{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T15:38:44Z","timestamp":1649173124849},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,1,1]]},"abstract":"<jats:title>Probability on Finite Set and Real-Valued Random Variables<\/jats:title>\n        <jats:p>In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The <jats:italic>Formalized Mathematics<\/jats:italic> has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.<\/jats:p>","DOI":"10.2478\/v10037-009-0014-x","type":"journal-article","created":{"date-parts":[[2009,7,15]],"date-time":"2009-07-15T01:04:59Z","timestamp":1247619899000},"page":"129-136","source":"Crossref","is-referenced-by-count":2,"title":["Probability on Finite Set and Real-Valued Random Variables"],"prefix":"10.2478","volume":"17","author":[{"given":"Hiroyuki","family":"Okazaki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"key":"1","first-page":"377","article-title":"Cardinal numbers","volume":"2","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"key":"2","first-page":"41","article-title":"The fundamental properties of natural numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"key":"3","first-page":"91","article-title":"The ordinal numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"key":"4","first-page":"107","article-title":"Segments of natural numbers and finite sequences","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"key":"5","first-page":"163","article-title":"Infimum and supremum of the set of real numbers. 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