{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:15Z","timestamp":1740491295286,"version":"3.38.0"},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"A","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p>The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, \u03c0). To fix the scale, one side chosen arbitrarily is assigned unit length; let <jats:italic>a<\/jats:italic> and <jats:italic>b<\/jats:italic> denote the lengths of the other sides. We find the density functions of <jats:italic>a<\/jats:italic> \/ <jats:italic>b<\/jats:italic>, max{<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>}, min{<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, \u221e) and (\u00bd, \u221e), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).<\/jats:p>","DOI":"10.1017\/s0021900200021331","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T16:25:31Z","timestamp":1459355131000},"page":"283-295","source":"Crossref","is-referenced-by-count":0,"title":["Size distributions in random triangles"],"prefix":"10.1017","volume":"51","author":[{"given":"D. J.","family":"Daley","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sven","family":"Ebert","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. J.","family":"Swift","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,3,30]]},"reference":[{"volume-title":"Geometric Probability","year":"1963","key":"S0021900200021331_ref3"},{"volume-title":"An Introduction to Geometrical Probability: Distributional Aspects with Applications","year":"1999","key":"S0021900200021331_ref4"},{"first-page":"3","volume-title":"Stochastic Geometry","year":"1974","key":"S0021900200021331_ref2"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200021331","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,21]],"date-time":"2017-04-21T18:41:53Z","timestamp":1492800113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200021331\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12]]},"references-count":3,"journal-issue":{"issue":"A","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["S0021900200021331"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200021331","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"type":"print","value":"0021-9002"},{"type":"electronic","value":"1475-6072"}],"subject":[],"published":{"date-parts":[[2014,12]]}}}