{ localUrl: '../page/uncountable_sample_spaces_are_too_large.html', arbitalUrl: 'https://arbital.com/p/uncountable_sample_spaces_are_too_large', rawJsonUrl: '../raw/3tl.json', likeableId: '2550', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'ZackMDavis', 'EricRogstad' ], pageId: 'uncountable_sample_spaces_are_too_large', edit: '4', editSummary: '', prevEdit: '3', currentEdit: '4', wasPublished: 'true', type: 'wiki', title: 'Uncountable sample spaces are way too large', clickbait: 'We can't define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.', textLength: '1113', alias: 'uncountable_sample_spaces_are_too_large', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'TsviBT', editCreatedAt: '2016-06-16 09:05:14', pageCreatorId: 'TsviBT', pageCreatedAt: '2016-05-25 22:29:53', seeDomainId: '0', editDomainId: 'AlexeiAndreev', 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: '46', text: '[summary: \nIf the sample space $\\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\\Omega$ in the same way we defined\n[3tb probability distributions] over countable sample spaces, i.e. by just assigning numbers to each point in the sample space. Any function $f: \\Omega \\to [0,1]$ with $\\sum_{\\omega \\in \\Omega} f(\\omega) = 1$ can only assign positive values to at most countably many elements of $\\Omega$. ]\n\n\n\nIf the sample space $\\Omega$ is [2w0 uncountable], then in general we can't even define a probability distribution over $\\Omega$ in the same way we defined\n[3tb probability distributions] over countable sample spaces, i.e. by just assigning numbers to each point in the sample space. Any function $f: \\Omega \\to [0,1]$ with $\\sum_{\\omega \\in \\Omega} f(\\omega) = 1$ can only assign positive values to at most countably many elements of $\\Omega$. But this means we can't, for example, talk about a\n[uniform_distribution uniform distribution] over the interval $[0,2]$, which intuitively should assign equal probability to everywhere in the interval. \n', 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: [], parentIds: [ 'sample_space_probability' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], 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: '13286', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '4', type: 'newEdit', createdAt: '2016-06-16 09:05:14', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11156', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '3', type: 'newEdit', createdAt: '2016-05-27 08:47:01', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11054', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '2', type: 'newParent', createdAt: '2016-05-25 22:35:54', auxPageId: 'sample_space_probability', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11052', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '0', type: 'deleteParent', createdAt: '2016-05-25 22:35:45', auxPageId: 'probability_theory', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11043', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '2', type: 'newEdit', createdAt: '2016-05-25 22:32:01', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11040', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '1', type: 'newEdit', createdAt: '2016-05-25 22:29:53', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11035', pageId: 'uncountable_sample_spaces_are_too_large', userId: 'TsviBT', edit: '1', type: 'newParent', createdAt: '2016-05-25 22:26:21', auxPageId: 'probability_theory', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }