{ localUrl: '../page/joint_probability_distribution.html', arbitalUrl: 'https://arbital.com/p/joint_probability_distribution', rawJsonUrl: '../raw/467.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'joint_probability_distribution', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Joint probability distribution', clickbait: 'A probability distribution over the collection of joint configurations of all the variables you care about.', textLength: '1639', alias: 'joint_probability_distribution', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'TsviBT', editCreatedAt: '2016-06-11 06:42:46', pageCreatorId: 'TsviBT', pageCreatedAt: '2016-06-11 06:42:46', seeDomainId: '0', editDomainId: 'MaloBourgon', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '63', text: '[summary: \n$$\\newcommand{\\bR}{\\mathbb{R}}\n\\newcommand{\\bP}{\\mathbb{P}}\n\\newcommand{\\cS}{\\mathcal{S}}\n\\newcommand{\\cF}{\\mathcal{F}}\n\\newcommand{\\gO}{\\Omega}\n\\newcommand{\\go}{\\omega}\n\\newcommand{\\ts}{\\times}$$\n\nA joint probability distribution of real-valued random variables $X_1, X_2, \\cdots, X_n$ is a probability distribution $\\bP$ over $\\bR^n$. The probability of the event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ is $\\bP(A_1,A_2, \\cdots, A_n)$. \n]\n\n\n$$\n\\newcommand{\\bR}{\\mathbb{R}}\n\\newcommand{\\bP}{\\mathbb{P}}\n\\newcommand{\\cS}{\\mathcal{S}}\n\\newcommand{\\cF}{\\mathcal{F}}\n\\newcommand{\\gO}{\\Omega}\n\\newcommand{\\go}{\\omega}\n\\newcommand{\\ts}{\\times}\n$$\n\n\nA joint probability distribution of real-valued random variables $X_1, X_2, \\cdots, X_n$ is a probability distribution $\\bP$ over $\\bR^n$. The probability of the event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ is $\\bP(A_1,A_2, \\cdots, A_n)$. \n\n\n# Formal definition\n\nLet $\\{X_i \\}_{i \\in I}$ be a collection of random variables taking values in the spaces $(S_i, \\cS_i)$. Then a joint distribution of the $\\{X_i \\}_{i \\in I}$ is a probability distribution over $\\prod_{i \\in I} S_i$. \n\nIf the $\\{X_i \\}_{i \\in I}$ are defined on a probability space $(\\gO, \\cF, \\bP)$, then $\\bP$ induces a joint distribution of the $X_i$. The induced function is a distribution because an event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ with $A_k \\in \\cS_k$ can be viewed as the event $\\{ \\go \\in \\gO : X_1(\\go) \\in A_1, \\cdots, X_n(\\go) \\in A_n\\}$, which is in $\\cF$ because $\\go \\mapsto (X_1(\\go), \\cdots, X_n(\\go))$ is a measurable map $\\gO \\to S_1 \\ts \\cdots \\ts S_k$.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'TsviBT' ], childIds: [ '468' ], parentIds: [], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [ { id: '40', pageId: 'joint_probability_distribution', lensId: '468', lensIndex: '0', lensName: '(Motivation) coherent probabilit', lensSubtitle: '', createdBy: '1', createdAt: '2016-06-17 21:58:56', updatedBy: '1', updatedAt: '2016-06-17 21:58:56' } ], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12400', pageId: 'joint_probability_distribution', userId: 'TsviBT', edit: '1', type: 'newChild', createdAt: '2016-06-11 06:48:51', auxPageId: '468', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12398', pageId: 'joint_probability_distribution', userId: 'TsviBT', edit: '1', type: 'newEdit', createdAt: '2016-06-11 06:42:46', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'true', hasParents: 'false', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }