{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:25:33Z","timestamp":1760145933842,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T00:00:00Z","timestamp":1727049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, Portugal","award":["UIDB\/00006\/2020","UIDB\/00685\/2020"],"award-info":[{"award-number":["UIDB\/00006\/2020","UIDB\/00685\/2020"]}]},{"name":"BCEEAplA\u2014Centre of Applied Economics Studies of the Atlantic, School of Business and Economics of the University of the Azores","award":["UIDB\/00006\/2020","UIDB\/00685\/2020"],"award-info":[{"award-number":["UIDB\/00006\/2020","UIDB\/00685\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Encyclopedia"],"abstract":"<jats:p>The observation of randomness patterns serves as guidance for the craft of probabilistic modelling. The most used count models\u2014Binomial, Poisson, Negative Binomial\u2014are the discrete Morris\u2019 natural exponential families whose variance is at most quadratic on the mean, and the solutions of Katz\u2013Panjer recurrence relation, aside from being members of the generalised power series and hypergeometric distribution families, and this accounts for their many advantageous characteristics. Some other basic count models are also described, as well as models with less obvious but useful randomness patterns in connection with maximum entropy characterisations, such as Zipf and Good models. Simple tools, such as truncation, thinning, or parameter randomisation, are straightforward ways of constructing other count models.<\/jats:p>","DOI":"10.3390\/encyclopedia4030089","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T07:49:24Z","timestamp":1727423364000},"page":"1367-1380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Count Random Variables"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3364-0357","authenticated-orcid":false,"given":"Sandra","family":"Mendon\u00e7a","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica\u2014FCEE, Universidade da Madeira, Campus Universit\u00e1rio da Penteada, 9020-105 Funchal, Portugal"},{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa (CEAUL), Campo Grande, 1749-016 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-0480-3038","authenticated-orcid":false,"given":"Ant\u00f3nio Alberto","family":"Oliveira","sequence":"additional","affiliation":[{"name":"Instituto Superior de Administra\u00e7\u00e3o e Finan\u00e7as (ISAF), Luanda, Angola"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8999-1354","authenticated-orcid":false,"given":"Dinis","family":"Pestana","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa (CEAUL), Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Faculdade de Ci\u00eancias da Universidade de Lisboa (DEIO), Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Instituto de Investiga\u00e7\u00e3o Cient\u00edfica Bento da Rocha Cabral, Cal\u00e7ada Bento da Rocha Cabral 14, 1250-012 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9966-2271","authenticated-orcid":false,"given":"Maria Lu\u00edsa","family":"Rocha","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa (CEAUL), Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Faculdade de Economia e Gest\u00e3o, Universidade dos A\u00e7ores, Rua da M\u00e3e de Deus, 9500-321 Ponta Delgada, Portugal"},{"name":"CEEAplA\u2014Centro de Estudos de Economia Aplicada do Atl\u00e2ntico, Universidade dos A\u00e7ores, Rua da M\u00e3e de Deus, 9500-321 Ponta Delgada, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gani, J. 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