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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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策略全流程","\u002Fzh\u002Fquant\u002F09-case-study","zh\u002Fquant\u002F09-case-study",{"title":1779,"path":1780,"stem":1781},"量化投资文献与软件图谱","\u002Fzh\u002Fquant\u002F10-reading-software-map","zh\u002Fquant\u002F10-reading-software-map",null,{"id":1784,"title":1663,"body":1785,"description":6193,"extension":6194,"features":1782,"hero":1782,"layout":1782,"locale":1782,"meta":6195,"navigation":1782,"path":1664,"published":6198,"seo":6199,"stem":1665,"__hash__":6200},"docs\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F03-moment\u002Findex.md",{"type":1786,"value":1787,"toc":6179},"minimark",[1788,1792,1803,1806,1810,1813,1832,1836,1839,2125,2128,2396,2488,2491,2710,2820,2824,3136,3300,3303,3533,3536,3540,3736,4072,4335,4339,4409,4528,4531,4655,4658,4867,4870,4874,4877,4884,4887,4891,5078,5087,5090,5265,5502,5618,5622,5679,5861,5864,6095,6098,6102,6122,6125,6128,6136,6139,6142,6166],[1789,1790,1663],"h1",{"id":1791},"第三章期望方差与条件矩",[1793,1794,1795],"blockquote",{},[1796,1797,1798,1802],"p",{},[1799,1800,1801],"strong",{},"案例："," 保险赔付的总体波动，来自每类客户内部的随机性，还是高低风险客户之间的差异？",[1796,1804,1805],{},"矩把完整分布压缩成可解释的数值。压缩会丢失信息，因此应知道每个矩回答什么、不能回答什么。",[1807,1808,1809],"h2",{"id":1809},"学习目标",[1796,1811,1812],{},"你应能：",[1814,1815,1816,1820,1823,1826,1829],"ol",{},[1817,1818,1819],"li",{},"计算离散与连续随机变量的期望；",[1817,1821,1822],{},"使用期望的线性性与 LOTUS；",[1817,1824,1825],{},"分解方差、协方差和相关系数；",[1817,1827,1828],{},"运用条件期望、全期望与全方差；",[1817,1830,1831],{},"说明矩母函数何时存在、何时不能使用。",[1807,1833,1835],{"id":1834},"_1-期望是概率加权平均","1. 期望是概率加权平均",[1796,1837,1838],{},"离散情形：",[1840,1841,1844],"span",{"className":1842},[1843],"katex-display",[1840,1845,1848,1929],{"className":1846},[1847],"katex",[1840,1849,1852],{"className":1850},[1851],"katex-mathml",[1853,1854,1857],"math",{"xmlns":1855,"display":1856},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[1858,1859,1860,1924],"semantics",{},[1861,1862,1863,1867,1872,1875,1878,1881,1884,1887,1890,1899,1901,1903,1905,1907,1914,1916,1918,1920],"mrow",{},[1864,1865,1866],"mi",{},"E",[1868,1869,1871],"mo",{"stretchy":1870},"false","[",[1864,1873,1874],{},"g",[1868,1876,1877],{"stretchy":1870},"(",[1864,1879,1880],{},"X",[1868,1882,1883],{"stretchy":1870},")",[1868,1885,1886],{"stretchy":1870},"]",[1868,1888,1889],{},"=",[1891,1892,1893,1896],"munder",{},[1868,1894,1895],{},"∑",[1864,1897,1898],{},"x",[1864,1900,1874],{},[1868,1902,1877],{"stretchy":1870},[1864,1904,1898],{},[1868,1906,1883],{"stretchy":1870},[1908,1909,1910,1912],"msub",{},[1864,1911,1796],{},[1864,1913,1880],{},[1868,1915,1877],{"stretchy":1870},[1864,1917,1898],{},[1868,1919,1883],{"stretchy":1870},[1864,1921,1923],{"mathvariant":1922},"normal",".",[1925,1926,1928],"annotation",{"encoding":1927},"application\u002Fx-tex","E[g(X)]=\\sum_xg(x)p_X(x).",[1840,1930,1934,1981],{"className":1931,"ariaHidden":1933},[1932],"katex-html","true",[1840,1935,1938,1943,1949,1953,1957,1960,1964,1969,1974,1978],{"className":1936},[1937],"base",[1840,1939],{"className":1940,"style":1942},[1941],"strut","height:1em;vertical-align:-0.25em;",[1840,1944,1866],{"className":1945,"style":1948},[1946,1947],"mord","mathnormal","margin-right:0.0576em;",[1840,1950,1871],{"className":1951},[1952],"mopen",[1840,1954,1874],{"className":1955,"style":1956},[1946,1947],"margin-right:0.0359em;",[1840,1958,1877],{"className":1959},[1952],[1840,1961,1880],{"className":1962,"style":1963},[1946,1947],"margin-right:0.0785em;",[1840,1965,1968],{"className":1966},[1967],"mclose",")]",[1840,1970],{"className":1971,"style":1973},[1972],"mspace","margin-right:0.2778em;",[1840,1975,1889],{"className":1976},[1977],"mrel",[1840,1979],{"className":1980,"style":1973},[1972],[1840,1982,1984,1988,2052,2056,2059,2062,2065,2068,2113,2116,2119,2122],{"className":1983},[1937],[1840,1985],{"className":1986,"style":1987},[1941],"height:2.3em;vertical-align:-1.25em;",[1840,1989,1993],{"className":1990},[1991,1992],"mop","op-limits",[1840,1994,1998,2043],{"className":1995},[1996,1997],"vlist-t","vlist-t2",[1840,1999,2002,2038],{"className":2000},[2001],"vlist-r",[1840,2003,2007,2025],{"className":2004,"style":2006},[2005],"vlist","height:1.05em;",[1840,2008,2010,2015],{"style":2009},"top:-1.9em;margin-left:0em;",[1840,2011],{"className":2012,"style":2014},[2013],"pstrut","height:3.05em;",[1840,2016,2022],{"className":2017},[2018,2019,2020,2021],"sizing","reset-size6","size3","mtight",[1840,2023,1898],{"className":2024},[1946,1947,2021],[1840,2026,2028,2031],{"style":2027},"top:-3.05em;",[1840,2029],{"className":2030,"style":2014},[2013],[1840,2032,2033],{},[1840,2034,1895],{"className":2035},[1991,2036,2037],"op-symbol","large-op",[1840,2039,2042],{"className":2040},[2041],"vlist-s","​",[1840,2044,2046],{"className":2045},[2001],[1840,2047,2050],{"className":2048,"style":2049},[2005],"height:1.25em;",[1840,2051],{},[1840,2053],{"className":2054,"style":2055},[1972],"margin-right:0.1667em;",[1840,2057,1874],{"className":2058,"style":1956},[1946,1947],[1840,2060,1877],{"className":2061},[1952],[1840,2063,1898],{"className":2064},[1946,1947],[1840,2066,1883],{"className":2067},[1967],[1840,2069,2071,2074],{"className":2070},[1946],[1840,2072,1796],{"className":2073},[1946,1947],[1840,2075,2078],{"className":2076},[2077],"msupsub",[1840,2079,2081,2104],{"className":2080},[1996,1997],[1840,2082,2084,2101],{"className":2083},[2001],[1840,2085,2088],{"className":2086,"style":2087},[2005],"height:0.3283em;",[1840,2089,2091,2095],{"style":2090},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1840,2092],{"className":2093,"style":2094},[2013],"height:2.7em;",[1840,2096,2098],{"className":2097},[2018,2019,2020,2021],[1840,2099,1880],{"className":2100,"style":1963},[1946,1947,2021],[1840,2102,2042],{"className":2103},[2041],[1840,2105,2107],{"className":2106},[2001],[1840,2108,2111],{"className":2109,"style":2110},[2005],"height:0.15em;",[1840,2112],{},[1840,2114,1877],{"className":2115},[1952],[1840,2117,1898],{"className":2118},[1946,1947],[1840,2120,1883],{"className":2121},[1967],[1840,2123,1923],{"className":2124},[1946],[1796,2126,2127],{},"连续情形：",[1840,2129,2131],{"className":2130},[1843],[1840,2132,2134,2210],{"className":2133},[1847],[1840,2135,2137],{"className":2136},[1851],[1853,2138,2139],{"xmlns":1855,"display":1856},[1858,2140,2141,2207],{},[1861,2142,2143,2145,2147,2149,2151,2153,2155,2157,2159,2175,2177,2179,2181,2183,2190,2192,2194,2196,2200,2203,2205],{},[1864,2144,1866],{},[1868,2146,1871],{"stretchy":1870},[1864,2148,1874],{},[1868,2150,1877],{"stretchy":1870},[1864,2152,1880],{},[1868,2154,1883],{"stretchy":1870},[1868,2156,1886],{"stretchy":1870},[1868,2158,1889],{},[2160,2161,2162,2165,2173],"msubsup",{},[1868,2163,2164],{},"∫",[1861,2166,2167,2170],{},[1868,2168,2169],{},"−",[1864,2171,2172],{"mathvariant":1922},"∞",[1864,2174,2172],{"mathvariant":1922},[1864,2176,1874],{},[1868,2178,1877],{"stretchy":1870},[1864,2180,1898],{},[1868,2182,1883],{"stretchy":1870},[1908,2184,2185,2188],{},[1864,2186,2187],{},"f",[1864,2189,1880],{},[1868,2191,1877],{"stretchy":1870},[1864,2193,1898],{},[1868,2195,1883],{"stretchy":1870},[2197,2198,2199],"mtext",{}," ",[1864,2201,2202],{},"d",[1864,2204,1898],{},[1864,2206,1923],{"mathvariant":1922},[1925,2208,2209],{"encoding":1927},"E[g(X)]=\\int_{-\\infty}^{\\infty}g(x)f_X(x)\\,dx.",[1840,2211,2213,2246],{"className":2212,"ariaHidden":1933},[1932],[1840,2214,2216,2219,2222,2225,2228,2231,2234,2237,2240,2243],{"className":2215},[1937],[1840,2217],{"className":2218,"style":1942},[1941],[1840,2220,1866],{"className":2221,"style":1948},[1946,1947],[1840,2223,1871],{"className":2224},[1952],[1840,2226,1874],{"className":2227,"style":1956},[1946,1947],[1840,2229,1877],{"className":2230},[1952],[1840,2232,1880],{"className":2233,"style":1963},[1946,1947],[1840,2235,1968],{"className":2236},[1967],[1840,2238],{"className":2239,"style":1973},[1972],[1840,2241,1889],{"className":2242},[1977],[1840,2244],{"className":2245,"style":1973},[1972],[1840,2247,2249,2253,2318,2321,2324,2327,2330,2333,2375,2378,2381,2384,2387,2390,2393],{"className":2248},[1937],[1840,2250],{"className":2251,"style":2252},[1941],"height:2.3846em;vertical-align:-0.9703em;",[1840,2254,2256,2260],{"className":2255},[1991],[1840,2257,2164],{"className":2258,"style":2259},[1991,2036,2037],"margin-right:0.4445em;position:relative;top:-0.0011em;",[1840,2261,2263],{"className":2262},[2077],[1840,2264,2266,2309],{"className":2265},[1996,1997],[1840,2267,2269,2306],{"className":2268},[2001],[1840,2270,2273,2291],{"className":2271,"style":2272},[2005],"height:1.4143em;",[1840,2274,2276,2279],{"style":2275},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[1840,2277],{"className":2278,"style":2094},[2013],[1840,2280,2282],{"className":2281},[2018,2019,2020,2021],[1840,2283,2285,2288],{"className":2284},[1946,2021],[1840,2286,2169],{"className":2287},[1946,2021],[1840,2289,2172],{"className":2290},[1946,2021],[1840,2292,2294,2297],{"style":2293},"top:-3.8129em;margin-right:0.05em;",[1840,2295],{"className":2296,"style":2094},[2013],[1840,2298,2300],{"className":2299},[2018,2019,2020,2021],[1840,2301,2303],{"className":2302},[1946,2021],[1840,2304,2172],{"className":2305},[1946,2021],[1840,2307,2042],{"className":2308},[2041],[1840,2310,2312],{"className":2311},[2001],[1840,2313,2316],{"className":2314,"style":2315},[2005],"height:0.9703em;",[1840,2317],{},[1840,2319],{"className":2320,"style":2055},[1972],[1840,2322,1874],{"className":2323,"style":1956},[1946,1947],[1840,2325,1877],{"className":2326},[1952],[1840,2328,1898],{"className":2329},[1946,1947],[1840,2331,1883],{"className":2332},[1967],[1840,2334,2336,2340],{"className":2335},[1946],[1840,2337,2187],{"className":2338,"style":2339},[1946,1947],"margin-right:0.1076em;",[1840,2341,2343],{"className":2342},[2077],[1840,2344,2346,2367],{"className":2345},[1996,1997],[1840,2347,2349,2364],{"className":2348},[2001],[1840,2350,2352],{"className":2351,"style":2087},[2005],[1840,2353,2355,2358],{"style":2354},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[1840,2356],{"className":2357,"style":2094},[2013],[1840,2359,2361],{"className":2360},[2018,2019,2020,2021],[1840,2362,1880],{"className":2363,"style":1963},[1946,1947,2021],[1840,2365,2042],{"className":2366},[2041],[1840,2368,2370],{"className":2369},[2001],[1840,2371,2373],{"className":2372,"style":2110},[2005],[1840,2374],{},[1840,2376,1877],{"className":2377},[1952],[1840,2379,1898],{"className":2380},[1946,1947],[1840,2382,1883],{"className":2383},[1967],[1840,2385],{"className":2386,"style":2055},[1972],[1840,2388,2202],{"className":2389},[1946,1947],[1840,2391,1898],{"className":2392},[1946,1947],[1840,2394,1923],{"className":2395},[1946],[1796,2397,2398,2399,2443,2444,2487],{},"这就是 LOTUS：计算 ",[1840,2400,2402,2422],{"className":2401},[1847],[1840,2403,2405],{"className":2404},[1851],[1853,2406,2407],{"xmlns":1855},[1858,2408,2409,2419],{},[1861,2410,2411,2413,2415,2417],{},[1864,2412,1874],{},[1868,2414,1877],{"stretchy":1870},[1864,2416,1880],{},[1868,2418,1883],{"stretchy":1870},[1925,2420,2421],{"encoding":1927},"g(X)",[1840,2423,2425],{"className":2424,"ariaHidden":1933},[1932],[1840,2426,2428,2431,2434,2437,2440],{"className":2427},[1937],[1840,2429],{"className":2430,"style":1942},[1941],[1840,2432,1874],{"className":2433,"style":1956},[1946,1947],[1840,2435,1877],{"className":2436},[1952],[1840,2438,1880],{"className":2439,"style":1963},[1946,1947],[1840,2441,1883],{"className":2442},[1967]," 的期望不必先求 ",[1840,2445,2447,2466],{"className":2446},[1847],[1840,2448,2450],{"className":2449},[1851],[1853,2451,2452],{"xmlns":1855},[1858,2453,2454,2464],{},[1861,2455,2456,2458,2460,2462],{},[1864,2457,1874],{},[1868,2459,1877],{"stretchy":1870},[1864,2461,1880],{},[1868,2463,1883],{"stretchy":1870},[1925,2465,2421],{"encoding":1927},[1840,2467,2469],{"className":2468,"ariaHidden":1933},[1932],[1840,2470,2472,2475,2478,2481,2484],{"className":2471},[1937],[1840,2473],{"className":2474,"style":1942},[1941],[1840,2476,1874],{"className":2477,"style":1956},[1946,1947],[1840,2479,1877],{"className":2480},[1952],[1840,2482,1880],{"className":2483,"style":1963},[1946,1947],[1840,2485,1883],{"className":2486},[1967]," 的分布。存在性要求相应绝对矩有限；柯西分布的“对称中心”不能当作有限期望。",[1796,2489,2490],{},"期望的线性性不要求独立：",[1840,2492,2494],{"className":2493},[1843],[1840,2495,2497,2564],{"className":2496},[1847],[1840,2498,2500],{"className":2499},[1851],[1853,2501,2502],{"xmlns":1855,"display":1856},[1858,2503,2504,2561],{},[1861,2505,2506,2508,2510,2513,2515,2518,2521,2524,2526,2529,2531,2533,2535,2537,2539,2541,2543,2545,2547,2549,2551,2553,2555,2557,2559],{},[1864,2507,1866],{},[1868,2509,1871],{"stretchy":1870},[1864,2511,2512],{},"a",[1864,2514,1880],{},[1868,2516,2517],{},"+",[1864,2519,2520],{},"b",[1864,2522,2523],{},"Y",[1868,2525,2517],{},[1864,2527,2528],{},"c",[1868,2530,1886],{"stretchy":1870},[1868,2532,1889],{},[1864,2534,2512],{},[1864,2536,1866],{},[1868,2538,1871],{"stretchy":1870},[1864,2540,1880],{},[1868,2542,1886],{"stretchy":1870},[1868,2544,2517],{},[1864,2546,2520],{},[1864,2548,1866],{},[1868,2550,1871],{"stretchy":1870},[1864,2552,2523],{},[1868,2554,1886],{"stretchy":1870},[1868,2556,2517],{},[1864,2558,2528],{},[1864,2560,1923],{"mathvariant":1922},[1925,2562,2563],{"encoding":1927},"E[aX+bY+c]=aE[X]+bE[Y]+c.",[1840,2565,2567,2596,2616,2637,2667,2697],{"className":2566,"ariaHidden":1933},[1932],[1840,2568,2570,2573,2576,2579,2582,2585,2589,2593],{"className":2569},[1937],[1840,2571],{"className":2572,"style":1942},[1941],[1840,2574,1866],{"className":2575,"style":1948},[1946,1947],[1840,2577,1871],{"className":2578},[1952],[1840,2580,2512],{"className":2581},[1946,1947],[1840,2583,1880],{"className":2584,"style":1963},[1946,1947],[1840,2586],{"className":2587,"style":2588},[1972],"margin-right:0.2222em;",[1840,2590,2517],{"className":2591},[2592],"mbin",[1840,2594],{"className":2595,"style":2588},[1972],[1840,2597,2599,2603,2607,2610,2613],{"className":2598},[1937],[1840,2600],{"className":2601,"style":2602},[1941],"height:0.7778em;vertical-align:-0.0833em;",[1840,2604,2606],{"className":2605,"style":2588},[1946,1947],"bY",[1840,2608],{"className":2609,"style":2588},[1972],[1840,2611,2517],{"className":2612},[2592],[1840,2614],{"className":2615,"style":2588},[1972],[1840,2617,2619,2622,2625,2628,2631,2634],{"className":2618},[1937],[1840,2620],{"className":2621,"style":1942},[1941],[1840,2623,2528],{"className":2624},[1946,1947],[1840,2626,1886],{"className":2627},[1967],[1840,2629],{"className":2630,"style":1973},[1972],[1840,2632,1889],{"className":2633},[1977],[1840,2635],{"className":2636,"style":1973},[1972],[1840,2638,2640,2643,2646,2649,2652,2655,2658,2661,2664],{"className":2639},[1937],[1840,2641],{"className":2642,"style":1942},[1941],[1840,2644,2512],{"className":2645},[1946,1947],[1840,2647,1866],{"className":2648,"style":1948},[1946,1947],[1840,2650,1871],{"className":2651},[1952],[1840,2653,1880],{"className":2654,"style":1963},[1946,1947],[1840,2656,1886],{"className":2657},[1967],[1840,2659],{"className":2660,"style":2588},[1972],[1840,2662,2517],{"className":2663},[2592],[1840,2665],{"className":2666,"style":2588},[1972],[1840,2668,2670,2673,2676,2679,2682,2685,2688,2691,2694],{"className":2669},[1937],[1840,2671],{"className":2672,"style":1942},[1941],[1840,2674,2520],{"className":2675},[1946,1947],[1840,2677,1866],{"className":2678,"style":1948},[1946,1947],[1840,2680,1871],{"className":2681},[1952],[1840,2683,2523],{"className":2684,"style":2588},[1946,1947],[1840,2686,1886],{"className":2687},[1967],[1840,2689],{"className":2690,"style":2588},[1972],[1840,2692,2517],{"className":2693},[2592],[1840,2695],{"className":2696,"style":2588},[1972],[1840,2698,2700,2704,2707],{"className":2699},[1937],[1840,2701],{"className":2702,"style":2703},[1941],"height:0.4306em;",[1840,2705,2528],{"className":2706},[1946,1947],[1840,2708,1923],{"className":2709},[1946],[1796,2711,2712,2713,2819],{},"独立性在计算乘积期望 ",[1840,2714,2716,2756],{"className":2715},[1847],[1840,2717,2719],{"className":2718},[1851],[1853,2720,2721],{"xmlns":1855},[1858,2722,2723,2753],{},[1861,2724,2725,2727,2729,2731,2733,2735,2737,2739,2741,2743,2745,2747,2749,2751],{},[1864,2726,1866],{},[1868,2728,1871],{"stretchy":1870},[1864,2730,1880],{},[1864,2732,2523],{},[1868,2734,1886],{"stretchy":1870},[1868,2736,1889],{},[1864,2738,1866],{},[1868,2740,1871],{"stretchy":1870},[1864,2742,1880],{},[1868,2744,1886],{"stretchy":1870},[1864,2746,1866],{},[1868,2748,1871],{"stretchy":1870},[1864,2750,2523],{},[1868,2752,1886],{"stretchy":1870},[1925,2754,2755],{"encoding":1927},"E[XY]=E[X]E[Y]",[1840,2757,2759,2789],{"className":2758,"ariaHidden":1933},[1932],[1840,2760,2762,2765,2768,2771,2774,2777,2780,2783,2786],{"className":2761},[1937],[1840,2763],{"className":2764,"style":1942},[1941],[1840,2766,1866],{"className":2767,"style":1948},[1946,1947],[1840,2769,1871],{"className":2770},[1952],[1840,2772,1880],{"className":2773,"style":1963},[1946,1947],[1840,2775,2523],{"className":2776,"style":2588},[1946,1947],[1840,2778,1886],{"className":2779},[1967],[1840,2781],{"className":2782,"style":1973},[1972],[1840,2784,1889],{"className":2785},[1977],[1840,2787],{"className":2788,"style":1973},[1972],[1840,2790,2792,2795,2798,2801,2804,2807,2810,2813,2816],{"className":2791},[1937],[1840,2793],{"className":2794,"style":1942},[1941],[1840,2796,1866],{"className":2797,"style":1948},[1946,1947],[1840,2799,1871],{"className":2800},[1952],[1840,2802,1880],{"className":2803,"style":1963},[1946,1947],[1840,2805,1886],{"className":2806},[1967],[1840,2808,1866],{"className":2809,"style":1948},[1946,1947],[1840,2811,1871],{"className":2812},[1952],[1840,2814,2523],{"className":2815,"style":2588},[1946,1947],[1840,2817,1886],{"className":2818},[1967]," 时才关键。",[1807,2821,2823],{"id":2822},"_2-方差度量围绕均值的平方波动","2. 方差度量围绕均值的平方波动",[1840,2825,2827],{"className":2826},[1843],[1840,2828,2830,2915],{"className":2829},[1847],[1840,2831,2833],{"className":2832},[1851],[1853,2834,2835],{"xmlns":1855,"display":1856},[1858,2836,2837,2912],{},[1861,2838,2839,2842,2845,2847,2849,2851,2853,2855,2857,2859,2861,2863,2865,2867,2869,2871,2880,2882,2884,2886,2888,2894,2896,2898,2900,2902,2904,2910],{},[1864,2840,2841],{"mathvariant":1922},"Var",[1868,2843,2844],{},"⁡",[1868,2846,1877],{"stretchy":1870},[1864,2848,1880],{},[1868,2850,1883],{"stretchy":1870},[1868,2852,1889],{},[1864,2854,1866],{},[1868,2856,1871],{"stretchy":1870},[1868,2858,1877],{"stretchy":1870},[1864,2860,1880],{},[1868,2862,2169],{},[1864,2864,1866],{},[1868,2866,1871],{"stretchy":1870},[1864,2868,1880],{},[1868,2870,1886],{"stretchy":1870},[2872,2873,2874,2876],"msup",{},[1868,2875,1883],{"stretchy":1870},[2877,2878,2879],"mn",{},"2",[1868,2881,1886],{"stretchy":1870},[1868,2883,1889],{},[1864,2885,1866],{},[1868,2887,1871],{"stretchy":1870},[2872,2889,2890,2892],{},[1864,2891,1880],{},[2877,2893,2879],{},[1868,2895,1886],{"stretchy":1870},[1868,2897,2169],{},[1864,2899,1866],{},[1868,2901,1871],{"stretchy":1870},[1864,2903,1880],{},[2872,2905,2906,2908],{},[1868,2907,1886],{"stretchy":1870},[2877,2909,2879],{},[1864,2911,1923],{"mathvariant":1922},[1925,2913,2914],{"encoding":1927},"\\operatorname{Var}(X)\n=E[(X-E[X])^2]\n=E[X^2]-E[X]^2.",[1840,2916,2918,2949,2974,3036,3089],{"className":2917,"ariaHidden":1933},[1932],[1840,2919,2921,2924,2931,2934,2937,2940,2943,2946],{"className":2920},[1937],[1840,2922],{"className":2923,"style":1942},[1941],[1840,2925,2927],{"className":2926},[1991],[1840,2928,2841],{"className":2929},[1946,2930],"mathrm",[1840,2932,1877],{"className":2933},[1952],[1840,2935,1880],{"className":2936,"style":1963},[1946,1947],[1840,2938,1883],{"className":2939},[1967],[1840,2941],{"className":2942,"style":1973},[1972],[1840,2944,1889],{"className":2945},[1977],[1840,2947],{"className":2948,"style":1973},[1972],[1840,2950,2952,2955,2958,2962,2965,2968,2971],{"className":2951},[1937],[1840,2953],{"className":2954,"style":1942},[1941],[1840,2956,1866],{"className":2957,"style":1948},[1946,1947],[1840,2959,2961],{"className":2960},[1952],"[(",[1840,2963,1880],{"className":2964,"style":1963},[1946,1947],[1840,2966],{"className":2967,"style":2588},[1972],[1840,2969,2169],{"className":2970},[2592],[1840,2972],{"className":2973,"style":2588},[1972],[1840,2975,2977,2981,2984,2987,2990,2993,3024,3027,3030,3033],{"className":2976},[1937],[1840,2978],{"className":2979,"style":2980},[1941],"height:1.1141em;vertical-align:-0.25em;",[1840,2982,1866],{"className":2983,"style":1948},[1946,1947],[1840,2985,1871],{"className":2986},[1952],[1840,2988,1880],{"className":2989,"style":1963},[1946,1947],[1840,2991,1886],{"className":2992},[1967],[1840,2994,2996,2999],{"className":2995},[1967],[1840,2997,1883],{"className":2998},[1967],[1840,3000,3002],{"className":3001},[2077],[1840,3003,3005],{"className":3004},[1996],[1840,3006,3008],{"className":3007},[2001],[1840,3009,3012],{"className":3010,"style":3011},[2005],"height:0.8641em;",[1840,3013,3015,3018],{"style":3014},"top:-3.113em;margin-right:0.05em;",[1840,3016],{"className":3017,"style":2094},[2013],[1840,3019,3021],{"className":3020},[2018,2019,2020,2021],[1840,3022,2879],{"className":3023},[1946,2021],[1840,3025,1886],{"className":3026},[1967],[1840,3028],{"className":3029,"style":1973},[1972],[1840,3031,1889],{"className":3032},[1977],[1840,3034],{"className":3035,"style":1973},[1972],[1840,3037,3039,3042,3045,3048,3077,3080,3083,3086],{"className":3038},[1937],[1840,3040],{"className":3041,"style":2980},[1941],[1840,3043,1866],{"className":3044,"style":1948},[1946,1947],[1840,3046,1871],{"className":3047},[1952],[1840,3049,3051,3054],{"className":3050},[1946],[1840,3052,1880],{"className":3053,"style":1963},[1946,1947],[1840,3055,3057],{"className":3056},[2077],[1840,3058,3060],{"className":3059},[1996],[1840,3061,3063],{"className":3062},[2001],[1840,3064,3066],{"className":3065,"style":3011},[2005],[1840,3067,3068,3071],{"style":3014},[1840,3069],{"className":3070,"style":2094},[2013],[1840,3072,3074],{"className":3073},[2018,2019,2020,2021],[1840,3075,2879],{"className":3076},[1946,2021],[1840,3078,1886],{"className":3079},[1967],[1840,3081],{"className":3082,"style":2588},[1972],[1840,3084,2169],{"className":3085},[2592],[1840,3087],{"className":3088,"style":2588},[1972],[1840,3090,3092,3095,3098,3101,3104,3133],{"className":3091},[1937],[1840,3093],{"className":3094,"style":2980},[1941],[1840,3096,1866],{"className":3097,"style":1948},[1946,1947],[1840,3099,1871],{"className":3100},[1952],[1840,3102,1880],{"className":3103,"style":1963},[1946,1947],[1840,3105,3107,3110],{"className":3106},[1967],[1840,3108,1886],{"className":3109},[1967],[1840,3111,3113],{"className":3112},[2077],[1840,3114,3116],{"className":3115},[1996],[1840,3117,3119],{"className":3118},[2001],[1840,3120,3122],{"className":3121,"st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0 时方差才直接相加。独立且二阶矩存在会推出零协方差，反向一般不成立。",[1807,3537,3539],{"id":3538},"_3-协方差与相关","3. 协方差与相关",[1840,3541,3543],{"className":3542},[1843],[1840,3544,3546,3614],{"className":3545},[1847],[1840,3547,3549],{"className":3548},[1851],[1853,3550,3551],{"xmlns":1855,"display":1856},[1858,3552,3553,3611],{},[1861,3554,3555,3557,3559,3561,3563,3565,3567,3569,3571,3573,3575,3577,3579,3581,3583,3585,3587,3589,3591,3593,3595,3597,3599,3601,3603,3605,3607,3609],{},[1864,3556,3362],{"mathvariant":1922},[1868,3558,2844],{},[1868,3560,1877],{"stretchy":1870},[1864,3562,1880],{},[1868,3564,3371],{"separator":1933},[1864,3566,2523],{},[1868,3568,1883],{"stretchy":1870},[1868,3570,1889],{},[1864,3572,1866],{},[1868,3574,1871],{"stretchy":1870},[1868,3576,1877],{"stretchy":1870},[1864,3578,1880],{},[1868,3580,2169],{},[1864,3582,1866],{},[1868,3584,1871],{"stretchy":1870},[1864,3586,1880],{},[1868,3588,1886],{"stretchy":1870},[1868,3590,1883],{"stretchy":1870},[1868,3592,1877],{"stretchy":1870},[1864,3594,2523],{},[1868,3596,2169],{},[1864,3598,1866],{},[1868,3600,1871],{"stretchy":1870},[1864,3602,2523],{},[1868,3604,1886],{"stretchy":1870},[1868,3606,1883],{"stretchy":1870},[1868,3608,1886],{"stretchy":1870},[1864,3610,1923],{"mathvariant":1922},[1925,3612,3613],{"encoding":1927},"\\operatorname{Cov}(X,Y)\n=E[(X-E[X])(Y-E[Y])].",[1840,3615,3617,3656,3680,3714],{"className":3616,"ariaHidden":1933},[1932],[1840,3618,3620,3623,3629,3632,3635,3638,3641,3644,3647,3650,3653],{"className":3619},[1937],[1840,3621],{"className":3622,"style":1942},[1941],[1840,3624,3626],{"className":3625},[1991],[1840,3627,3362],{"className":3628,"style":3510},[1946,2930],[1840,3630,1877],{"className":3631},[1952],[1840,3633,1880],{"className":3634,"style":1963},[1946,1947],[1840,3636,3371],{"className":3637},[3520],[1840,3639],{"className":3640,"style":2055},[1972],[1840,3642,2523],{"className":3643,"style":2588},[1946,1947],[1840,3645,1883],{"className":3646},[1967],[1840,3648],{"className":3649,"style":1973},[1972],[1840,3651,1889],{"className":3652},[1977],[1840,3654],{"className":3655,"style":1973},[1972],[1840,3657,3659,3662,3665,3668,3671,3674,3677],{"className":3658},[1937],[1840,3660],{"className":3661,"style":1942},[1941],[1840,3663,1866],{"className":3664,"style":1948},[1946,1947],[1840,3666,2961],{"className":3667},[1952],[1840,3669,1880],{"className":3670,"style":1963},[1946,1947],[1840,3672],{"className":3673,"style":2588},[1972],[1840,3675,2169],{"className":3676},[2592],[1840,3678],{"className":3679,"style":2588},[1972],[1840,3681,3683,3686,3689,3692,3695,3699,3702,3705,3708,3711],{"className":3682},[1937],[1840,3684],{"className":3685,"style":1942},[1941],[1840,3687,1866],{"className":3688,"style":1948},[1946,1947],[1840,3690,1871],{"className":3691},[1952],[1840,3693,1880],{"className":3694,"style":1963},[1946,1947],[1840,3696,3698],{"className":3697},[1967],"])",[1840,3700,1877],{"className":3701},[1952],[1840,3703,2523],{"className":3704,"style":2588},[1946,1947],[1840,3706],{"className":3707,"style":2588},[1972],[1840,3709,2169],{"className":3710},[2592],[1840,3712],{"className":3713,"style":2588},[1972],[1840,3715,3717,3720,3723,3726,3729,3733],{"className":3716},[1937],[1840,3718],{"className":3719,"style":1942},[1941],[1840,3721,1866],{"className":3722,"style":1948},[1946,1947],[1840,3724,1871],{"className":3725},[1952],[1840,3727,2523],{"className":3728,"style":2588},[1946,1947],[1840,3730,3732],{"className":3731},[1967],"])]",[1840,3734,1923],{"className":3735},[1946],[1840,3737,3739],{"className":3738},[1843],[1840,3740,3742,3813],{"className":3741},[1847],[1840,3743,3745],{"className":3744},[1851],[1853,3746,3747],{"xmlns":1855,"display":1856},[1858,3748,3749,3810],{},[1861,3750,3751,3762,3764,3808],{},[1908,3752,3753,3756],{},[1864,3754,3755],{},"ρ",[1861,3757,3758,3760],{},[1864,3759,1880],{},[1864,3761,2523],{},[1868,3763,1889],{},[3765,3766,3767,3783],"mfrac",{},[1861,3768,3769,3771,3773,3775,3777,3779,3781],{},[1864,3770,3362],{"mathvariant":1922},[1868,3772,2844],{},[1868,3774,1877],{"stretchy":1870},[1864,3776,1880],{},[1868,3778,3371],{"separator":1933},[1864,3780,2523],{},[1868,3782,1883],{"stretchy":1870},[3784,3785,3786],"msqrt",{},[1861,3787,3788,3790,3792,3794,3796,3798,3800,3802,3804,3806],{},[1864,3789,2841],{"mathvariant":1922},[1868,3791,2844],{},[1868,3793,1877],{"stretchy":1870},[1864,3795,1880],{},[1868,3797,1883],{"stretchy":1870},[1864,3799,2841],{"mathvariant":1922},[1868,3801,2844],{},[1868,3803,1877],{"stretchy":1870},[1864,3805,2523],{},[1868,3807,1883],{"stretchy":1870},[1864,3809,1923],{"mathvariant":1922},[1925,3811,3812],{"encoding":1927},"\\rho_{XY}\n=\\frac{\\operatorname{Cov}(X,Y)}\n{\\sqrt{\\operatorname{Var}(X)\\operatorname{Var}(Y)}}.",[1840,3814,3816,3878],{"className":3815,"ariaHidden":1933},[1932],[1840,3817,3819,3823,3869,3872,3875],{"className":3818},[1937],[1840,3820],{"className":3821,"style":3822},[1941],"height:0.625em;vertical-align:-0.1944em;",[1840,3824,3826,3829],{"className":3825},[1946],[1840,3827,3755],{"className":3828},[1946,1947],[1840,3830,3832],{"class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0 400000 1296","xMinYMin slice",[3995,3996],"path",{"d":3997},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 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",[1840,4076,4078,4103],{"className":4077},[1847],[1840,4079,4081],{"className":4080},[1851],[1853,4082,4083],{"xmlns":1855},[1858,4084,4085,4100],{},[1861,4086,4087,4089,4091,4094,4096,4098],{},[1868,4088,1871],{"stretchy":1870},[1868,4090,2169],{},[2877,4092,4093],{},"1",[1868,4095,3371],{"separator":1933},[2877,4097,4093],{},[1868,4099,1886],{"stretchy":1870},[1925,4101,4102],{"encoding":1927},"[-1,1]",[1840,4104,4106],{"className":4105,"ariaHidden":1933},[1932],[1840,4107,4109,4112,4115,4118,4121,4124,4127,4130],{"className":4108},[1937],[1840,4110],{"className":4111,"style":1942},[1941],[1840,4113,1871],{"className":4114},[1952],[1840,4116,2169],{"className":4117},[1946],[1840,4119,4093],{"className":4120},[1946],[1840,4122,3371],{"className":4123},[3520],[1840,4125],{"className":4126,"style":2055},[1972],[1840,4128,4093],{"className":4129},[1946],[1840,4131,1886],{"className":4132},[1967],"，但只度量线性关联。若 ",[1840,4135,4137,4150],{"className":4136},[1847],[1840,4138,4140],{"className":4139},[1851],[1853,4141,4142],{"xmlns":1855},[1858,4143,4144,4148],{},[1861,4145,4146],{},[1864,4147,1880],{},[1925,4149,1880],{"encoding":1927},[1840,4151,4153],{"className":4152,"ariaHidden":1933},[1932],[1840,4154,4156,4160],{"className":4155},[1937],[1840,4157],{"className":4158,"style":4159},[1941],"height:0.6833em;",[1840,4161,1880],{"className":4162,"style":1963},[1946,1947]," 关于 0 对称且 ",[1840,4165,4167,4189],{"className":4166},[1847],[1840,4168,4170],{"className":4169},[1851],[1853,4171,4172],{"xmlns":1855},[1858,4173,4174,4186],{},[1861,4175,4176,4178,4180],{},[1864,4177,2523],{},[1868,4179,1889],{},[2872,4181,4182,4184],{},[1864,4183,1880],{},[2877,4185,2879],{},[1925,4187,4188],{"encoding":1927},"Y=X^2",[1840,4190,4192,4210],{"className":4191,"ariaHidden":1933},[1932],[1840,4193,4195,4198,4201,4204,4207],{"className":4194},[1937],[1840,4196],{"className":4197,"style":4159},[1941],[1840,4199,2523],{"className":4200,"style":2588},[1946,1947],[1840,4202],{"className":4203,"style":1973},[1972],[1840,4205,1889],{"className":4206},[1977],[1840,4208],{"className":4209,"style":1973},[1972],[1840,4211,4213,4217],{"className":4212},[1937],[1840,4214],{"className":4215,"style":4216},[1941],"height:0.8141em;",[1840,4218,4220,4223],{"className":4219},[1946],[1840,4221,1880],{"className":4222,"style":1963},[1946,1947],[1840,4224,4226],{"className":4225},[2077],[1840,4227,4229],{"className":4228},[1996],[1840,4230,4232],{"className":4231},[2001],[1840,4233,4235],{"className":4234,"style":4216},[2005],[1840,4236,4238,4241],{"style":4237},"top:-3.063em;margin-right:0.05em;",[1840,4239],{"className":4240,"style":2094},[2013],[1840,4242,4244],{"className":4243},[2018,2019,2020,2021],[1840,4245,2879],{"className":4246},[1946,2021],"，常有 ",[1840,4249,4251,4282],{"className":4250},[1847],[1840,4252,4254],{"className":4253},[1851],[1853,4255,4256],{"xmlns":1855},[1858,4257,4258,4279],{},[1861,4259,4260,4262,4264,4266,4268,4270,4272,4274,4276],{},[1864,4261,3362],{"mathvariant":1922},[1868,4263,2844],{},[1868,4265,1877],{"stretchy":1870},[1864,4267,1880],{},[1868,4269,3371],{"separator":1933},[1864,4271,2523],{},[1868,4273,1883],{"stretchy":1870},[1868,4275,1889],{},[2877,4277,4278],{},"0",[1925,4280,4281],{"encoding":1927},"\\operatorname{Cov}(X,Y)=0",[1840,4283,4285,4324],{"className":4284,"ariaHidden":1933},[1932],[1840,4286,4288,4291,4297,4300,4303,4306,4309,4312,4315,4318,4321],{"className":4287},[1937],[1840,4289],{"className":4290,"style":1942},[1941],[1840,4292,4294],{"className":4293},[1991],[1840,4295,3362],{"className":4296,"style":3510},[1946,2930],[1840,4298,1877],{"className":4299},[1952],[1840,4301,1880],{"className":4302,"style":1963},[1946,1947],[1840,4304,3371],{"className":4305},[3520],[1840,4307],{"className":4308,"style":2055},[1972],[1840,4310,2523],{"className":4311,"style":2588},[1946,1947],[1840,4313,1883],{"className":4314},[1967],[1840,4316],{"className":4317,"style":1973},[1972],[1840,4319,1889],{"className":4320},[1977],[1840,4322],{"className":4323,"style":1973},[1972],[1840,4325,4327,4331],{"className":4326},[1937],[1840,4328],{"className":4329,"style":4330},[1941],"height:0.6444em;",[1840,4332,4278],{"className":4333},[1946],"，二者却完全依赖。",[1807,4336,4338],{"id":4337},"_4-条件期望是随机变量","4. 条件期望是随机变量",[1840,4340,4342],{"className":4341},[1843],[1840,4343,4345,4370],{"className":4344},[1847],[1840,4346,4348],{"className":4347},[1851],[1853,4349,4350],{"xmlns":1855,"display":1856},[1858,4351,4352,4367],{},[1861,4353,4354,4356,4358,4360,4363,4365],{},[1864,4355,1866],{},[1868,4357,1871],{"stretchy":1870},[1864,4359,2523],{},[1868,4361,4362],{},"∣",[1864,4364,1880],{},[1868,4366,1886],{"stretchy":1870},[1925,4368,4369],{"encoding":1927},"E[Y\\mid X]",[1840,4371,4373,4397],{"className":4372,"ariaHidden":1933},[1932],[1840,4374,4376,4379,4382,4385,4388,4391,4394],{"className":4375},[1937],[1840,4377],{"className":4378,"style":1942},[1941],[1840,4380,1866],{"className":4381,"style":1948},[1946,1947],[1840,4383,1871],{"className":4384},[1952],[1840,4386,2523],{"className":4387,"style":2588},[1946,1947],[1840,4389],{"className":4390,"style":1973},[1972],[1840,4392,4362],{"className":4393},[1977],[1840,4395],{"className":4396,"style":1973},[1972],[1840,4398,4400,4403,4406],{"className":4399},[1937],[1840,4401],{"className":4402,"style":1942},[1941],[1840,4404,1880],{"className":4405,"style":1963},[1946,1947],[1840,4407,1886],{"className":4408},[1967],[1796,4410,4411,4412,4440,4441,4469,4470,4498,4499,4527],{},"是 ",[1840,4413,4415,4428],{"className":4414},[1847],[1840,4416,4418],{"className":4417},[1851],[1853,4419,4420],{"xmlns":1855},[1858,4421,4422,4426],{},[1861,4423,4424],{},[1864,4425,1880],{},[1925,4427,1880],{"encoding":1927},[1840,4429,4431],{"className":4430,"ariaHidden":1933},[1932],[1840,4432,4434,4437],{"className":4433},[1937],[1840,4435],{"className":4436,"style":4159},[1941],[1840,4438,1880],{"className":4439,"style":1963},[1946,1947]," 的函数：观察到不同 ",[1840,4442,4444,4457],{"className":4443},[1847],[1840,4445,4447],{"className":4446},[1851],[1853,4448,4449],{"xmlns":1855},[1858,4450,4451,4455],{},[1861,4452,4453],{},[1864,4454,1880],{},[1925,4456,1880],{"encoding":1927},[1840,4458,4460],{"className":4459,"ariaHidden":1933},[1932],[1840,4461,4463,4466],{"className":4462},[1937],[1840,4464],{"className":4465,"style":4159},[1941],[1840,4467,1880],{"className":4468,"style":1963},[1946,1947]," 会得到不同的条件均值。它是均方误差下预测 ",[1840,4471,4473,4486],{"className":4472},[1847],[1840,4474,4476],{"className":4475},[1851],[1853,4477,4478],{"xmlns":1855},[1858,4479,4480,4484],{},[1861,4481,4482],{},[1864,4483,2523],{},[1925,4485,2523],{"encoding":1927},[1840,4487,4489],{"className":4488,"ariaHidden":1933},[1932],[1840,4490,4492,4495],{"className":4491},[1937],[1840,4493],{"className":4494,"style":4159},[1941],[1840,4496,2523],{"className":4497,"style":2588},[1946,1947]," 的最佳 ",[1840,4500,4502,4515],{"className":4501},[1847],[1840,4503,4505],{"className":4504},[1851],[1853,4506,4507],{"xmlns":1855},[1858,4508,4509,4513],{},[1861,4510,4511],{},[1864,4512,1880],{},[1925,4514,1880],{"encoding":1927},[1840,4516,4518],{"className":4517,"ariaHidden":1933},[1932],[1840,4519,4521,4524],{"className":4520},[1937],[1840,4522],{"className":4523,"style":4159},[1941],[1840,4525,1880],{"className":4526,"style":1963},[1946,1947],"-可测函数。",[1796,4529,4530],{},"塔式法则：",[1840,4532,4534],{"className":4533},[1843],[1840,4535,4537,4579],{"className":4536},[1847],[1840,4538,4540],{"className":4539},[1851],[1853,4541,4542],{"xmlns":1855,"display":1856},[1858,4543,4544,4576],{},[1861,4545,4546,4548,4550,4552,4554,4556,4558,4560,4562,4564,4566,4568,4570,4572,4574],{},[1864,4547,1866],{},[1868,4549,1871],{"stretchy":1870},[1864,4551,1866],{},[1868,4553,1871],{"stretchy":1870},[1864,4555,2523],{},[1868,4557,4362],{},[1864,4559,1880],{},[1868,4561,1886],{"stretchy":1870},[1868,4563,1886],{"stretchy":1870},[1868,4565,1889],{},[1864,4567,1866],{},[1868,4569,1871],{"stretchy":1870},[1864,4571,2523],{},[1868,4573,1886],{"stretchy":1870},[1864,4575,1923],{"mathvariant":1922},[1925,4577,4578],{"encoding":1927},"E[E[Y\\mid X]]=E[Y].",[1840,4580,4582,4612,4634],{"className":4581,"ariaHidden":1933},[1932],[1840,4583,4585,4588,4591,4594,4597,4600,4603,4606,4609],{"className":4584},[1937],[1840,4586],{"className":4587,"style":1942},[1941],[1840,4589,1866],{"className":4590,"style":1948},[1946,1947],[1840,4592,1871],{"className":4593},[1952],[1840,4595,1866],{"className":4596,"style":1948},[1946,1947],[1840,4598,1871],{"className":4599},[1952],[1840,4601,2523],{"className":4602,"style":2588},[1946,1947],[1840,4604],{"className":4605,"style":1973},[1972],[1840,4607,4362],{"className":4608},[1977],[1840,4610],{"className":4611,"style":1973},[1972],[1840,4613,4615,4618,4621,4625,4628,4631],{"className":4614},[1937],[1840,4616],{"className":4617,"style":1942},[1941],[1840,4619,1880],{"className":4620,"style":1963},[1946,1947],[1840,4622,4624],{"className":4623},[1967],"]]",[1840,4626],{"className":4627,"style":1973},[1972],[1840,4629,1889],{"className":4630},[1977],[1840,4632],{"className":4633,"style":1973},[1972],[1840,4635,4637,4640,4643,4646,4649,4652],{"className":4636},[1937],[1840,4638],{"className":4639,"style":1942},[1941],[1840,4641,1866],{"className":4642,"style":1948},[1946,1947],[1840,4644,1871],{"className":4645},[1952],[1840,4647,2523],{"className":4648,"style":2588},[1946,1947],[1840,4650,1886],{"className":4651},[1967],[1840,4653,1923],{"className":4654},[1946],[1796,4656,4657],{},"全方差公式：",[1840,4659,4661],{"className":4660},[1843],[1840,4662,4664,4732],{"className":4663},[1847],[1840,4665,4667],{"className":4666},[1851],[1853,4668,4669],{"xmlns":1855,"display":1856},[1858,4670,4671,4729],{},[1861,4672,4673,4675,4677,4679,4681,4683,4685,4687,4689,4691,4693,4695,4697,4699,4701,4703,4705,4707,4709,4711,4713,4715,4717,4719,4721,4723,4725,4727],{},[1864,4674,2841],{"mathvariant":1922},[1868,4676,2844],{},[1868,4678,1877],{"stretchy":1870},[1864,4680,2523],{},[1868,4682,1883],{"stretchy":1870},[1868,4684,1889],{},[1864,4686,1866],{},[1868,4688,1871],{"stretchy":1870},[1864,4690,2841],{"mathvariant":1922},[1868,4692,2844],{},[1868,4694,1877],{"stretchy":1870},[1864,4696,2523],{},[1868,4698,4362],{},[1864,4700,1880],{},[1868,4702,1883],{"stretchy":1870},[1868,4704,1886],{"stretchy":1870},[1868,4706,2517],{},[1864,4708,2841],{"mathvariant":1922},[1868,4710,2844],{},[1868,4712,1877],{"stretchy":1870},[1864,4714,1866],{},[1868,4716,1871],{"stretchy":1870},[1864,4718,2523],{},[1868,4720,4362],{},[1864,4722,1880],{},[1868,4724,1886],{"stretchy":1870},[1868,4726,1883],{"stretchy":1870},[1864,4728,1923],{"mathvariant":1922},[1925,4730,4731],{"encoding":1927},"\\operatorname{Var}(Y)\n=E[\\operatorname{Var}(Y\\mid X)]\n+\\operatorname{Var}(E[Y\\mid X]).",[1840,4733,4735,4765,4798,4819,4852],{"className":4734,"ariaHidden":1933},[1932],[1840,4736,4738,4741,4747,4750,4753,4756,4759,4762],{"className":4737},[1937],[1840,4739],{"className":4740,"style":1942},[1941],[1840,4742,4744],{"className":4743},[1991],[1840,4745,2841],{"className":4746},[1946,2930],[1840,4748,1877],{"className":4749},[1952],[1840,4751,2523],{"className":4752,"style":2588},[1946,1947],[1840,4754,1883],{"className":4755},[1967],[1840,4757],{"className":4758,"style":1973},[1972],[1840,4760,1889],{"className":4761},[1977],[1840,4763],{"className":4764,"style":1973},[1972],[1840,4766,4768,4771,4774,4777,4783,4786,4789,4792,4795],{"className":4767},[1937],[1840,4769],{"className":4770,"style":1942},[1941],[1840,4772,1866],{"className":4773,"style":1948},[1946,1947],[1840,4775,1871],{"className":4776},[1952],[1840,4778,4780],{"className":4779},[1991],[1840,4781,2841],{"className":4782},[1946,2930],[1840,4784,1877],{"className":4785},[1952],[1840,4787,2523],{"className":4788,"style":2588},[1946,1947],[1840,4790],{"className":4791,"style":1973},[1972],[1840,4793,4362],{"className":4794},[1977],[1840,4796],{"className":4797,"style":1973},[1972],[1840,4799,4801,4804,4807,4810,4813,4816],{"className":4800},[1937],[1840,4802],{"className":4803,"style":1942},[1941],[1840,4805,1880],{"className":4806,"style":1963},[1946,1947],[1840,4808,1968],{"className":4809},[1967],[1840,4811],{"className":4812,"style":2588},[1972],[1840,4814,2517],{"className":4815},[2592],[1840,4817],{"className":4818,"style":2588},[1972],[1840,4820,4822,4825,4831,4834,4837,4840,4843,4846,4849],{"className":4821},[1937],[1840,4823],{"className":4824,"style":1942},[1941],[1840,4826,4828],{"className":4827},[1991],[1840,4829,2841],{"className":4830},[1946,2930],[1840,4832,1877],{"className":4833},[1952],[1840,4835,1866],{"className":4836,"style":1948},[1946,1947],[1840,4838,1871],{"className":4839},[1952],[1840,4841,2523],{"className":4842,"style":2588},[1946,1947],[1840,4844],{"className":4845,"style":1973},[1972],[1840,4847,4362],{"className":4848},[1977],[1840,4850],{"className":4851,"style":1973},[1972],[1840,4853,4855,4858,4861,4864],{"className":4854},[1937],[1840,4856],{"className":4857,"style":1942},[1941],[1840,4859,1880],{"className":4860,"style":1963},[1946,1947],[1840,4862,3698],{"className":4863},[1967],[1840,4865,1923],{"className":4866},[1946],[1796,4868,4869],{},"第一项是组内波动，第二项是组间均值差异。",[1807,4871,4873],{"id":4872},"_5-可运行案例保险赔付的组内与组间方差","5. 可运行案例：保险赔付的组内与组间方差",[1796,4875,4876],{},"设 20% 客户属于高风险组。低风险与高风险客户的赔付分别服从均值为 1,000 和 5,000 的指数分布。代码核对全期望和全方差。",[4878,4879],"pyodide",{"code64":4880,"layout":4881,"locale":7,"packages":4882,"title":4883},"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","vertical","numpy","Python：全期望与全方差分解",[1796,4885,4886],{},"若只报告总体方差，会看不到风险组构成。反过来，即使组间差异很大，每个组内仍可能有大量不可预测波动。",[1807,4888,4890],{"id":4889},"_6-高阶矩与矩母函数","6. 高阶矩与矩母函数",[1796,4892,4893,4894,4972,4973,5077],{},"原点矩 ",[1840,4895,4897,4922],{"className":4896},[1847],[1840,4898,4900],{"className":4899},[1851],[1853,4901,4902],{"xmlns":1855},[1858,4903,4904,4919],{},[1861,4905,4906,4908,4910,4917],{},[1864,4907,1866],{},[1868,4909,1871],{"stretchy":1870},[2872,4911,4912,4914],{},[1864,4913,1880],{},[1864,4915,4916],{},"k",[1868,4918,1886],{"stretchy":1870},[1925,4920,4921],{"encoding":1927},"E[X^k]",[1840,4923,4925],{"className":4924,"ariaHidden":1933},[1932],[1840,4926,4928,4932,4935,4938,4969],{"className":4927},[1937],[1840,4929],{"className":4930,"style":4931},[1941],"height:1.0991em;vertical-align:-0.25em;",[1840,4933,1866],{"className":4934,"style":1948},[1946,1947],[1840,4936,1871],{"className":4937},[1952],[1840,4939,4941,4944],{"className":4940},[1946],[1840,4942,1880],{"className":4943,"style":1963},[1946,1947],[1840,4945,4947],{"className":4946},[2077],[1840,4948,4950],{"className":4949},[1996],[1840,4951,4953],{"className":4952},[2001],[1840,4954,4957],{"className":4955,"style":4956},[2005],"height:0.8491em;",[1840,4958,4959,4962],{"style":4237},[1840,4960],{"className":4961,"style":2094},[2013],[1840,4963,4965],{"className":4964},[2018,2019,2020,2021],[1840,4966,4916],{"className":4967,"style":4968},[1946,1947,2021],"margin-right:0.0315em;",[1840,4970,1886],{"className":4971},[1967]," 与中心矩 ",[1840,4974,4976,5009],{"className":4975},[1847],[1840,4977,4979],{"className":4978},[1851],[1853,4980,4981],{"xmlns":1855},[1858,4982,4983,5006],{},[1861,4984,4985,4987,4989,4991,4993,4995,4998,5004],{},[1864,4986,1866],{},[1868,4988,1871],{"stretchy":1870},[1868,4990,1877],{"stretchy":1870},[1864,4992,1880],{},[1868,4994,2169],{},[1864,4996,4997],{},"μ",[2872,4999,5000,5002],{},[1868,5001,1883],{"stretchy":1870},[1864,5003,4916],{},[1868,5005,1886],{"stretchy":1870},[1925,5007,5008],{"encoding":1927},"E[(X-\\mu)^k]",[1840,5010,5012,5036],{"className":5011,"ariaHidden":1933},[1932],[1840,5013,5015,5018,5021,5024,5027,5030,5033],{"className":5014},[1937],[1840,5016],{"className":5017,"style":1942},[1941],[1840,5019,1866],{"className":5020,"style":1948},[1946,1947],[1840,5022,2961],{"className":5023},[1952],[1840,5025,1880],{"className":5026,"style":1963},[1946,1947],[1840,5028],{"className":5029,"style":2588},[1972],[1840,5031,2169],{"className":5032},[2592],[1840,5034],{"className":5035,"style":2588},[1972],[1840,5037,5039,5042,5045,5074],{"className":5038},[1937],[1840,5040],{"className":5041,"style":4931},[1941],[1840,5043,4997],{"className":5044},[1946,1947],[1840,5046,5048,5051],{"className":5047},[1967],[1840,5049,1883],{"className":5050},[1967],[1840,5052,5054],{"className":5053},[2077],[1840,5055,5057],{"className":5056},[1996],[1840,5058,5060],{"className":5059},[2001],[1840,5061,5063],{"className":5062,"style":4956},[2005],[1840,5064,5065,5068],{"style":4237},[1840,5066],{"className":5067,"style":2094},[2013],[1840,5069,5071],{"className":5070},[2018,2019,2020,2021],[1840,5072,4916],{"className":5073,"style":4968},[1946,1947,2021],[1840,5075,1886],{"className":5076},[1967]," 描述形状：",[5079,5080,5081,5084],"ul",{},[1817,5082,5083],{},"三阶标准化矩对应偏度；",[1817,5085,5086],{},"四阶标准化矩与尾部\u002F峰度有关。",[1796,5088,5089],{},"矩母函数：",[1840,5091,5093],{"className":5092},[1843],[1840,5094,5096,5143],{"className":5095},[1847],[1840,5097,5099],{"className":5098},[1851],[1853,5100,5101],{"xmlns":1855,"display":1856},[1858,5102,5103,5140],{},[1861,5104,5105,5112,5114,5117,5119,5121,5123,5125,5136,5138],{},[1908,5106,5107,5110],{},[1864,5108,5109],{},"M",[1864,5111,1880],{},[1868,5113,1877],{"stretchy":1870},[1864,5115,5116],{},"t",[1868,5118,1883],{"stretchy":1870},[1868,5120,1889],{},[1864,5122,1866],{},[1868,5124,1871],{"stretchy":1870},[2872,5126,5127,5130],{},[1864,5128,5129],{},"e",[1861,5131,5132,5134],{},[1864,5133,5116],{},[1864,5135,1880],{},[1868,5137,1886],{"stretchy":1870},[1864,5139,1923],{"mathvariant":1922},[1925,5141,5142],{"encoding":1927},"M_X(t)=E[e^{tX}].",[1840,5144,5146,5212],{"className":5145,"ariaHidden":1933},[1932],[1840,5147,5149,5152,5194,5197,5200,5203,5206,5209],{"className":5148},[1937],[1840,5150],{"className":5151,"style":1942},[1941],[1840,5153,5155,5159],{"className":5154},[1946],[1840,5156,5109],{"className":5157,"style":5158},[1946,1947],"margin-right:0.109em;",[1840,5160,5162],{"className":5161},[2077],[1840,5163,5165,5186],{"className":5164},[1996,1997],[1840,5166,5168,5183],{"className":5167},[2001],[1840,5169,5171],{"className":5170,"style":2087},[2005],[1840,5172,5174,5177],{"style":5173},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[1840,5175],{"className":5176,"style":2094},[2013],[1840,5178,5180],{"className":5179},[2018,2019,2020,2021],[1840,5181,1880],{"className":5182,"style":1963},[1946,1947,2021],[1840,5184,2042],{"className":5185},[2041],[1840,5187,5189],{"className":5188},[2001],[1840,5190,5192],{"className":5191,"style":2110},[2005],[1840,5193],{},[1840,5195,1877],{"className":5196},[1952],[1840,5198,5116],{"className":5199},[1946,1947],[1840,5201,1883],{"className":5202},[1967],[1840,5204],{"className":5205,"style":1973},[1972],[1840,5207,1889],{"className":5208},[1977],[1840,5210],{"className":5211,"style":1973},[1972],[1840,5213,5215,5219,5222,5225,5259,5262],{"className":5214},[1937],[1840,5216],{"className":5217,"style":5218},[1941],"height:1.1413em;vertical-align:-0.25em;",[1840,5220,1866],{"className":5221,"style":1948},[1946,1947],[1840,5223,1871],{"className":5224},[1952],[1840,5226,5228,5231],{"className":5227},[1946],[1840,5229,5129],{"className":5230},[1946,1947],[1840,5232,5234],{"className":5233},[2077],[1840,5235,5237],{"className":5236},[1996],[1840,5238,5240],{"className":5239},[2001],[1840,5241,5244],{"className":5242,"style":5243},[2005],"height:0.8913em;",[1840,5245,5246,5249],{"style":3014},[1840,5247],{"className":5248,"style":2094},[2013],[1840,5250,5252],{"className":5251},[2018,2019,2020,2021],[1840,5253,5255],{"className":5254},[1946,2021],[1840,5256,5258],{"className":5257,"style":1963},[1946,1947,2021],"tX",[1840,5260,1886],{"className":5261},[1967],[1840,5263,1923],{"className":5264},[1946],[1796,5266,5267,5268,5320,5321,5501],{},"若它在 ",[1840,5269,5271,5289],{"className":5270},[1847],[1840,5272,5274],{"className":5273},[1851],[1853,5275,5276],{"xmlns":1855},[1858,5277,5278,5286],{},[1861,5279,5280,5282,5284],{},[1864,5281,5116],{},[1868,5283,1889],{},[2877,5285,4278],{},[1925,5287,5288],{"encoding":1927},"t=0",[1840,5290,5292,5311],{"className":5291,"ariaHidden":1933},[1932],[1840,5293,5295,5299,5302,5305,5308],{"className":5294},[1937],[1840,5296],{"className":5297,"style":5298},[1941],"height:0.6151em;",[1840,5300,5116],{"className":5301},[1946,1947],[1840,5303],{"className":5304,"style":1973},[1972],[1840,5306,1889],{"className":5307},[1977],[1840,5309],{"className":5310,"style":1973},[1972],[1840,5312,5314,5317],{"className":5313},[1937],[1840,5315],{"className":5316,"style":4330},[1941],[1840,5318,4278],{"className":5319},[1946]," 的邻域存在，可通过求导得到各阶矩，并在常见条件下唯一确定分布。对数矩母函数 ",[1840,5322,5324,5368],{"className":5323},[1847],[1840,5325,5327],{"className":5326},[1851],[1853,5328,5329],{"xmlns":1855},[1858,5330,5331,5365],{},[1861,5332,5333,5340,5342,5344,5346,5348,5351,5353,5359,5361,5363],{},[1908,5334,5335,5338],{},[1864,5336,5337],{},"K",[1864,5339,1880],{},[1868,5341,1877],{"stretchy":1870},[1864,5343,5116],{},[1868,5345,1883],{"stretchy":1870},[1868,5347,1889],{},[1864,5349,5350],{},"log",[1868,5352,2844],{},[1908,5354,5355,5357],{},[1864,5356,5109],{},[1864,5358,1880],{},[1868,5360,1877],{"stretchy":1870},[1864,5362,5116],{},[1868,5364,1883],{"stretchy":1870},[1925,5366,5367],{"encoding":1927},"K_X(t)=\\log M_X(t)",[1840,5369,5371,5437],{"className":5370,"ariaHidden":1933},[1932],[1840,5372,5374,5377,5419,5422,5425,5428,5431,5434],{"className":5373},[1937],[1840,5375],{"className":5376,"style":1942},[1941],[1840,5378,5380,5384],{"className":5379},[1946],[1840,5381,5337],{"className":5382,"style":5383},[1946,1947],"margin-right:0.0715em;",[1840,5385,5387],{"className":5386},[2077],[1840,5388,5390,5411],{"className":5389},[1996,1997],[1840,5391,5393,5408],{"className":5392},[2001],[1840,5394,5396],{"className":5395,"style":2087},[2005],[1840,5397,5399,5402],{"style":5398},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[1840,5400],{"className":5401,"style":2094},[2013],[1840,5403,5405],{"className":5404},[2018,2019,2020,2021],[1840,5406,1880],{"className":5407,"style":1963},[1946,1947,2021],[1840,5409,2042],{"className":5410},[2041],[1840,5412,5414],{"className":5413},[2001],[1840,5415,5417],{"className":5416,"style":2110},[2005],[1840,5418],{},[1840,5420,1877],{"className":5421},[1952],[1840,5423,5116],{"className":5424},[1946,1947],[1840,5426,1883],{"className":5427},[1967],[1840,5429],{"className":5430,"style":1973},[1972],[1840,5432,1889],{"className":5433},[1977],[1840,5435],{"className":5436,"style":1973},[1972],[1840,5438,5440,5443,5449,5452,5492,5495,5498],{"className":5439},[1937],[1840,5441],{"className":5442,"style":1942},[1941],[1840,5444,5446,5447],{"className":5445},[1991],"lo",[1840,5448,1874],{"style":3510},[1840,5450],{"className":5451,"style":2055},[1972],[1840,5453,5455,5458],{"className":5454},[1946],[1840,5456,5109],{"className":5457,"style":5158},[1946,1947],[1840,5459,5461],{"className":5460},[2077],[1840,5462,5464,5484],{"className":5463},[1996,1997],[1840,5465,5467,5481],{"className":5466},[2001],[1840,5468,5470],{"className":5469,"style":2087},[2005],[1840,5471,5472,5475],{"style":5173},[1840,5473],{"className":5474,"style":2094},[2013],[1840,5476,5478],{"className":5477},[2018,2019,2020,2021],[1840,5479,1880],{"className":5480,"style":1963},[1946,1947,2021],[1840,5482,2042],{"className":5483},[2041],[1840,5485,5487],{"className":5486},[2001],[1840,5488,5490],{"className":5489,"style":2110},[2005],[1840,5491],{},[1840,5493,1877],{"className":5494},[1952],[1840,5496,5116],{"className":5497},[1946,1947],[1840,5499,1883],{"className":5500},[1967]," 的导数给出累积量。",[1796,5503,5504,5505,5533,5534,5617],{},"但对数正态等分布的 MGF 在正 ",[1840,5506,5508,5521],{"className":5507},[1847],[1840,5509,5511],{"className":5510},[1851],[1853,5512,5513],{"xmlns":1855},[1858,5514,5515,5519],{},[1861,5516,5517],{},[1864,5518,5116],{},[1925,5520,5116],{"encoding":1927},[1840,5522,5524],{"className":5523,"ariaHidden":1933},[1932],[1840,5525,5527,5530],{"className":5526},[1937],[1840,5528],{"className":5529,"style":5298},[1941],[1840,5531,5116],{"className":5532},[1946,1947]," 处可能发散。不要因为形式上能写 ",[1840,5535,5537,5565],{"className":5536},[1847],[1840,5538,5540],{"className":5539},[1851],[1853,5541,5542],{"xmlns":1855},[1858,5543,5544,5562],{},[1861,5545,5546,5548,5550,5560],{},[1864,5547,1866],{},[1868,5549,1871],{"stretchy":1870},[2872,5551,5552,5554],{},[1864,5553,5129],{},[1861,5555,5556,5558],{},[1864,5557,5116],{},[1864,5559,1880],{},[1868,5561,1886],{"stretchy":1870},[1925,5563,5564],{"encoding":1927},"E[e^{tX}]",[1840,5566,5568],{"className":5567,"ariaHidden":1933},[1932],[1840,5569,5571,5575,5578,5581,5614],{"className":5570},[1937],[1840,5572],{"className":5573,"style":5574},[1941],"height:1.0913em;vertical-align:-0.25em;",[1840,5576,1866],{"className":5577,"style":1948},[1946,1947],[1840,5579,1871],{"className":5580},[1952],[1840,5582,5584,5587],{"className":5583},[1946],[1840,5585,5129],{"className":5586},[1946,1947],[1840,5588,5590],{"className":5589},[2077],[1840,5591,5593],{"className":5592},[1996],[1840,5594,5596],{"className":5595},[2001],[1840,5597,5600],{"className":5598,"style":5599},[2005],"height:0.8413em;",[1840,5601,5602,5605],{"style":4237},[1840,5603],{"className":5604,"style":2094},[2013],[1840,5606,5608],{"className":5607},[2018,2019,2020,2021],[1840,5609,5611],{"className":5610},[1946,2021],[1840,5612,5258],{"className":5613,"style":1963},[1946,1947,2021],[1840,5615,1886],{"className":5616},[1967]," 就默认有限。",[1807,5619,5621],{"id":5620},"_7-不等式提供分布无关界","7. 不等式提供分布无关界",[1796,5623,5624,5625,5678],{},"Markov 不等式（",[1840,5626,5628,5647],{"className":5627},[1847],[1840,5629,5631],{"className":5630},[1851],[1853,5632,5633],{"xmlns":1855},[1858,5634,5635,5644],{},[1861,5636,5637,5639,5642],{},[1864,5638,1880],{},[1868,5640,5641],{},"≥",[2877,5643,4278],{},[1925,5645,5646],{"encoding":1927},"X\\ge0",[1840,5648,5650,5669],{"className":5649,"ariaHidden":1933},[1932],[1840,5651,5653,5657,5660,5663,5666],{"className":5652},[1937],[1840,5654],{"className":5655,"style":5656},[1941],"height:0.8193em;vertical-align:-0.136em;",[1840,5658,1880],{"className":5659,"style":1963},[1946,1947],[1840,5661],{"className":5662,"style":1973},[1972],[1840,5664,5641],{"className":5665},[1977],[1840,5667],{"className":5668,"style":1973},[1972],[1840,5670,5672,5675],{"className":5671},[1937],[1840,5673],{"className":5674,"style":4330},[1941],[1840,5676,4278],{"className":5677},[1946],"）：",[1840,5680,5682],{"className":5681},[1843],[1840,5683,5685,5729],{"className":5684},[1847],[1840,5686,5688],{"className":5687},[1851],[1853,5689,5690],{"xmlns":1855,"display":1856},[1858,5691,5692,5726],{},[1861,5693,5694,5697,5699,5701,5703,5705,5707,5710,5724],{},[1864,5695,5696],{},"P",[1868,5698,1877],{"stretchy":1870},[1864,5700,1880],{},[1868,5702,5641],{},[1864,5704,2512],{},[1868,5706,1883],{"stretchy":1870},[1868,5708,5709],{},"≤",[3765,5711,5712,5722],{},[1861,5713,5714,5716,5718,5720],{},[1864,5715,1866],{},[1868,5717,1871],{"stretchy":1870},[1864,5719,1880],{},[1868,5721,1886],{"stretchy":1870},[1864,5723,2512],{},[1864,5725,1923],{"mathvariant":1922},[1925,5727,5728],{"encoding":1927},"P(X\\ge a)\\le\\frac{E[X]}{a}.",[1840,5730,5732,5757,5778],{"className":5731,"ariaHidden":1933},[1932],[1840,5733,5735,5738,5742,5745,5748,5751,5754],{"className":5734},[1937],[1840,5736],{"className":5737,"style":1942},[1941],[1840,5739,5696],{"className":5740,"style":5741},[1946,1947],"margin-right:0.1389em;",[1840,5743,1877],{"className":5744},[1952],[1840,5746,1880],{"className":5747,"style":1963},[1946,1947],[1840,5749],{"className":5750,"style":1973},[1972],[1840,5752,5641],{"className":5753},[1977],[1840,5755],{"className":5756,"style":1973},[1972],[1840,5758,5760,5763,5766,5769,5772,5775],{"className":5759},[1937],[1840,5761],{"className":5762,"style":1942},[1941],[1840,5764,2512],{"className":5765},[1946,1947],[1840,5767,1883],{"className":5768},[1967],[1840,5770],{"className":5771,"style":1973},[1972],[1840,5773,5709],{"className":5774},[1977],[1840,5776],{"className":5777,"style":1973},[1972],[1840,5779,5781,5785,5858],{"className":5780},[1937],[1840,5782],{"className":5783,"style":5784},[1941],"height:2.113em;vertical-align:-0.686em;",[1840,5786,5788,5791,5855],{"className":5787},[1946],[1840,5789],{"className":5790},[1952,3891],[1840,5792,5794],{"className":5793},[3765],[1840,5795,5797,5846],{"className":5796},[1996,1997],[1840,5798,5800,5843],{"className":5799},[2001],[1840,5801,5803,5815,5823],{"className":5802,"style":3904},[2005],[1840,5804,5806,5809],{"style":5805},"top:-2.314em;",[1840,5807],{"className":5808,"style":3911},[2013],[1840,5810,5812],{"className":5811},[1946],[1840,5813,2512],{"className":5814},[1946,1947],[1840,5816,5817,5820],{"style":4012},[1840,5818],{"className":5819,"style":3911},[2013],[1840,5821],{"className":5822,"style":4020},[4019],[1840,5824,5825,5828],{"style":4023},[1840,5826],{"className":5827,"style":3911},[2013],[1840,5829,5831,5834,5837,5840],{"className":5830},[1946],[1840,5832,1866],{"className":5833,"style":1948},[1946,1947],[1840,5835,1871],{"className":5836},[1952],[1840,5838,1880],{"className":5839,"style":1963},[1946,1947],[1840,5841,1886],{"className":5842},[1967],[1840,5844,2042],{"className":5845},[2041],[1840,5847,5849],{"className":5848},[2001],[1840,5850,5853],{"className":5851,"style":5852},[2005],"height:0.686em;",[1840,5854],{},[1840,5856],{"className":5857},[1967,3891],[1840,5859,1923],{"className":5860},[1946],[1796,5862,5863],{},"Chebyshev 不等式：",[1840,5865,5867],{"className":5866},[1843],[1840,5868,5870,5919],{"className":5869},[1847],[1840,5871,5873],{"className":5872},[1851],[1853,5874,5875],{"xmlns":1855,"display":1856},[1858,5876,5877,5916],{},[1861,5878,5879,5881,5883,5885,5887,5889,5891,5893,5895,5897,5900,5902,5904,5914],{},[1864,5880,5696],{},[1868,5882,1877],{"stretchy":1870},[1864,5884,4362],{"mathvariant":1922},[1864,5886,1880],{},[1868,5888,2169],{},[1864,5890,4997],{},[1864,5892,4362],{"mathvariant":1922},[1868,5894,5641],{},[1864,5896,4916],{},[1864,5898,5899],{},"σ",[1868,5901,1883],{"stretchy":1870},[1868,5903,5709],{},[3765,5905,5906,5908],{},[2877,5907,4093],{},[2872,5909,5910,5912],{},[1864,5911,4916],{},[2877,5913,2879],{},[1864,5915,1923],{"mathvariant":1922},[1925,5917,5918],{"encoding":1927},"P(|X-\\mu|\\ge k\\sigma)\\le\\frac1{k^2}.",[1840,5920,5922,5949,5970,5994],{"className":5921,"ariaHidden":1933},[1932],[1840,5923,5925,5928,5931,5934,5937,5940,5943,5946],{"className":5924},[1937],[1840,5926],{"className":5927,"style":1942},[1941],[1840,5929,5696],{"className":5930,"style":5741},[1946,1947],[1840,5932,1877],{"className":5933},[1952],[1840,5935,4362],{"className":5936},[1946],[1840,5938,1880],{"className":5939,"style":1963},[1946,1947],[1840,5941],{"className":5942,"style":2588},[1972],[1840,5944,2169],{"className":5945},[2592],[1840,5947],{"className":5948,"style":2588},[1972],[1840,5950,5952,5955,5958,5961,5964,5967],{"className":5951},[1937],[1840,5953],{"className":5954,"style":1942},[1941],[1840,5956,4997],{"className":5957},[1946,1947],[1840,5959,4362],{"className":5960},[1946],[1840,5962],{"className":5963,"style":1973},[1972],[1840,5965,5641],{"className":5966},[1977],[1840,5968],{"className":5969,"style":1973},[1972],[1840,5971,5973,5976,5979,5982,5985,5988,5991],{"className":5972},[1937],[1840,5974],{"className":5975,"style":1942},[1941],[1840,5977,4916],{"className":5978,"style":4968},[1946,1947],[1840,5980,5899],{"className":5981,"style":1956},[1946,1947],[1840,5983,1883],{"className":5984},[1967],[1840,5986],{"className":5987,"style":1973},[1972],[1840,5989,5709],{"className":5990},[1977],[1840,5992],{"className":5993,"style":1973},[1972],[1840,5995,5997,6001,6092],{"className":5996},[1937],[1840,5998],{"className":5999,"style":6000},[1941],"height:2.0074em;vertical-align:-0.686em;",[1840,6002,6004,6007,6089],{"className":6003},[1946],[1840,6005],{"className":6006},[1952,3891],[1840,6008,6010],{"className":6009},[3765],[1840,6011,6013,6081],{"className":6012},[1996,1997],[1840,6014,6016,6078],{"className":6015},[2001],[1840,6017,6020,6059,6067],{"className":6018,"style":6019},[2005],"height:1.3214em;",[1840,6021,6022,6025],{"style":5805},[1840,6023],{"className":6024,"style":3911},[2013],[1840,6026,6028],{"className":6027},[1946],[1840,6029,6031,6034],{"className":6030},[1946],[1840,6032,4916],{"className":6033,"style":4968},[1946,1947],[1840,6035,6037],{"className":6036},[2077],[1840,6038,6040],{"className":6039},[1996],[1840,6041,6043],{"className":6042},[2001],[1840,6044,6047],{"className":6045,"style":6046},[2005],"height:0.7401em;",[1840,6048,6050,6053],{"style":6049},"top:-2.989em;margin-right:0.05em;",[1840,6051],{"className":6052,"style":2094},[2013],[1840,6054,6056],{"className":6055},[2018,2019,2020,2021],[1840,6057,2879],{"className":6058},[1946,2021],[1840,6060,6061,6064],{"style":4012},[1840,6062],{"className":6063,"style":3911},[2013],[1840,6065],{"className":6066,"style":4020},[4019],[1840,6068,6069,6072],{"style":4023},[1840,6070],{"className":6071,"style":3911},[2013],[1840,6073,6075],{"className":6074},[1946],[1840,6076,4093],{"className":6077},[1946],[1840,6079,2042],{"className":6080},[2041],[1840,6082,6084],{"className":6083},[2001],[1840,6085,6087],{"className":6086,"style":5852},[2005],[1840,6088],{},[1840,6090],{"className":6091},[1967,3891],[1840,6093,1923],{"className":6094},[1946],[1796,6096,6097],{},"界通常保守，但只依赖少量矩。它们是大数定律、集中不等式和风险控制的入口。",[1807,6099,6101],{"id":6100},"_8-诊断清单","8. 诊断清单",[5079,6103,6104,6107,6110,6113,6116,6119],{},[1817,6105,6106],{},"所需矩是否存在？",[1817,6108,6109],{},"单位是否正确：方差单位是原单位平方？",[1817,6111,6112],{},"是否把独立性当作期望线性性的条件？",[1817,6114,6115],{},"聚合方差是否漏掉协方差项？",[1817,6117,6118],{},"相关接近 0 是否仍可能存在非线性关系？",[1817,6120,6121],{},"总体异质性是否应使用条件矩分解？",[1807,6123,6124],{"id":6124},"课堂任务",[1796,6126,6127],{},"某学校学生成绩来自两类课程：",[5079,6129,6130,6133],{},[1817,6131,6132],{},"A 类占 70%，均值 75、方差 64；",[1817,6134,6135],{},"B 类占 30%，均值 85、方差 100。",[1796,6137,6138],{},"计算总体均值、组内方差、组间方差与总体方差，并解释“总体方差大”有多少来自课程构成。",[1807,6140,6141],{"id":6141},"核心阅读",[5079,6143,6144,6152,6159],{},[1817,6145,6146,6147,6151],{},"Ross, ",[6148,6149,6150],"em",{},"A First Course in Probability","，期望与条件期望章节。",[1817,6153,6154,6155,6158],{},"Casella & Berger, ",[6148,6156,6157],{},"Statistical Inference","，第 1–2 章。",[1817,6160,6161,6162,6165],{},"Grimmett & Stirzaker, ",[6148,6163,6164],{},"Probability and Random Processes","，条件期望章节。",[1796,6167,6168,6169,6173,6174,6178],{},"上一章：",[2512,6170,6172],{"href":6171},"..\u002F02-random-variables\u002F","随机变量","｜下一章：",[2512,6175,6177],{"href":6176},"..\u002F04-families\u002F","常见分布族","。",{"title":10,"searchDepth":6180,"depth":6180,"links":6181},2,[6182,6183,6184,6185,6186,6187,6188,6189,6190,6191,6192],{"id":1809,"depth":6180,"text":1809},{"id":1834,"depth":6180,"text":1835},{"id":2822,"depth":6180,"text":2823},{"id":3538,"depth":6180,"text":3539},{"id":4337,"depth":6180,"text":4338},{"id":4872,"depth":6180,"text":4873},{"id":4889,"depth":6180,"text":4890},{"id":5620,"depth":6180,"text":5621},{"id":6100,"depth":6180,"text":6101},{"id":6124,"depth":6180,"text":6124},{"id":6141,"depth":6180,"text":6141},"用矩概括分布，并通过全期望、全方差和协方差分解理解异质总体。","md",{"sidebar":6196},{"order":6197},4,true,{"title":1663,"description":6193},"I4oMfUvM10VoQksodTMpMqmj2T-YApPjN5ULQ-6HAkI",[6202,6204],{"title":1657,"path":1658,"stem":1659,"description":6203,"children":-1},"统一理解 PMF、PDF、CDF、分位数、联合分布、条件分布与变量变换。",{"title":1669,"path":1670,"stem":1671,"description":6205,"children":-1},"按重复次数、事件率、等待时间和比例机制选择常见离散与连续分布。",1785754756537]