{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T02:56:48Z","timestamp":1649041008277},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2010,1,1]]},"abstract":"<jats:title>Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables<\/jats:title>\n        <jats:p>In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.<\/jats:p>","DOI":"10.2478\/v10037-010-0026-6","type":"journal-article","created":{"date-parts":[[2011,1,6]],"date-time":"2011-01-06T02:22:47Z","timestamp":1294280567000},"page":"213-217","source":"Crossref","is-referenced-by-count":0,"title":["Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables"],"prefix":"10.2478","volume":"18","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":[{"issue":"1","key":"1","first-page":"107","article-title":"Segments of natural numbers and finite sequences","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"2","first-page":"173","article-title":"Series of positive real numbers. Measure theory","volume":"2","author":"J\u00f3zef Bia\u0142as","year":"1991","journal-title":"Formalized Mathematics"},{"issue":"1","key":"3","first-page":"175","article-title":"Binary operations","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"4","first-page":"55","article-title":"Functions and their basic properties","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"5","first-page":"153","article-title":"Functions from a set to a set","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"2","key":"6","first-page":"357","article-title":"Partial functions","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"7","first-page":"47","article-title":"Some basic properties of sets","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"4","key":"8","first-page":"661","article-title":"The sum and product of finite sequences of real numbers","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"9","first-page":"165","article-title":"Finite sets","volume":"1","author":"Agata Darmochwa\u0142","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"2","key":"10","first-page":"269","article-title":"Real sequences and basic operations on them","volume":"1","author":"Jaros\u0142aw Kotowicz","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"319","DOI":"10.2478\/v10037-008-0039-6","article-title":"Integral of complex-valued measurable function","volume":"16","author":"Keiko Narita","year":"2008","journal-title":"Formalized Mathematics"},{"issue":"2","key":"12","first-page":"401","article-title":"s-fields and probability","volume":"1","author":"Andrzej N\u0119dzusiak","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"129","DOI":"10.2478\/v10037-009-0014-x","article-title":"Probability on finite set and real-valued random variables","volume":"17","author":"Hiroyuki Okazaki","year":"2009","journal-title":"Formalized Mathematics"},{"issue":"3","key":"14","first-page":"555","article-title":"Real functions spaces","volume":"1","author":"Henryk Oryszczyszyn","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"143","DOI":"10.2478\/v10037-006-0018-8","article-title":"Integral of real-valued measurable function","volume":"14","author":"Yasunari Shidama","year":"2006","journal-title":"Formalized Mathematics"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"115","DOI":"10.2478\/v10037-008-0017-z","article-title":"Banach algebra of bounded functionals","volume":"16","author":"Yasunari Shidama","year":"2008","journal-title":"Formalized Mathematics"},{"issue":"2","key":"17","first-page":"329","article-title":"Binary operations applied to functions","volume":"1","author":"Andrzej Trybulec","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"18","first-page":"67","article-title":"Properties of subsets","volume":"1","author":"Zinaida Trybulec","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"19","first-page":"73","article-title":"Relations and their basic properties","volume":"1","author":"Edmund Woronowicz","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"20","first-page":"181","article-title":"Relations defined on sets","volume":"1","author":"Edmund Woronowicz","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"2","key":"21","first-page":"169","article-title":"Bessel's inequality","volume":"11","author":"Hiroshi Yamazaki","year":"2003","journal-title":"Formalized Mathematics"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"225","DOI":"10.2478\/v10037-006-0026-8","article-title":"The relevance of measure and probability, and definition of completeness of probability","volume":"14","author":"Bo Zhang","year":"2006","journal-title":"Formalized Mathematics"}],"container-title":["Formalized Mathematics"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/18\/4\/article-p213.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/forma.2010.18.issue-4\/v10037-010-0026-6\/v10037-010-0026-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T15:28:08Z","timestamp":1590852488000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10037-010-0026-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,1]]},"references-count":22,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.2478\/v10037-010-0026-6","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2010,1,1]]}}}