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This alternative expectation formula has a wide spectrum of applications. It is natural to ask whether there is a multivariate version of this formula. This note gives an affirmative answer by establishing such a formula using two different approaches. The two approaches employed in this note correspond to the two approaches for the univariate case. Supplementary materials for this article are available online.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"https://tandf.figshare.com/collections/Another_Remark_on_the_Alternative_Expectation_Formula/2078330/1","contentUrl":null,"metadataVersion":2,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2015-12-05T20:15:19.000Z","registered":"2015-12-05T20:15:20.000Z","published":"2015","updated":"2024-03-21T08:45:14.000Z"},"relationships":{"client":{"data":{"id":"figshare.ars","type":"clients"}},"provider":{"data":{"id":"otjm","type":"providers"}},"media":{"data":{"id":"10.6084/m9.figshare.c.2078330.v1","type":"media"}},"references":{"data":[]},"citations":{"data":[]},"parts":{"data":[]},"partOf":{"data":[]},"versions":{"data":[]},"versionOf":{"data":[]}}}}