{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:17:24Z","timestamp":1740107844663,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"23","license":[{"start":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T00:00:00Z","timestamp":1595376000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T00:00:00Z","timestamp":1595376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000866","name":"Cardiff University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000866","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2020,12]]},"abstract":"Abstract<\/jats:title>We study properties of two probability distributions defined on the infinite set$$\\{0,1,2, \\ldots \\}$$<\/jats:tex-math>{<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>2<\/mml:mn>,<\/mml:mo>\u2026<\/mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses$$1\/\\textcircled {1}$$<\/jats:tex-math>1<\/mml:mn>\/<\/mml:mo>1<\/mml:mn><\/mml:menclose><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>to all points in the set$$ \\{0,1,\\ldots ,\\textcircled {1}-1\\}$$<\/jats:tex-math>{<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>1<\/mml:mn><\/mml:menclose>-<\/mml:mo>1<\/mml:mn>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where$$\\textcircled {1}$$<\/jats:tex-math>1<\/mml:mn><\/mml:menclose><\/mml:math><\/jats:alternatives><\/jats:inline-formula>denotes the grossone. For this distribution, we study the problem of decomposing a random variable$$\\xi $$<\/jats:tex-math>\u03be<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with this distribution as a sum$$\\xi {\\mathop {=}\\limits ^\\mathrm{d}} \\xi _1 + \\cdots + \\xi _m$$<\/jats:tex-math>\u03be<\/mml:mi>=<\/mml:mo>d<\/mml:mi><\/mml:mover>\u03be<\/mml:mi>1<\/mml:mn><\/mml:msub>+<\/mml:mo>\u22ef<\/mml:mo>+<\/mml:mo>\u03be<\/mml:mi>m<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where$$\\xi _1 , \\ldots , \\xi _m$$<\/jats:tex-math>\u03be<\/mml:mi>1<\/mml:mn><\/mml:msub>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>\u03be<\/mml:mi>m<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin$$(\\textcircled {1},p)$$<\/jats:tex-math>(<\/mml:mo>1<\/mml:mn><\/mml:menclose>,<\/mml:mo>p<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with$$p=c\/\\textcircled {1}^{\\alpha }$$<\/jats:tex-math>p<\/mml:mi>=<\/mml:mo>c<\/mml:mi>\/<\/mml:mo>1<\/mml:mn><\/mml:menclose>\u03b1<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with$$1\/2<\\alpha \\le 1$$<\/jats:tex-math>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><<\/mml:mo>\u03b1<\/mml:mi>\u2264<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The accuracy of this approximation is assessed using a numerical study.<\/jats:p>","DOI":"10.1007\/s00500-020-05190-2","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T13:03:57Z","timestamp":1595423037000},"page":"17517-17524","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Discrete uniform and binomial distributions with infinite support"],"prefix":"10.1007","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5634-5559","authenticated-orcid":false,"given":"Andrey","family":"Pepelyshev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anatoly","family":"Zhigljavsky","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,22]]},"reference":[{"key":"5190_CR1","volume-title":"A primer on statistical distributions","author":"N Balakrishnan","year":"2004","unstructured":"Balakrishnan N, Nevzorov V (2004) A primer on statistical distributions. 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