{
  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: {}
}