The r‐hypergeometric distribution: Characterization, mathematical methods, simulations, and applications in sciences and engineering
In this article, we introduce a novel univariate discrete distribution called the r‐hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no‐order sampling of the hypergeometric distribution. Mathematical expressions are obtained f...
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| Published in | Mathematical methods in the applied sciences Vol. 46; no. 5; pp. 5208 - 5233 |
|---|---|
| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Freiburg
Wiley Subscription Services, Inc
30.03.2023
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0170-4214 1099-1476 |
| DOI | 10.1002/mma.8826 |
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| Abstract | In this article, we introduce a novel univariate discrete distribution called the r‐hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no‐order sampling of the hypergeometric distribution. Mathematical expressions are obtained for computing probabilities, the mode, and moments of the new distribution. Two simulation algorithms are proposed using the acceptance‐rejection and inverse‐transform methods and computational features to generate values of an r‐hypergeometric distributed random variable. Python and R codes are implemented to perform computational experiments. Applications of mathematical methods based on the new distribution in sciences and engineering employing simulated and real‐world data sets are provided. A comparison with existing distributions is also included. In addition to the mathematical results, some findings obtained from our study are related to a better computational performance of the inverse‐transform method. Also, we identify applications of our model that are not covered by the traditional count distributions. In addition, we establish distinct probabilities for the same event under the binomial, hypergeometric, and r‐hypergeometric distributions, with their means also being distinct, but their variances, skewness, and kurtosis converge to the same value. |
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| AbstractList | In this article, we introduce a novel univariate discrete distribution called the r‐hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no‐order sampling of the hypergeometric distribution. Mathematical expressions are obtained for computing probabilities, the mode, and moments of the new distribution. Two simulation algorithms are proposed using the acceptance‐rejection and inverse‐transform methods and computational features to generate values of an r‐hypergeometric distributed random variable. Python and R codes are implemented to perform computational experiments. Applications of mathematical methods based on the new distribution in sciences and engineering employing simulated and real‐world data sets are provided. A comparison with existing distributions is also included. In addition to the mathematical results, some findings obtained from our study are related to a better computational performance of the inverse‐transform method. Also, we identify applications of our model that are not covered by the traditional count distributions. In addition, we establish distinct probabilities for the same event under the binomial, hypergeometric, and r‐hypergeometric distributions, with their means also being distinct, but their variances, skewness, and kurtosis converge to the same value. In this article, we introduce a novel univariate discrete distribution called the r‐hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no‐order sampling of the hypergeometric distribution. Mathematical expressions are obtained for computing probabilities, the mode, and moments of the new distribution. Two simulation algorithms are proposed using the acceptance‐rejection and inverse‐transform methods and computational features to generate values of an r‐hypergeometric distributed random variable. Python and R codes are implemented to perform computational experiments. Applications of mathematical methods based on the new distribution in sciences and engineering employing simulated and real‐world data sets are provided. A comparison with existing distributions is also included. In addition to the mathematical results, some findings obtained from our study are related to a better computational performance of the inverse‐transform method. Also, we identify applications of our model that are not covered by the traditional count distributions. In addition, we establish distinct probabilities for the same event under the binomial, hypergeometric, and r‐hypergeometric distributions, with their means also being distinct, but their variances, skewness, and kurtosis converge to the same value. |
| Author | Mahdi, Esam Leiva, Víctor Díaz‐Rodríguez, Martín Martin‐Barreiro, Carlos Cabezas, Xavier |
| Author_xml | – sequence: 1 givenname: Martín orcidid: 0000-0003-4783-0063 surname: Díaz‐Rodríguez fullname: Díaz‐Rodríguez, Martín organization: Universidad del Norte – sequence: 2 givenname: Víctor orcidid: 0000-0003-4755-3270 surname: Leiva fullname: Leiva, Víctor email: victorleivasanchez@gmail.com organization: School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso – sequence: 3 givenname: Carlos orcidid: 0000-0002-8797-681X surname: Martin‐Barreiro fullname: Martin‐Barreiro, Carlos organization: Universidad Espíritu Santo – sequence: 4 givenname: Xavier orcidid: 0000-0003-3128-001X surname: Cabezas fullname: Cabezas, Xavier organization: Escuela Superior Politécnica del Litoral ESPOL – sequence: 5 givenname: Esam surname: Mahdi fullname: Mahdi, Esam organization: University of Toronto |
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| Cites_doi | 10.2466/pms.1996.83.3f.1216 10.1016/j.trb.2011.09.007 10.1007/BF02613383 10.1007/s00362-021-01260-1 10.1080/0020739930240403 10.1007/s00362-006-0376-x 10.1002/sim.7929 10.1007/BF02480295 10.1016/j.jspi.2010.12.005 10.1109/TR.2015.2499964 10.1002/0471715816 10.3390/math9172058 10.1007/s00477-020-01961-3 10.1007/978-3-642-04898-2_291 10.3906/mat-1906-6 10.1002/mma.5463 10.1080/02664763.2018.1531978 10.1081/QEN-200059865 10.1002/asmb.1944 10.1080/03610926.2019.1682163 10.1002/env.2349 10.1007/s00362-017-0888-6 10.1080/0020739870180316 10.1080/03610919208813021 10.4028/www.scientific.net/AMM.427-429.2549 10.1002/mma.3986 10.3390/s21124094 10.1081/STA-120018555 10.1016/j.jspi.2010.06.029 10.1007/978-3-031-02425-2 10.1016/j.csda.2012.05.007 10.1201/9781315120744 10.3390/s21155198 10.1002/mma.855 10.1016/j.chemolab.2018.03.012 10.3390/sym13060926 10.1016/j.chemolab.2019.04.013 10.3390/s21165352 |
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| SubjectTerms | Algorithms applied sciences binomial Binomial distribution f‐binomial and hypergeometric distributions Kurtosis Mathematical analysis no‐order sampling python and R computer languages random numbers Random variables Sampling sampling with replacement Simulation |
| Title | The r‐hypergeometric distribution: Characterization, mathematical methods, simulations, and applications in sciences and engineering |
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