an element once or repeatedly. All empirical psychometric functions were approximated by a theoretical model showing that the ability to keep track of the already tagged elements is not an inflexible part of the mental architecture but rather an individually variable strategy which also depends on conspicuity of perceptual attributes. Strong evidence is provided that in a considerable number of trials, observers tagged the same element repeatedly which can only be done serially at two separate time moments.

Some properties of Kemp family of distributions

This paper studies some important properties of the generalized hypergeometric probability distribution (GHPD) of Kemp (Sankhya-Series A, 1968) by establishing the existence of all the moments of the distribution and by deriving a formula for raw moments. Here we also obtain certain recurrence relations for probabilities, raw moments and factorial moments of the GHPD.