{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:27Z","timestamp":1760146407020,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,31]],"date-time":"2024-10-31T00:00:00Z","timestamp":1730332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Write pbeta(\u00b7,\u03b1,\u03b2) for the distribution function of the Beta distribution with parameters \u03b1 and \u03b2. We show that \u03b1\u21a6pbeta(\u03b1\/(\u03b1+\u03b2),\u03b1,\u03b2) is decreasing and \u03b1\u21a6pbeta(\u03b1\/(\u03b1+\u03b2),\u03b1+1,\u03b2) is increasing over the positive reals, with the common limit for \u03b1\u2192\u221e expressible in terms of the Gamma distribution functions, and discuss implications for the distribution functions of the Gamma, Poisson and Binomial distributions.<\/jats:p>","DOI":"10.3390\/e26110938","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T13:09:27Z","timestamp":1730466567000},"page":"938","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two Monotonicity Results for Beta Distribution Functions"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4198-9911","authenticated-orcid":false,"given":"Kurt","family":"Hornik","sequence":"first","affiliation":[{"name":"Institute for Statistics and Mathematics, WU Wirtschaftsuniversit\u00e4t Wien, Welthandelsplatz 1, A-1020 Wien, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"468","DOI":"10.2307\/2325105","article-title":"A Monotonic Function","volume":"99","author":"Letac","year":"1992","journal-title":"Am. Math. Mon."},{"key":"ref_2","first-page":"85","article-title":"\u00dcber gewisse die unvollst\u00e4ndige Betafunktion betreffende Ungleichungen","volume":"191","author":"Vietoris","year":"1982","journal-title":"Sitzungsberichte Der\u00f6sterreischischen Akad. Wiss. Abt. II"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/BF01540779","article-title":"Die Ungleichungen von Vietoris","volume":"98","author":"Raab","year":"1984","journal-title":"Monatshefte Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/0167-7152(86)90044-1","article-title":"Bounds for the Difference between Median and Mean of Gamma and Poisson Distributions","volume":"4","author":"Chen","year":"1986","journal-title":"Stat. Probab. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1214\/aoms\/1177728608","article-title":"An Inequality on Poisson Probabilities","volume":"26","author":"Teicher","year":"1955","journal-title":"Ann. Math. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF02614111","article-title":"Bounds for the Difference Between Median and Mean of Beta and Negative Binomial Distributions","volume":"36","author":"Payton","year":"1989","journal-title":"Metrika"},{"key":"ref_7","unstructured":"Kerman, J. (2011). A closed-form approximation for the median of the Beta distribution. arXiv."},{"key":"ref_8","unstructured":"Olver, F.W.J., Daalhuis, A.B.O., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., and McClain, M.A. (2024, October 28). NIST Digital Library of Mathematical Functions. Release 1.2.2 of 15 September 2024, Available online: https:\/\/dlmf.nist.gov\/."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/938\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:25:51Z","timestamp":1760113551000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/938"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,31]]},"references-count":8,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["e26110938"],"URL":"https:\/\/doi.org\/10.3390\/e26110938","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,10,31]]}}}