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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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",[1804,1852,1854,1886],{"className":1853},[1807],[1804,1855,1857],{"className":1856},[1811],[1813,1858,1859],{"xmlns":1815},[1817,1860,1861,1883],{},[1820,1862,1863,1868,1871,1875,1878,1880],{},[1864,1865,1867],"mo",{"stretchy":1866},"false","[",[1864,1869,1870],{},"−",[1872,1873,1874],"mn",{},"1",[1864,1876,1877],{"separator":1834},",",[1872,1879,1874],{},[1864,1881,1882],{"stretchy":1866},"]",[1827,1884,1885],{"encoding":1829},"[-1,1]",[1804,1887,1889],{"className":1888,"ariaHidden":1834},[1833],[1804,1890,1892,1896,1900,1903,1906,1910,1915,1918],{"className":1891},[1838],[1804,1893],{"className":1894,"style":1895},[1842],"height:1em;vertical-align:-0.25em;",[1804,1897,1867],{"className":1898},[1899],"mopen",[1804,1901,1870],{"className":1902},[1847],[1804,1904,1874],{"className":1905},[1847],[1804,1907,1877],{"className":1908},[1909],"mpunct",[1804,1911],{"className":1912,"style":1914},[1913],"mspace","margin-right:0.1667em;",[1804,1916,1874],{"className":1917},[1847],[1804,1919,1882],{"className":1920},[1921],"mclose"," 上均匀，距离平方 ",[1804,1924,1926,1952],{"className":1925},[1807],[1804,1927,1929],{"className":1928},[1811],[1813,1930,1931],{"xmlns":1815},[1817,1932,1933,1949],{},[1820,1934,1935,1938,1941],{},[1823,1936,1937],{},"Y",[1864,1939,1940],{},"=",[1942,1943,1944,1946],"msup",{},[1823,1945,1825],{},[1872,1947,1948],{},"2",[1827,1950,1951],{"encoding":1829},"Y=X^2",[1804,1953,1955,1976],{"className":1954,"ariaHidden":1834},[1833],[1804,1956,1958,1961,1965,1969,1973],{"className":1957},[1838],[1804,1959],{"className":1960,"style":1843},[1842],[1804,1962,1937],{"className":1963,"style":1964},[1847,1848],"margin-right:0.2222em;",[1804,1966],{"className":1967,"style":1968},[1913],"margin-right:0.2778em;",[1804,1970,1940],{"className":1971},[1972],"mrel",[1804,1974],{"className":1975,"style":1968},[1913],[1804,1977,1979,1983],{"className":1978},[1838],[1804,1980],{"className":1981,"style":1982},[1842],"height:0.8141em;",[1804,1984,1986,1989],{"className":1985},[1847],[1804,1987,1825],{"className":1988,"style":1849},[1847,1848],[1804,1990,1993],{"className":1991},[1992],"msupsub",[1804,1994,1997],{"className":1995},[1996],"vlist-t",[1804,1998,2001],{"className":1999},[2000],"vlist-r",[1804,2002,2005],{"className":2003,"style":1982},[2004],"vlist",[1804,2006,2008,2013],{"style":2007},"top:-3.063em;margin-right:0.05em;",[1804,2009],{"className":2010,"style":2012},[2011],"pstrut","height:2.7em;",[1804,2014,2020],{"className":2015},[2016,2017,2018,2019],"sizing","reset-size6","size3","mtight",[1804,2021,1948],{"className":2022},[1847,2019]," 仍然均匀吗？",[1796,2025,2026],{},"随机变量是从样本空间到数轴的函数。分布描述它把概率质量放在数轴何处。",[2028,2029,2030],"h2",{"id":2030},"学习目标",[1796,2032,2033],{},"你应能：",[2035,2036,2037,2041,2044,2047,2050],"ol",{},[2038,2039,2040],"li",{},"区分离散、连续与混合随机变量；",[2038,2042,2043],{},"在 PMF、PDF 与 CDF 之间转换；",[2038,2045,2046],{},"计算分位数与尾概率；",[2038,2048,2049],{},"使用联合、边际和条件分布；",[2038,2051,2052],{},"对单调与多对一变换求分布。",[2028,2054,2056],{"id":2055},"_1-cdf-是统一语言","1. CDF 是统一语言",[1804,2058,2061],{"className":2059},[2060],"katex-display",[1804,2062,2064,2114],{"className":2063},[1807],[1804,2065,2067],{"className":2066},[1811],[1813,2068,2070],{"xmlns":1815,"display":2069},"block",[1817,2071,2072,2111],{},[1820,2073,2074,2082,2085,2088,2091,2093,2096,2098,2100,2103,2105,2107],{},[2075,2076,2077,2080],"msub",{},[1823,2078,2079],{},"F",[1823,2081,1825],{},[1864,2083,2084],{"stretchy":1866},"(",[1823,2086,2087],{},"x",[1864,2089,2090],{"stretchy":1866},")",[1864,2092,1940],{},[1823,2094,2095],{},"P",[1864,2097,2084],{"stretchy":1866},[1823,2099,1825],{},[1864,2101,2102],{},"≤",[1823,2104,2087],{},[1864,2106,2090],{"stretchy":1866},[1823,2108,2110],{"mathvariant":2109},"normal",".",[1827,2112,2113],{"encoding":1829},"F_X(x)=P(X\\le x).",[1804,2115,2117,2188,2212],{"className":2116,"ariaHidden":1834},[1833],[1804,2118,2120,2123,2170,2173,2176,2179,2182,2185],{"className":2119},[1838],[1804,2121],{"className":2122,"style":1895},[1842],[1804,2124,2126,2130],{"className":2125},[1847],[1804,2127,2079],{"className":2128,"style":2129},[1847,1848],"margin-right:0.1389em;",[1804,2131,2133],{"className":2132},[1992],[1804,2134,2137,2161],{"className":2135},[1996,2136],"vlist-t2",[1804,2138,2140,2156],{"className":2139},[2000],[1804,2141,2144],{"className":2142,"style":2143},[2004],"height:0.3283em;",[1804,2145,2147,2150],{"style":2146},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1804,2148],{"className":2149,"style":2012},[2011],[1804,2151,2153],{"className":2152},[2016,2017,2018,2019],[1804,2154,1825],{"className":2155,"style":1849},[1847,1848,2019],[1804,2157,2160],{"className":2158},[2159],"vlist-s","​",[1804,2162,2164],{"className":2163},[2000],[1804,2165,2168],{"className":2166,"style":2167},[2004],"height:0.15em;",[1804,2169],{},[1804,2171,2084],{"className":2172},[1899],[1804,2174,2087],{"className":2175},[1847,1848],[1804,2177,2090],{"className":2178},[1921],[1804,2180],{"className":2181,"style":1968},[1913],[1804,2183,1940],{"className":2184},[1972],[1804,2186],{"className":2187,"style":1968},[1913],[1804,2189,2191,2194,2197,2200,2203,2206,2209],{"className":2190},[1838],[1804,2192],{"className":2193,"style":1895},[1842],[1804,2195,2095],{"className":2196,"style":2129},[1847,1848],[1804,2198,2084],{"className":2199},[1899],[1804,2201,1825],{"className":2202,"style":1849},[1847,1848],[1804,2204],{"className":2205,"style":1968},[1913],[1804,2207,2102],{"className":2208},[1972],[1804,2210],{"className":2211,"style":1968},[1913],[1804,2213,2215,2218,2221,2224],{"className":2214},[1838],[1804,2216],{"className":2217,"style":1895},[1842],[1804,2219,2087],{"className":2220},[1847,1848],[1804,2222,2090],{"className":2223},[1921],[1804,2225,2110],{"className":2226},[1847],[1796,2228,2229],{},"任何 CDF 都满足：",[2231,2232,2233,2236,2239],"ul",{},[2038,2234,2235],{},"单调不减；",[2038,2237,2238],{},"右连续；",[2038,2240,2241,2454],{},[1804,2242,2244,2298],{"className":2243},[1807],[1804,2245,2247],{"className":2246},[1811],[1813,2248,2249],{"xmlns":1815},[1817,2250,2251,2295],{},[1820,2252,2253,2259,2261,2263,2266,2268,2270,2273,2275,2279,2285,2287,2289,2291,2293],{},[2075,2254,2255,2257],{},[1823,2256,2079],{},[1823,2258,1825],{},[1864,2260,2084],{"stretchy":1866},[1864,2262,1870],{},[1823,2264,2265],{"mathvariant":2109},"∞",[1864,2267,2090],{"stretchy":1866},[1864,2269,1940],{},[1872,2271,2272],{},"0",[1864,2274,1877],{"separator":1834},[2276,2277,2278],"mtext",{},"  ",[2075,2280,2281,2283],{},[1823,2282,2079],{},[1823,2284,1825],{},[1864,2286,2084],{"stretchy":1866},[1823,2288,2265],{"mathvariant":2109},[1864,2290,2090],{"stretchy":1866},[1864,2292,1940],{},[1872,2294,1874],{},[1827,2296,2297],{"encoding":1829},"F_X(-\\infty)=0,\\;F_X(\\infty)=1",[1804,2299,2301,2368,2444],{"className":2300,"ariaHidden":1834},[1833],[1804,2302,2304,2307,2347,2350,2353,2356,2359,2362,2365],{"className":2303},[1838],[1804,2305],{"className":2306,"style":1895},[1842],[1804,2308,2310,2313],{"className":2309},[1847],[1804,2311,2079],{"className":2312,"style":2129},[1847,1848],[1804,2314,2316],{"className":2315},[1992],[1804,2317,2319,2339],{"className":2318},[1996,2136],[1804,2320,2322,2336],{"className":2321},[2000],[1804,2323,2325],{"className":2324,"style":2143},[2004],[1804,2326,2327,2330],{"style":2146},[1804,2328],{"className":2329,"style":2012},[2011],[1804,2331,2333],{"className":2332},[2016,2017,2018,2019],[1804,2334,1825],{"className":2335,"style":1849},[1847,1848,2019],[1804,2337,2160],{"className":2338},[2159],[1804,2340,2342],{"className":2341},[2000],[1804,2343,2345],{"className":2344,"style":2167},[2004],[1804,2346],{},[1804,2348,2084],{"className":2349},[1899],[1804,2351,1870],{"className":2352},[1847],[1804,2354,2265],{"className":2355},[1847],[1804,2357,2090],{"className":2358},[1921],[1804,2360],{"className":2361,"style":1968},[1913],[1804,2363,1940],{"className":2364},[1972],[1804,2366],{"className":2367,"style":1968},[1913],[1804,2369,2371,2374,2377,2380,2383,2386,2426,2429,2432,2435,2438,2441],{"className":2370},[1838],[1804,2372],{"className":2373,"style":1895},[1842],[1804,2375,2272],{"className":2376},[1847],[1804,2378,1877],{"className":2379},[1909],[1804,2381],{"className":2382,"style":1968},[1913],[1804,2384],{"className":2385,"style":1914},[1913],[1804,2387,2389,2392],{"className":2388},[1847],[1804,2390,2079],{"className":2391,"style":2129},[1847,1848],[1804,2393,2395],{"className":2394},[1992],[1804,2396,2398,2418],{"className":2397},[1996,2136],[1804,2399,2401,2415],{"className":2400},[2000],[1804,2402,2404],{"className":2403,"style":2143},[2004],[1804,2405,2406,2409],{"style":2146},[1804,2407],{"className":2408,"style":2012},[2011],[1804,2410,2412],{"className":2411},[2016,2017,2018,2019],[1804,2413,1825],{"className":2414,"style":1849},[1847,1848,2019],[1804,2416,2160],{"className":2417},[2159],[1804,2419,2421],{"className":2420},[2000],[1804,2422,2424],{"className":2423,"style":2167},[2004],[1804,2425],{},[1804,2427,2084],{"className":2428},[1899],[1804,2430,2265],{"className":2431},[1847],[1804,2433,2090],{"className":2434},[1921],[1804,2436],{"className":2437,"style":1968},[1913],[1804,2439,1940],{"className":2440},[1972],[1804,2442],{"className":2443,"style":1968},[1913],[1804,2445,2447,2451],{"className":2446},[1838],[1804,2448],{"className":2449,"style":2450},[1842],"height:0.6444em;",[1804,2452,1874],{"className":2453},[1847],"。",[1796,2456,2457],{},"离散变量用概率质量函数",[1804,2459,2461],{"className":2460},[2060],[1804,2462,2464,2532],{"className":2463},[1807],[1804,2465,2467],{"className":2466},[1811],[1813,2468,2469],{"xmlns":1815,"display":2069},[1817,2470,2471,2529],{},[1820,2472,2473,2479,2481,2483,2485,2487,2489,2491,2493,2495,2497,2499,2501,2504,2512,2518,2520,2522,2524,2526],{},[2075,2474,2475,2477],{},[1823,2476,1796],{},[1823,2478,1825],{},[1864,2480,2084],{"stretchy":1866},[1823,2482,2087],{},[1864,2484,2090],{"stretchy":1866},[1864,2486,1940],{},[1823,2488,2095],{},[1864,2490,2084],{"stretchy":1866},[1823,2492,1825],{},[1864,2494,1940],{},[1823,2496,2087],{},[1864,2498,2090],{"stretchy":1866},[1864,2500,1877],{"separator":1834},[1913,2502],{"width":2503},"2em",[2505,2506,2507,2510],"munder",{},[1864,2508,2509],{},"∑",[1823,2511,2087],{},[2075,2513,2514,2516],{},[1823,2515,1796],{},[1823,2517,1825],{},[1864,2519,2084],{"stretchy":1866},[1823,2521,2087],{},[1864,2523,2090],{"stretchy":1866},[1864,2525,1940],{},[1872,2527,2528],{},"1.",[1827,2530,2531],{"encoding":1829},"p_X(x)=P(X=x),\n\\qquad \\sum_xp_X(x)=1.",[1804,2533,2535,2600,2624,2761],{"className":2534,"ariaHidden":1834},[1833],[1804,2536,2538,2541,2582,2585,2588,2591,2594,2597],{"className":2537},[1838],[1804,2539],{"className":2540,"style":1895},[1842],[1804,2542,2544,2547],{"className":2543},[1847],[1804,2545,1796],{"className":2546},[1847,1848],[1804,2548,2550],{"className":2549},[1992],[1804,2551,2553,2574],{"className":2552},[1996,2136],[1804,2554,2556,2571],{"className":2555},[2000],[1804,2557,2559],{"className":2558,"style":2143},[2004],[1804,2560,2562,2565],{"style":2561},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1804,2563],{"className":2564,"style":2012},[2011],[1804,2566,2568],{"className":2567},[2016,2017,2018,2019],[1804,2569,1825],{"className":2570,"style":1849},[1847,1848,2019],[1804,2572,2160],{"className":2573},[2159],[1804,2575,2577],{"className":2576},[2000],[1804,2578,2580],{"className":2579,"style":2167},[2004],[1804,2581],{},[1804,2583,2084],{"className":2584},[1899],[1804,2586,2087],{"className":2587},[1847,1848],[1804,2589,2090],{"className":2590},[1921],[1804,2592],{"className":2593,"style":1968},[1913],[1804,2595,1940],{"className":2596},[1972],[1804,2598],{"className":2599,"style":1968},[1913],[1804,2601,2603,2606,2609,2612,2615,2618,2621],{"className":2602},[1838],[1804,2604],{"className":2605,"style":1895},[1842],[1804,2607,2095],{"className":2608,"style":2129},[1847,1848],[1804,2610,2084],{"className":2611},[1899],[1804,2613,1825],{"className":2614,"style":1849},[1847,1848],[1804,2616],{"className":2617,"style":1968},[1913],[1804,2619,1940],{"className":2620},[1972],[1804,2622],{"className":2623,"style":1968},[1913],[1804,2625,2627,2631,2634,2637,2640,2644,2647,2700,2703,2743,2746,2749,2752,2755,2758],{"className":2626},[1838],[1804,2628],{"className":2629,"style":2630},[1842],"height:2.3em;vertical-align:-1.25em;",[1804,2632,2087],{"className":2633},[1847,1848],[1804,2635,2090],{"className":2636},[1921],[1804,2638,1877],{"className":2639},[1909],[1804,2641],{"className":2642,"style":2643},[1913],"margin-right:2em;",[1804,2645],{"className":2646,"style":1914},[1913],[1804,2648,2652],{"className":2649},[2650,2651],"mop","op-limits",[1804,2653,2655,2691],{"className":2654},[1996,2136],[1804,2656,2658,2688],{"className":2657},[2000],[1804,2659,2662,2675],{"className":2660,"style":2661},[2004],"height:1.05em;",[1804,2663,2665,2669],{"style":2664},"top:-1.9em;margin-left:0em;",[1804,2666],{"className":2667,"style":2668},[2011],"height:3.05em;",[1804,2670,2672],{"className":2671},[2016,2017,2018,2019],[1804,2673,2087],{"className":2674},[1847,1848,2019],[1804,2676,2678,2681],{"style":2677},"top:-3.05em;",[1804,2679],{"className":2680,"style":2668},[2011],[1804,2682,2683],{},[1804,2684,2509],{"className":2685},[2650,2686,2687],"op-symbol","large-op",[1804,2689,2160],{"className":2690},[2159],[1804,2692,2694],{"className":2693},[2000],[1804,2695,2698],{"className":2696,"style":2697},[2004],"height:1.25em;",[1804,2699],{},[1804,2701],{"className":2702,"style":1914},[1913],[1804,2704,2706,2709],{"className":2705},[1847],[1804,2707,1796],{"className":2708},[1847,1848],[1804,2710,2712],{"className":2711},[1992],[1804,2713,2715,2735],{"className":2714},[1996,2136],[1804,2716,2718,2732],{"className":2717},[2000],[1804,2719,2721],{"className":2720,"style":2143},[2004],[1804,2722,2723,2726],{"style":2561},[1804,2724],{"className":2725,"style":2012},[2011],[1804,2727,2729],{"className":2728},[2016,2017,2018,2019],[1804,2730,1825],{"className":2731,"style":1849},[1847,1848,2019],[1804,2733,2160],{"className":2734},[2159],[1804,2736,2738],{"className":2737},[2000],[1804,2739,2741],{"className":2740,"style":2167},[2004],[1804,2742],{},[1804,2744,2084],{"className":2745},[1899],[1804,2747,2087],{"className":2748},[1847,1848],[1804,2750,2090],{"className":2751},[1921],[1804,2753],{"className":2754,"style":1968},[1913],[1804,2756,1940],{"className":2757},[1972],[1804,2759],{"className":2760,"style":1968},[1913],[1804,2762,2764,2767],{"className":2763},[1838],[1804,2765],{"className":2766,"style":2450},[1842],[1804,2768,2528],{"className":2769},[1847],[1796,2771,2772],{},"绝对连续变量用密度",[1804,2774,2776],{"className":2775},[2060],[1804,2777,2779,2894],{"className":2778},[1807],[1804,2780,2782],{"className":2781},[1811],[1813,2783,2784],{"xmlns":1815,"display":2069},[1817,2785,2786,2891],{},[1820,2787,2788,2794,2796,2798,2800,2802,2816,2823,2825,2828,2830,2833,2836,2838,2840,2842,2844,2846,2849,2852,2854,2856,2859,2861,2863,2871,2877,2879,2881,2883,2885,2887,2889],{},[2075,2789,2790,2792],{},[1823,2791,2079],{},[1823,2793,1825],{},[1864,2795,2084],{"stretchy":1866},[1823,2797,2087],{},[1864,2799,2090],{"stretchy":1866},[1864,2801,1940],{},[2803,2804,2805,2808,2814],"msubsup",{},[1864,2806,2807],{},"∫",[1820,2809,2810,2812],{},[1864,2811,1870],{},[1823,2813,2265],{"mathvariant":2109},[1823,2815,2087],{},[2075,2817,2818,2821],{},[1823,2819,2820],{},"f",[1823,2822,1825],{},[1864,2824,2084],{"stretchy":1866},[1823,2826,2827],{},"t",[1864,2829,2090],{"stretchy":1866},[2276,2831,2832],{}," ",[1823,2834,2835],{},"d",[1823,2837,2827],{},[1864,2839,1877],{"separator":1834},[1913,2841],{"width":2503},[1823,2843,2095],{},[1864,2845,2084],{"stretchy":1866},[1823,2847,2848],{},"a",[1864,2850,2851],{},"\u003C",[1823,2853,1825],{},[1864,2855,2102],{},[1823,2857,2858],{},"b",[1864,2860,2090],{"stretchy":1866},[1864,2862,1940],{},[2803,2864,2865,2867,2869],{},[1864,2866,2807],{},[1823,2868,2848],{},[1823,2870,2858],{},[2075,2872,2873,2875],{},[1823,2874,2820],{},[1823,2876,1825],{},[1864,2878,2084],{"stretchy":1866},[1823,2880,2827],{},[1864,2882,2090],{"stretchy":1866},[2276,2884,2832],{},[1823,2886,2835],{},[1823,2888,2827],{},[1823,2890,2110],{"mathvariant":2109},[1827,2892,2893],{"encoding":1829},"F_X(x)=\\int_{-\\infty}^xf_X(t)\\,dt,\n\\qquad P(a\u003CX\\le b)=\\int_a^bf_X(t)\\,dt.",[1804,2895,2897,2961,3120,3139,3160],{"className":2896,"ariaHidden":1834},[1833],[1804,2898,2900,2903,2943,2946,2949,2952,2955,2958],{"className":2899},[1838],[1804,2901],{"className":2902,"style":1895},[1842],[1804,2904,2906,2909],{"className":2905},[1847],[1804,2907,2079],{"className":2908,"style":2129},[1847,1848],[1804,2910,2912],{"className":2911},[1992],[1804,2913,2915,2935],{"className":2914},[1996,2136],[1804,2916,2918,2932],{"className":2917},[2000],[1804,2919,2921],{"className":2920,"style":2143},[2004],[1804,2922,2923,2926],{"style":2146},[1804,2924],{"className":2925,"style":2012},[2011],[1804,2927,2929],{"className":2928},[2016,2017,2018,2019],[1804,2930,1825],{"className":2931,"style":1849},[1847,1848,2019],[1804,2933,2160],{"className":2934},[2159],[1804,2936,2938],{"className":2937},[2000],[1804,2939,2941],{"className":2940,"style":2167},[2004],[1804,2942],{},[1804,2944,2084],{"className":2945},[1899],[1804,2947,2087],{"className":2948},[1847,1848],[1804,2950,2090],{"className":2951},[1921],[1804,2953],{"className":2954,"style":1968},[1913],[1804,2956,1940],{"className":2957},[1972],[1804,2959],{"className":2960,"style":1968},[1913],[1804,2962,2964,2968,3030,3033,3075,3078,3081,3084,3087,3090,3093,3096,3099,3102,3105,3108,3111,3114,3117],{"className":2963},[1838],[1804,2965],{"className":2966,"style":2967},[1842],"height:2.3846em;vertical-align:-0.9703em;",[1804,2969,2971,2975],{"className":2970},[2650],[1804,2972,2807],{"className":2973,"style":2974},[2650,2686,2687],"margin-right:0.4445em;position:relative;top:-0.0011em;",[1804,2976,2978],{"className":2977},[1992],[1804,2979,2981,3021],{"className":2980},[1996,2136],[1804,2982,2984,3018],{"className":2983},[2000],[1804,2985,2988,3006],{"className":2986,"style":2987},[2004],"height:1.4143em;",[1804,2989,2991,2994],{"style":2990},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[1804,2992],{"className":2993,"style":2012},[2011],[1804,2995,2997],{"className":2996},[2016,2017,2018,2019],[1804,2998,3000,3003],{"className":2999},[1847,2019],[1804,3001,1870],{"className":3002},[1847,2019],[1804,3004,2265],{"className":3005},[1847,2019],[1804,3007,3009,3012],{"style":3008},"top:-3.8129em;margin-right:0.05em;",[1804,3010],{"className":3011,"style":2012},[2011],[1804,3013,3015],{"className":3014},[2016,2017,2018,2019],[1804,3016,2087],{"className":3017},[1847,1848,2019],[1804,3019,2160],{"className":3020},[2159],[1804,3022,3024],{"className":3023},[2000],[1804,3025,3028],{"className":3026,"style":3027},[2004],"height:0.9703em;",[1804,3029],{},[1804,3031],{"className":3032,"style":1914},[1913],[1804,3034,3036,3040],{"className":3035},[1847],[1804,3037,2820],{"className":3038,"style":3039},[1847,1848],"margin-right:0.1076em;",[1804,3041,3043],{"className":3042},[1992],[1804,3044,3046,3067],{"className":3045},[1996,2136],[1804,3047,3049,3064],{"className":3048},[2000],[1804,3050,3052],{"className":3051,"style":2143},[2004],[1804,3053,3055,3058],{"style":3054},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[1804,3056],{"className":3057,"style":2012},[2011],[1804,3059,3061],{"className":3060},[2016,2017,2018,2019],[1804,3062,1825],{"className":3063,"style":1849},[1847,1848,2019],[1804,3065,2160],{"className":3066},[2159],[1804,3068,3070],{"className":3069},[2000],[1804,3071,3073],{"className":3072,"style":2167},[2004],[1804,3074],{},[1804,3076,2084],{"className":3077},[1899],[1804,3079,2827],{"className":3080},[1847,1848],[1804,3082,2090],{"className":3083},[1921],[1804,3085],{"className":3086,"style":1914},[1913],[1804,3088,2835],{"className":3089},[1847,1848],[1804,3091,2827],{"className":3092},[1847,1848],[1804,3094,1877],{"className":3095},[1909],[1804,3097],{"className":3098,"style":2643},[1913],[1804,3100],{"className":3101,"style":1914},[1913],[1804,3103,2095],{"className":3104,"style":2129},[1847,1848],[1804,3106,2084],{"className":3107},[1899],[1804,3109,2848],{"className":3110},[1847,1848],[1804,3112],{"className":3113,"style":1968},[1913],[1804,3115,2851],{"className":3116},[1972],[1804,3118],{"className":3119,"style":1968},[1913],[1804,3121,3123,3127,3130,3133,3136],{"className":3122},[1838],[1804,3124],{"className":3125,"style":3126},[1842],"height:0.8193em;vertical-align:-0.136em;",[1804,3128,1825],{"className":3129,"style":1849},[1847,1848],[1804,3131],{"className":3132,"style":1968},[1913],[1804,3134,2102],{"className":3135},[1972],[1804,3137],{"className":3138,"style":1968},[1913],[1804,3140,3142,3145,3148,3151,3154,3157],{"className":3141},[1838],[1804,3143],{"className":3144,"style":1895},[1842],[1804,3146,2858],{"className":3147},[1847,1848],[1804,3149,2090],{"className":3150},[1921],[1804,3152],{"className":3153,"style":1968},[1913],[1804,3155,1940],{"className":3156},[1972],[1804,3158],{"className":3159,"style":1968},[1913],[1804,3161,3163,3167,3220,3223,3263,3266,3269,3272,3275,3278,3281],{"className":3162},[1838],[1804,3164],{"className":3165,"style":3166},[1842],"height:2.511em;vertical-align:-0.9119em;",[1804,3168,3170,3173],{"className":3169},[2650],[1804,3171,2807],{"className":3172,"style":2974},[2650,2686,2687],[1804,3174,3176],{"className":3175},[1992],[1804,3177,3179,3211],{"className":3178},[1996,2136],[1804,3180,3182,3208],{"className":3181},[2000],[1804,3183,3186,3197],{"className":3184,"style":3185},[2004],"height:1.599em;",[1804,3187,3188,3191],{"style":2990},[1804,3189],{"className":3190,"style":2012},[2011],[1804,3192,3194],{"className":3193},[2016,2017,2018,2019],[1804,3195,2848],{"className":3196},[1847,1848,2019],[1804,3198,3199,3202],{"style":3008},[1804,3200],{"className":3201,"style":2012},[2011],[1804,3203,3205],{"className":3204},[2016,2017,2018,2019],[1804,3206,2858],{"className":3207},[1847,1848,2019],[1804,3209,2160],{"className":3210},[2159],[1804,3212,3214],{"className":3213},[2000],[1804,3215,3218],{"className":3216,"style":3217},[2004],"height:0.9119em;",[1804,3219],{},[1804,3221],{"className":3222,"style":1914},[1913],[1804,3224,3226,3229],{"className":3225},[1847],[1804,3227,2820],{"className":3228,"style":3039},[1847,1848],[1804,3230,3232],{"className":3231},[1992],[1804,3233,3235,3255],{"className":3234},[1996,2136],[1804,3236,3238,3252],{"className":3237},[2000],[1804,3239,3241],{"className":3240,"style":2143},[2004],[1804,3242,3243,3246],{"style":3054},[1804,3244],{"className":3245,"style":2012},[2011],[1804,3247,3249],{"className":3248},[2016,2017,2018,2019],[1804,3250,1825],{"className":3251,"style":1849},[1847,1848,2019],[1804,3253,2160],{"className":3254},[2159],[1804,3256,3258],{"className":3257},[2000],[1804,3259,3261],{"className":3260,"style":2167},[2004],[1804,3262],{},[1804,3264,2084],{"className":3265},[1899],[1804,3267,2827],{"className":3268},[1847,1848],[1804,3270,2090],{"className":3271},[1921],[1804,3273],{"className":3274,"style":1914},[1913],[1804,3276,2835],{"className":3277},[1847,1848],[1804,3279,2827],{"className":3280},[1847,1848],[1804,3282,2110],{"className":3283},[1847],[1796,3285,3286,3287,3339],{},"密度值可以大于 1；只有积分后的概率必须位于 ",[1804,3288,3290,3312],{"className":3289},[1807],[1804,3291,3293],{"className":3292},[1811],[1813,3294,3295],{"xmlns":1815},[1817,3296,3297,3309],{},[1820,3298,3299,3301,3303,3305,3307],{},[1864,3300,1867],{"stretchy":1866},[1872,3302,2272],{},[1864,3304,1877],{"separator":1834},[1872,3306,1874],{},[1864,3308,1882],{"stretchy":1866},[1827,3310,3311],{"encoding":1829},"[0,1]",[1804,3313,3315],{"className":3314,"ariaHidden":1834},[1833],[1804,3316,3318,3321,3324,3327,3330,3333,3336],{"className":3317},[1838],[1804,3319],{"className":3320,"style":1895},[1842],[1804,3322,1867],{"className":3323},[1899],[1804,3325,2272],{"className":3326},[1847],[1804,3328,1877],{"className":3329},[1909],[1804,3331],{"className":3332,"style":1914},[1913],[1804,3334,1874],{"className":3335},[1847],[1804,3337,1882],{"className":3338},[1921],"。连续变量在单点的概率为 0，不表示该数值“不可能出现”。",[2028,3341,3343],{"id":3342},"_2-分位数反转-cdf","2. 分位数反转 CDF",[1796,3345,3346,3347,3378],{},"广义 ",[1804,3348,3350,3364],{"className":3349},[1807],[1804,3351,3353],{"className":3352},[1811],[1813,3354,3355],{"xmlns":1815},[1817,3356,3357,3362],{},[1820,3358,3359],{},[1823,3360,3361],{},"q",[1827,3363,3361],{"encoding":1829},[1804,3365,3367],{"className":3366,"ariaHidden":1834},[1833],[1804,3368,3370,3374],{"className":3369},[1838],[1804,3371],{"className":3372,"style":3373},[1842],"height:0.625em;vertical-align:-0.1944em;",[1804,3375,3361],{"className":3376,"style":3377},[1847,1848],"margin-right:0.0359em;"," 分位数：",[1804,3380,3382],{"className":3381},[2060],[1804,3383,3385,3444],{"className":3384},[1807],[1804,3386,3388],{"className":3387},[1811],[1813,3389,3390],{"xmlns":1815,"display":2069},[1817,3391,3392,3441],{},[1820,3393,3394,3397,3399,3401,3403,3405,3408,3411,3414,3416,3419,3425,3427,3429,3431,3434,3436,3439],{},[1823,3395,3396],{},"Q",[1864,3398,2084],{"stretchy":1866},[1823,3400,3361],{},[1864,3402,2090],{"stretchy":1866},[1864,3404,1940],{},[1823,3406,3407],{},"inf",[1864,3409,3410],{},"⁡",[1864,3412,3413],{"stretchy":1866},"{",[1823,3415,2087],{},[1864,3417,3418],{},":",[2075,3420,3421,3423],{},[1823,3422,2079],{},[1823,3424,1825],{},[1864,3426,2084],{"stretchy":1866},[1823,3428,2087],{},[1864,3430,2090],{"stretchy":1866},[1864,3432,3433],{},"≥",[1823,3435,3361],{},[1864,3437,3438],{"stretchy":1866},"}",[1823,3440,2110],{"mathvariant":2109},[1827,3442,3443],{"encoding":1829},"Q(q)=\\inf\\{x:F_X(x)\\ge q\\}.",[1804,3445,3447,3474,3502,3566],{"className":3446,"ariaHidden":1834},[1833],[1804,3448,3450,3453,3456,3459,3462,3465,3468,3471],{"className":3449},[1838],[1804,3451],{"className":3452,"style":1895},[1842],[1804,3454,3396],{"className":3455},[1847,1848],[1804,3457,2084],{"className":3458},[1899],[1804,3460,3361],{"className":3461,"style":3377},[1847,1848],[1804,3463,2090],{"className":3464},[1921],[1804,3466],{"className":3467,"style":1968},[1913],[1804,3469,1940],{"className":3470},[1972],[1804,3472],{"className":3473,"style":1968},[1913],[1804,3475,3477,3480,3487,3490,3493,3496,3499],{"className":3476},[1838],[1804,3478],{"className":3479,"style":1895},[1842],[1804,3481,3483,3484],{"className":3482},[2650],"in",[1804,3485,2820],{"style":3486},"margin-right:0.0778em;",[1804,3488,3413],{"className":3489},[1899],[1804,3491,2087],{"className":3492},[1847,1848],[1804,3494],{"className":3495,"style":1968},[1913],[1804,3497,3418],{"className":3498},[1972],[1804,3500],{"className":3501,"style":1968},[1913],[1804,3503,3505,3508,3548,3551,3554,3557,3560,3563],{"className":3504},[1838],[1804,3506],{"className":3507,"style":1895},[1842],[1804,3509,3511,3514],{"className":3510},[1847],[1804,3512,2079],{"className":3513,"style":2129},[1847,1848],[1804,3515,3517],{"className":3516},[1992],[1804,3518,3520,3540],{"className":3519},[1996,2136],[1804,3521,3523,3537],{"className":3522},[2000],[1804,3524,3526],{"className":3525,"style":2143},[2004],[1804,3527,3528,3531],{"style":2146},[1804,3529],{"className":3530,"style":2012},[2011],[1804,3532,3534],{"className":3533},[2016,2017,2018,2019],[1804,3535,1825],{"className":3536,"style":1849},[1847,1848,2019],[1804,3538,2160],{"className":3539},[2159],[1804,3541,3543],{"className":3542},[2000],[1804,3544,3546],{"className":3545,"style":2167},[2004],[1804,3547],{},[1804,3549,2084],{"className":3550},[1899],[1804,3552,2087],{"className":3553},[1847,1848],[1804,3555,2090],{"className":3556},[1921],[1804,3558],{"className":3559,"style":1968},[1913],[1804,3561,3433],{"className":3562},[1972],[1804,3564],{"className":3565,"style":1968},[1913],[1804,3567,3569,3572,3575,3578],{"className":3568},[1838],[1804,3570],{"className":3571,"style":1895},[1842],[1804,3573,3361],{"className":3574,"style":3377},[1847,1848],[1804,3576,3438],{"className":3577},[1921],[1804,3579,2110],{"className":3580},[1847],[1796,3582,3583,3584,3629,3630,3696],{},"中位数是 ",[1804,3585,3587,3608],{"className":3586},[1807],[1804,3588,3590],{"className":3589},[1811],[1813,3591,3592],{"xmlns":1815},[1817,3593,3594,3605],{},[1820,3595,3596,3598,3600,3603],{},[1823,3597,3396],{},[1864,3599,2084],{"stretchy":1866},[1872,3601,3602],{},"0.5",[1864,3604,2090],{"stretchy":1866},[1827,3606,3607],{"encoding":1829},"Q(0.5)",[1804,3609,3611],{"className":3610,"ariaHidden":1834},[1833],[1804,3612,3614,3617,3620,3623,3626],{"className":3613},[1838],[1804,3615],{"className":3616,"style":1895},[1842],[1804,3618,3396],{"className":3619},[1847,1848],[1804,3621,2084],{"className":3622},[1899],[1804,3624,3602],{"className":3625},[1847],[1804,3627,2090],{"className":3628},[1921],"。离散分布可能没有满足 ",[1804,3631,3633,3657],{"className":3632},[1807],[1804,3634,3636],{"className":3635},[1811],[1813,3637,3638],{"xmlns":1815},[1817,3639,3640,3654],{},[1820,3641,3642,3644,3646,3648,3650,3652],{},[1823,3643,2079],{},[1864,3645,2084],{"stretchy":1866},[1823,3647,2087],{},[1864,3649,2090],{"stretchy":1866},[1864,3651,1940],{},[1823,3653,3361],{},[1827,3655,3656],{"encoding":1829},"F(x)=q",[1804,3658,3660,3687],{"className":3659,"ariaHidden":1834},[1833],[1804,3661,3663,3666,3669,3672,3675,3678,3681,3684],{"className":3662},[1838],[1804,3664],{"className":3665,"style":1895},[1842],[1804,3667,2079],{"className":3668,"style":2129},[1847,1848],[1804,3670,2084],{"className":3671},[1899],[1804,3673,2087],{"className":3674},[1847,1848],[1804,3676,2090],{"className":3677},[1921],[1804,3679],{"className":3680,"style":1968},[1913],[1804,3682,1940],{"className":3683},[1972],[1804,3685],{"className":3686,"style":1968},[1913],[1804,3688,3690,3693],{"className":3689},[1838],[1804,3691],{"className":3692,"style":3373},[1842],[1804,3694,3361],{"className":3695,"style":3377},[1847,1848]," 的精确点，因此使用“首次达到”定义。",[1796,3698,3699],{},"尾概率最好用补集：",[1804,3701,3703],{"className":3702},[2060],[1804,3704,3706,3751],{"className":3705},[1807],[1804,3707,3709],{"className":3708},[1811],[1813,3710,3711],{"xmlns":1815,"display":2069},[1817,3712,3713,3748],{},[1820,3714,3715,3717,3719,3721,3724,3726,3728,3730,3732,3734,3740,3742,3744,3746],{},[1823,3716,2095],{},[1864,3718,2084],{"stretchy":1866},[1823,3720,1825],{},[1864,3722,3723],{},">",[1823,3725,2087],{},[1864,3727,2090],{"stretchy":1866},[1864,3729,1940],{},[1872,3731,1874],{},[1864,3733,1870],{},[2075,3735,3736,3738],{},[1823,3737,2079],{},[1823,3739,1825],{},[1864,3741,2084],{"stretchy":1866},[1823,3743,2087],{},[1864,3745,2090],{"stretchy":1866},[1864,3747,1877],{"separator":1834},[1827,3749,3750],{"encoding":1829},"P(X>x)=1-F_X(x),",[1804,3752,3754,3778,3799,3819],{"className":3753,"ariaHidden":1834},[1833],[1804,3755,3757,3760,3763,3766,3769,3772,3775],{"className":3756},[1838],[1804,3758],{"className":3759,"style":1895},[1842],[1804,3761,2095],{"className":3762,"style":2129},[1847,1848],[1804,3764,2084],{"className":3765},[1899],[1804,3767,1825],{"className":3768,"style":1849},[1847,1848],[1804,3770],{"className":3771,"style":1968},[1913],[1804,3773,3723],{"className":3774},[1972],[1804,3776],{"className":3777,"style":1968},[1913],[1804,3779,3781,3784,3787,3790,3793,3796],{"className":3780},[1838],[1804,3782],{"className":3783,"style":1895},[1842],[1804,3785,2087],{"className":3786},[1847,1848],[1804,3788,2090],{"className":3789},[1921],[1804,3791],{"className":3792,"style":1968},[1913],[1804,3794,1940],{"className":3795},[1972],[1804,3797],{"className":3798,"style":1968},[1913],[1804,3800,3802,3806,3809,3812,3816],{"className":3801},[1838],[1804,3803],{"className":3804,"style":3805},[1842],"height:0.7278em;vertical-align:-0.0833em;",[1804,3807,1874],{"className":3808},[1847],[1804,3810],{"className":3811,"style":1964},[1913],[1804,3813,1870],{"className":3814},[3815],"mbin",[1804,3817],{"className":3818,"style":1964},[1913],[1804,3820,3822,3825,3865,3868,3871,3874],{"className":3821},[1838],[1804,3823],{"className":3824,"style":1895},[1842],[1804,3826,3828,3831],{"className":3827},[1847],[1804,3829,2079],{"className":3830,"style":2129},[1847,1848],[1804,3832,3834],{"className":3833},[1992],[1804,3835,3837,3857],{"className":3836},[1996,2136],[1804,3838,3840,3854],{"className":3839},[2000],[1804,3841,3843],{"className":3842,"style":2143},[2004],[1804,3844,3845,3848],{"style":2146},[1804,3846],{"className":3847,"style":2012},[2011],[1804,3849,3851],{"className":3850},[2016,2017,2018,2019],[1804,3852,1825],{"className":3853,"style":1849},[1847,1848,2019],[1804,3855,2160],{"className":3856},[2159],[1804,3858,3860],{"className":3859},[2000],[1804,3861,3863],{"className":3862,"style":2167},[2004],[1804,3864],{},[1804,3866,2084],{"className":3867},[1899],[1804,3869,2087],{"className":3870},[1847,1848],[1804,3872,2090],{"className":3873},[1921],[1804,3875,1877],{"className":3876},[1909],[1796,3878,3879,3880,3946,3947,4013,4014,2454],{},"但离散情形要小心 ",[1804,3881,3883,3907],{"className":3882},[1807],[1804,3884,3886],{"className":3885},[1811],[1813,3887,3888],{"xmlns":1815},[1817,3889,3890,3904],{},[1820,3891,3892,3894,3896,3898,3900,3902],{},[1823,3893,2095],{},[1864,3895,2084],{"stretchy":1866},[1823,3897,1825],{},[1864,3899,3433],{},[1823,3901,2087],{},[1864,3903,2090],{"stretchy":1866},[1827,3905,3906],{"encoding":1829},"P(X\\ge x)",[1804,3908,3910,3934],{"className":3909,"ariaHidden":1834},[1833],[1804,3911,3913,3916,3919,3922,3925,3928,3931],{"className":3912},[1838],[1804,3914],{"className":3915,"style":1895},[1842],[1804,3917,2095],{"className":3918,"style":2129},[1847,1848],[1804,3920,2084],{"className":3921},[1899],[1804,3923,1825],{"className":3924,"style":1849},[1847,1848],[1804,3926],{"className":3927,"style":1968},[1913],[1804,3929,3433],{"className":3930},[1972],[1804,3932],{"className":3933,"style":1968},[1913],[1804,3935,3937,3940,3943],{"className":3936},[1838],[1804,3938],{"className":3939,"style":1895},[1842],[1804,3941,2087],{"className":3942},[1847,1848],[1804,3944,2090],{"className":3945},[1921]," 与 ",[1804,3948,3950,3974],{"className":3949},[1807],[1804,3951,3953],{"className":3952},[1811],[1813,3954,3955],{"xmlns":1815},[1817,3956,3957,3971],{},[1820,3958,3959,3961,3963,3965,3967,3969],{},[1823,3960,2095],{},[1864,3962,2084],{"stretchy":1866},[1823,3964,1825],{},[1864,3966,3723],{},[1823,3968,2087],{},[1864,3970,2090],{"stretchy":1866},[1827,3972,3973],{"encoding":1829},"P(X>x)",[1804,3975,3977,4001],{"className":3976,"ariaHidden":1834},[1833],[1804,3978,3980,3983,3986,3989,3992,3995,3998],{"className":3979},[1838],[1804,3981],{"className":3982,"style":1895},[1842],[1804,3984,2095],{"className":3985,"style":2129},[1847,1848],[1804,3987,2084],{"className":3988},[1899],[1804,3990,1825],{"className":3991,"style":1849},[1847,1848],[1804,3993],{"className":3994,"style":1968},[1913],[1804,3996,3723],{"className":3997},[1972],[1804,3999],{"className":4000,"style":1968},[1913],[1804,4002,4004,4007,4010],{"className":4003},[1838],[1804,4005],{"className":4006,"style":1895},[1842],[1804,4008,2087],{"className":4009},[1847,1848],[1804,4011,2090],{"className":4012},[1921]," 相差 ",[1804,4015,4017,4041],{"className":4016},[1807],[1804,4018,4020],{"className":4019},[1811],[1813,4021,4022],{"xmlns":1815},[1817,4023,4024,4038],{},[1820,4025,4026,4028,4030,4032,4034,4036],{},[1823,4027,2095],{},[1864,4029,2084],{"stretchy":1866},[1823,4031,1825],{},[1864,4033,1940],{},[1823,4035,2087],{},[1864,4037,2090],{"stretchy":1866},[1827,4039,4040],{"encoding":1829},"P(X=x)",[1804,4042,4044,4068],{"className":4043,"ariaHidden":1834},[1833],[1804,4045,4047,4050,4053,4056,4059,4062,4065],{"className":4046},[1838],[1804,4048],{"className":4049,"style":1895},[1842],[1804,4051,2095],{"className":4052,"style":2129},[1847,1848],[1804,4054,2084],{"className":4055},[1899],[1804,4057,1825],{"className":4058,"style":1849},[1847,1848],[1804,4060],{"className":4061,"style":1968},[1913],[1804,4063,1940],{"className":4064},[1972],[1804,4066],{"className":4067,"style":1968},[1913],[1804,4069,4071,4074,4077],{"className":4070},[1838],[1804,4072],{"className":4073,"style":1895},[1842],[1804,4075,2087],{"className":4076},[1847,1848],[1804,4078,2090],{"className":4079},[1921],[2028,4081,4083],{"id":4082},"_3-联合边际与条件分布","3. 联合、边际与条件分布",[1796,4085,4086],{},"联合 CDF：",[1804,4088,4090],{"className":4089},[2060],[1804,4091,4093,4152],{"className":4092},[1807],[1804,4094,4096],{"className":4095},[1811],[1813,4097,4098],{"xmlns":1815,"display":2069},[1817,4099,4100,4149],{},[1820,4101,4102,4114,4116,4118,4120,4123,4125,4127,4129,4131,4133,4135,4137,4139,4141,4143,4145,4147],{},[2075,4103,4104,4106],{},[1823,4105,2079],{},[1820,4107,4108,4110,4112],{},[1823,4109,1825],{},[1864,4111,1877],{"separator":1834},[1823,4113,1937],{},[1864,4115,2084],{"stretchy":1866},[1823,4117,2087],{},[1864,4119,1877],{"separator":1834},[1823,4121,4122],{},"y",[1864,4124,2090],{"stretchy":1866},[1864,4126,1940],{},[1823,4128,2095],{},[1864,4130,2084],{"stretchy":1866},[1823,4132,1825],{},[1864,4134,2102],{},[1823,4136,2087],{},[1864,4138,1877],{"separator":1834},[1823,4140,1937],{},[1864,4142,2102],{},[1823,4144,4122],{},[1864,4146,2090],{"stretchy":1866},[1823,4148,2110],{"mathvariant":2109},[1827,4150,4151],{"encoding":1829},"F_{X,Y}(x,y)=P(X\\le x,Y\\le y).",[1804,4153,4155,4239,4263,4291],{"className":4154,"ariaHidden":1834},[1833],[1804,4156,4158,4162,4212,4215,4218,4221,4224,4227,4230,4233,4236],{"className":4157},[1838],[1804,4159],{"className":4160,"style":4161},[1842],"height:1.0361em;vertical-align:-0.2861em;",[1804,4163,4165,4168],{"className":4164},[1847],[1804,4166,2079],{"className":4167,"style":2129},[1847,1848],[1804,4169,4171],{"className":4170},[1992],[1804,4172,4174,4203],{"className":4173},[1996,2136],[1804,4175,4177,4200],{"className":4176},[2000],[1804,4178,4180],{"className":4179,"style":2143},[2004],[1804,4181,4182,4185],{"style":2146},[1804,4183],{"className":4184,"style":2012},[2011],[1804,4186,4188],{"className":4187},[2016,2017,2018,2019],[1804,4189,4191,4194,4197],{"className":4190},[1847,2019],[1804,4192,1825],{"className":4193,"style":1849},[1847,1848,2019],[1804,4195,1877],{"className":4196},[1909,2019],[1804,4198,1937],{"className":4199,"style":1964},[1847,1848,2019],[1804,4201,2160],{"className":4202},[2159],[1804,4204,4206],{"className":4205},[2000],[1804,4207,4210],{"className":4208,"style":4209},[2004],"height:0.2861em;",[1804,4211],{},[1804,4213,2084],{"className":4214},[1899],[1804,4216,2087],{"className":4217},[1847,1848],[1804,4219,1877],{"className":4220},[1909],[1804,4222],{"className":4223,"style":1914},[1913],[1804,4225,4122],{"className":4226,"style":3377},[1847,1848],[1804,4228,2090],{"className":4229},[1921],[1804,4231],{"className":4232,"style":1968},[1913],[1804,4234,1940],{"className":4235},[1972],[1804,4237],{"className":4238,"style":1968},[1913],[1804,4240,4242,4245,4248,4251,4254,4257,4260],{"className":4241},[1838],[1804,4243],{"className":4244,"style":1895},[1842],[1804,4246,2095],{"className":4247,"style":2129},[1847,1848],[1804,4249,2084],{"className":4250},[1899],[1804,4252,1825],{"className":4253,"style":1849},[1847,1848],[1804,4255],{"className":4256,"style":1968},[1913],[1804,4258,2102],{"className":4259},[1972],[1804,4261],{"className":4262,"style":1968},[1913],[1804,4264,4266,4270,4273,4276,4279,4282,4285,4288],{"className":4265},[1838],[1804,4267],{"className":4268,"style":4269},[1842],"height:0.8778em;vertical-align:-0.1944em;",[1804,4271,2087],{"className":4272},[1847,1848],[1804,4274,1877],{"className":4275},[1909],[1804,4277],{"className":4278,"style":1914},[1913],[1804,4280,1937],{"className":4281,"style":1964},[1847,1848],[1804,4283],{"className":4284,"style":1968},[1913],[1804,4286,2102],{"className":4287},[1972],[1804,4289],{"className":4290,"style":1968},[1913],[1804,4292,4294,4297,4300,4303],{"className":4293},[1838],[1804,4295],{"className":4296,"style":1895},[1842],[1804,4298,4122],{"className":4299,"style":3377},[1847,1848],[1804,4301,2090],{"className":4302},[1921],[1804,4304,2110],{"className":4305},[1847],[1796,4307,4308],{},"连续情形中，从联合密度积分得到边际：",[1804,4310,4312],{"className":4311},[2060],[1804,4313,4315,4373],{"className":4314},[1807],[1804,4316,4318],{"className":4317},[1811],[1813,4319,4320],{"xmlns":1815,"display":2069},[1817,4321,4322,4370],{},[1820,4323,4324,4330,4332,4334,4336,4338,4340,4352,4354,4356,4358,4360,4362,4364,4366,4368],{},[2075,4325,4326,4328],{},[1823,4327,2820],{},[1823,4329,1825],{},[1864,4331,2084],{"stretchy":1866},[1823,4333,2087],{},[1864,4335,2090],{"stretchy":1866},[1864,4337,1940],{},[1864,4339,2807],{},[2075,4341,4342,4344],{},[1823,4343,2820],{},[1820,4345,4346,4348,4350],{},[1823,4347,1825],{},[1864,4349,1877],{"separator":1834},[1823,4351,1937],{},[1864,4353,2084],{"stretchy":1866},[1823,4355,2087],{},[1864,4357,1877],{"separator":1834},[1823,4359,4122],{},[1864,4361,2090],{"stretchy":1866},[2276,4363,2832],{},[1823,4365,2835],{},[1823,4367,4122],{},[1823,4369,2110],{"mathvariant":2109},[1827,4371,4372],{"encoding":1829},"f_X(x)=\\int f_{X,Y}(x,y)\\,dy.",[1804,4374,4376,4440],{"className":4375,"ariaHidden":1834},[1833],[1804,4377,4379,4382,4422,4425,4428,4431,4434,4437],{"className":4378},[1838],[1804,4380],{"className":4381,"style":1895},[1842],[1804,4383,4385,4388],{"className":4384},[1847],[1804,4386,2820],{"className":4387,"style":3039},[1847,1848],[1804,4389,4391],{"className":4390},[1992],[1804,4392,4394,4414],{"className":4393},[1996,2136],[1804,4395,4397,4411],{"className":4396},[2000],[1804,4398,4400],{"className":4399,"style":2143},[2004],[1804,4401,4402,4405],{"style":3054},[1804,4403],{"className":4404,"style":2012},[2011],[1804,4406,4408],{"className":4407},[2016,2017,2018,2019],[1804,4409,1825],{"className":4410,"style":1849},[1847,1848,2019],[1804,4412,2160],{"className":4413},[2159],[1804,4415,4417],{"className":4416},[2000],[1804,4418,4420],{"className":4419,"style":2167},[2004],[1804,4421],{},[1804,4423,2084],{"className":4424},[1899],[1804,4426,2087],{"className":4427},[1847,1848],[1804,4429,2090],{"className":4430},[1921],[1804,4432],{"className":4433,"style":1968},[1913],[1804,4435,1940],{"className":4436},[1972],[1804,4438],{"className":4439,"style":1968},[1913],[1804,4441,4443,4447,4450,4453,4502,4505,4508,4511,4514,4517,4520,4523,4526,4529],{"className":4442},[1838],[1804,4444],{"className":4445,"style":4446},[1842],"height:2.2222em;vertical-align:-0.8622em;",[1804,4448,2807],{"className":4449,"style":2974},[2650,2686,2687],[1804,4451],{"className":4452,"style":1914},[1913],[1804,4454,4456,4459],{"className":4455},[1847],[1804,4457,2820],{"className":4458,"style":3039},[1847,1848],[1804,4460,4462],{"className":4461},[1992],[1804,4463,4465,4494],{"className":4464},[1996,2136],[1804,4466,4468,4491],{"className":4467},[2000],[1804,4469,4471],{"className":4470,"style":2143},[2004],[1804,4472,4473,4476],{"style":3054},[1804,4474],{"className":4475,"style":2012},[2011],[1804,4477,4479],{"className":4478},[2016,2017,2018,2019],[1804,4480,4482,4485,4488],{"className":4481},[1847,2019],[1804,4483,1825],{"className":4484,"style":1849},[1847,1848,2019],[1804,4486,1877],{"className":4487},[1909,2019],[1804,4489,1937],{"className":4490,"style":1964},[1847,1848,2019],[1804,4492,2160],{"className":4493},[2159],[1804,4495,4497],{"className":4496},[2000],[1804,4498,4500],{"className":4499,"style":4209},[2004],[1804,4501],{},[1804,4503,2084],{"className":4504},[1899],[1804,4506,2087],{"className":4507},[1847,1848],[1804,4509,1877],{"className":4510},[1909],[1804,4512],{"className":4513,"style":1914},[1913],[1804,4515,4122],{"className":4516,"style":3377},[1847,1848],[1804,4518,2090],{"className":4519},[1921],[1804,4521],{"className":4522,"style":1914},[1913],[1804,4524,2835],{"className":4525},[1847,1848],[1804,4527,4122],{"className":4528,"style":3377},[1847,1848],[1804,4530,2110],{"className":4531},[1847],[1796,4533,4534],{},"条件密度：",[1804,4536,4538],{"className":4537},[2060],[1804,4539,4541,4640],{"className":4540},[1807],[1804,4542,4544],{"className":4543},[1811],[1813,4545,4546],{"xmlns":1815,"display":2069},[1817,4547,4548,4637],{},[1820,4549,4550,4563,4565,4567,4569,4571,4573,4575,4616,4618,4620,4626,4628,4630,4632,4634],{},[2075,4551,4552,4554],{},[1823,4553,2820],{},[1820,4555,4556,4558,4561],{},[1823,4557,1937],{},[1864,4559,4560],{},"∣",[1823,4562,1825],{},[1864,4564,2084],{"stretchy":1866},[1823,4566,4122],{},[1864,4568,4560],{},[1823,4570,2087],{},[1864,4572,2090],{"stretchy":1866},[1864,4574,1940],{},[4576,4577,4578,4602],"mfrac",{},[1820,4579,4580,4592,4594,4596,4598,4600],{},[2075,4581,4582,4584],{},[1823,4583,2820],{},[1820,4585,4586,4588,4590],{},[1823,4587,1825],{},[1864,4589,1877],{"separator":1834},[1823,4591,1937],{},[1864,4593,2084],{"stretchy":1866},[1823,4595,2087],{},[1864,4597,1877],{"separator":1834},[1823,4599,4122],{},[1864,4601,2090],{"stretchy":1866},[1820,4603,4604,4610,4612,4614],{},[2075,4605,4606,4608],{},[1823,4607,2820],{},[1823,4609,1825],{},[1864,4611,2084],{"stretchy":1866},[1823,4613,2087],{},[1864,4615,2090],{"stretchy":1866},[1864,4617,1877],{"separator":1834},[1913,4619],{"width":2503},[2075,4621,4622,4624],{},[1823,4623,2820],{},[1823,4625,1825],{},[1864,4627,2084],{"stretchy":1866},[1823,4629,2087],{},[1864,4631,2090],{"stretchy":1866},[1864,4633,3723],{},[1872,4635,4636],{},"0.",[1827,4638,4639],{"encoding":1829},"f_{Y\\mid X}(y\\mid x)=\\frac{f_{X,Y}(x,y)}{f_X(x)},\n\\qquad 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两个原像，不能只取一个分支。",[2028,6206,6208],{"id":6207},"_5-可运行案例均匀变量平方后为何不均匀","5. 可运行案例：均匀变量平方后为何不均匀",[1796,6210,5317,6211,6298,6299,6374],{},[1804,6212,6214,6246],{"className":6213},[1807],[1804,6215,6217],{"className":6216},[1811],[1813,6218,6219],{"xmlns":1815},[1817,6220,6221,6243],{},[1820,6222,6223,6225,6228,6231,6233,6235,6237,6239,6241],{},[1823,6224,1825],{},[1864,6226,6227],{},"∼",[1823,6229,6230],{},"U",[1864,6232,1867],{"stretchy":1866},[1864,6234,1870],{},[1872,6236,1874],{},[1864,6238,1877],{"separator":1834},[1872,6240,1874],{},[1864,6242,1882],{"stretchy":1866},[1827,6244,6245],{"encoding":1829},"X\\sim U[-1,1]",[1804,6247,6249,6267],{"className":6248,"ariaHidden":1834},[1833],[1804,6250,6252,6255,6258,6261,6264],{"className":6251},[1838],[1804,6253],{"className":6254,"style":1843},[1842],[1804,6256,1825],{"className":6257,"style":1849},[1847,1848],[1804,6259],{"className":6260,"style":1968},[1913],[1804,6262,6227],{"className":6263},[1972],[1804,6265],{"className":6266,"style":1968},[1913],[1804,6268,6270,6273,6277,6280,6283,6286,6289,6292,6295],{"className":6269},[1838],[1804,6271],{"className":6272,"style":1895},[1842],[1804,6274,6230],{"className":6275,"style":6276},[1847,1848],"margin-right:0.109em;",[1804,6278,1867],{"className":6279},[1899],[1804,6281,1870],{"className":6282},[1847],[1804,6284,1874],{"className":6285},[1847],[1804,6287,1877],{"className":6288},[1909],[1804,6290],{"className":6291,"style":1914},[1913],[1804,6293,1874],{"className":6294},[1847],[1804,6296,1882],{"className":6297},[1921],"，则对 ",[1804,6300,6302,6324],{"className":6301},[1807],[1804,6303,6305],{"className":6304},[1811],[1813,6306,6307],{"xmlns":1815},[1817,6308,6309,6321],{},[1820,6310,6311,6313,6315,6317,6319],{},[1872,6312,2272],{},[1864,6314,2102],{},[1823,6316,4122],{},[1864,6318,2102],{},[1872,6320,1874],{},[1827,6322,6323],{"encoding":1829},"0\\le y\\le1",[1804,6325,6327,6346,6365],{"className":6326,"ariaHidden":1834},[1833],[1804,6328,6330,6334,6337,6340,6343],{"className":6329},[1838],[1804,6331],{"className":6332,"style":6333},[1842],"height:0.7804em;vertical-align:-0.136em;",[1804,6335,2272],{"className":6336},[1847],[1804,6338],{"className":6339,"style":1968},[1913],[1804,6341,2102],{"className":6342},[1972],[1804,6344],{"className":6345,"style":1968},[1913],[1804,6347,6349,6353,6356,6359,6362],{"className":6348},[1838],[1804,6350],{"className":6351,"style":6352},[1842],"height:0.8304em;vertical-align:-0.1944em;",[1804,6354,4122],{"className":6355,"style":3377},[1847,1848],[1804,6357],{"className":6358,"style":1968},[1913],[1804,6360,2102],{"className":6361},[1972],[1804,6363],{"className":6364,"style":1968},[1913],[1804,6366,6368,6371],{"className":6367},[1838],[1804,6369],{"className":6370,"style":2450},[1842],[1804,6372,1874],{"className":6373},[1847],"：",[1804,6376,6378],{"className":6377},[2060],[1804,6379,6381,6455],{"className":6380},[1807],[1804,6382,6384],{"className":6383},[1811],[1813,6385,6386],{"xmlns":1815,"display":2069},[1817,6387,6388,6452],{},[1820,6389,6390,6396,6398,6400,6402,6404,6406,6408,6414,6416,6418,6420,6422,6424,6426,6428,6432,6434,6436,6438,6442,6444,6446,6450],{},[2075,6391,6392,6394],{},[1823,6393,2079],{},[1823,6395,1937],{},[1864,6397,2084],{"stretchy":1866},[1823,6399,4122],{},[1864,6401,2090],{"stretchy":1866},[1864,6403,1940],{},[1823,6405,2095],{},[1864,6407,2084],{"stretchy":1866},[1942,6409,6410,6412],{},[1823,6411,1825],{},[1872,6413,1948],{},[1864,6415,2102],{},[1823,6417,4122],{},[1864,6419,2090],{"stretchy":1866},[1864,6421,1940],{},[1823,6423,2095],{},[1864,6425,2084],{"stretchy":1866},[1864,6427,1870],{},[6011,6429,6430],{},[1823,6431,4122],{},[1864,6433,2102],{},[1823,6435,1825],{},[1864,6437,2102],{},[6011,6439,6440],{},[1823,6441,4122],{},[1864,6443,2090],{"stretchy":1866},[1864,6445,1940],{},[6011,6447,6448],{},[1823,6449,4122],{},[1823,6451,2110],{"mathvariant":2109},[1827,6453,6454],{"encoding":1829},"F_Y(y)=P(X^2\\le y)=P(-\\sqrt y\\le X\\le\\sqrt y)=\\sqrt y.",[1804,6456,6458,6522,6573,6594,6666,6684,6746],{"className":6457,"ariaHidden":1834},[1833],[1804,6459,6461,6464,6504,6507,6510,6513,6516,6519],{"className":6460},[1838],[1804,6462],{"className":6463,"style":1895},[1842],[1804,6465,6467,6470],{"className":6466},[1847],[1804,6468,2079],{"className":6469,"style":2129},[1847,1848],[1804,6471,6473],{"className":6472},[1992],[1804,6474,6476,6496],{"className":6475},[1996,2136],[1804,6477,6479,6493],{"className":6478},[2000],[1804,6480,6482],{"className":6481,"style":2143},[2004],[1804,6483,6484,6487],{"style":2146},[1804,6485],{"className":6486,"style":2012},[2011],[1804,6488,6490],{"className":6489},[2016,2017,2018,2019],[1804,6491,1937],{"className":6492,"style":1964},[1847,1848,2019],[1804,6494,2160],{"className":6495},[2159],[1804,6497,6499],{"className":6498},[2000],[1804,6500,6502],{"className":6501,"style":2167},[2004],[1804,6503],{},[1804,6505,2084],{"className":6506},[1899],[1804,6508,4122],{"className":6509,"style":3377},[1847,1848],[1804,6511,2090],{"className":6512},[1921],[1804,6514],{"className":6515,"style":1968},[1913],[1804,6517,1940],{"className":6518},[1972],[1804,6520],{"className":6521,"style":1968},[1913],[1804,6523,6525,6529,6532,6535,6564,6567,6570],{"className":6524},[1838],[1804,6526],{"className":6527,"style":6528},[1842],"height:1.1141em;vertical-align:-0.25em;",[1804,6530,2095],{"className":6531,"style":2129},[1847,1848],[1804,6533,2084],{"className":6534},[1899],[1804,6536,6538,6541],{"className":6537},[1847],[1804,6539,1825],{"className":6540,"style":1849},[1847,1848],[1804,6542,6544],{"className":6543},[1992],[1804,6545,6547],{"className":6546},[1996],[1804,6548,6550],{"className":6549},[2000],[1804,6551,6553],{"className":6552,"style":5613},[2004],[1804,6554,6555,6558],{"style":5616},[1804,6556],{"className":6557,"style":2012},[2011],[1804,6559,6561],{"className":6560},[2016,2017,2018,2019],[1804,6562,1948],{"className":6563},[1847,2019],[1804,6565],{"className":6566,"style":1968},[1913],[1804,6568,2102],{"className":6569},[1972],[1804,6571],{"className":6572,"style":1968},[1913],[1804,6574,6576,6579,6582,6585,6588,6591],{"className":6575},[1838],[1804,6577],{"className":6578,"style":1895},[1842],[1804,6580,4122],{"className":6581,"style":3377},[1847,1848],[1804,6583,2090],{"className":6584},[1921],[1804,6586],{"className":6587,"style":1968},[1913],[1804,6589,1940],{"className":6590},[1972],[1804,6592],{"className":6593,"style":1968},[1913],[1804,6595,6597,6601,6604,6607,6610,6657,6660,6663],{"className":6596},[1838],[1804,6598],{"className":6599,"style":6600},[1842],"height:1.04em;vertical-align:-0.2881em;",[1804,6602,2095],{"className":6603,"style":2129},[1847,1848],[1804,6605,2084],{"className":6606},[1899],[1804,6608,1870],{"className":6609},[1847],[1804,6611,6613],{"className":6612},[1847,6050],[1804,6614,6616,6648],{"className":6615},[1996,2136],[1804,6617,6619,6645],{"className":6618},[2000],[1804,6620,6623,6632],{"className":6621,"style":6622},[2004],"height:0.7519em;",[1804,6624,6626,6629],{"className":6625,"style":6065},[6064],[1804,6627],{"className":6628,"style":4771},[2011],[1804,6630,4122],{"className":6631,"style":6072},[1847,1848],[1804,6633,6635,6638],{"style":6634},"top:-2.7119em;",[1804,6636],{"className":6637,"style":4771},[2011],[1804,6639,6641],{"className":6640,"style":6083},[6082],[5678,6642,6643],{"xmlns":5680,"width":6086,"height":6087,"viewBox":6088,"preserveAspectRatio":6089},[5685,6644],{"d":6092},[1804,6646,2160],{"className":6647},[2159],[1804,6649,6651],{"className":6650},[2000],[1804,6652,6655],{"className":6653,"style":6654},[2004],"height:0.2881em;",[1804,6656],{},[1804,6658],{"className":6659,"style":1968},[1913],[1804,6661,2102],{"className":6662},[1972],[1804,6664],{"className":6665,"style":1968},[1913],[1804,6667,6669,6672,6675,6678,6681],{"className":6668},[1838],[1804,6670],{"className":6671,"style":3126},[1842],[1804,6673,1825],{"className":6674,"style":1849},[1847,1848],[1804,6676],{"className":6677,"style":1968},[1913],[1804,6679,2102],{"className":6680},[1972],[1804,6682],{"className":6683,"style":1968},[1913],[1804,6685,6687,6690,6734,6737,6740,6743],{"className":6686},[1838],[1804,6688],{"className":6689,"style":6600},[1842],[1804,6691,6693],{"className":6692},[1847,6050],[1804,6694,6696,6726],{"className":6695},[1996,2136],[1804,6697,6699,6723],{"className":6698},[2000],[1804,6700,6702,6711],{"className":6701,"style":6622},[2004],[1804,6703,6705,6708],{"className":6704,"style":6065},[6064],[1804,6706],{"className":6707,"style":4771},[2011],[1804,6709,4122],{"className":6710,"style":6072},[1847,1848],[1804,6712,6713,6716],{"style":6634},[1804,6714],{"className":6715,"style":4771},[2011],[1804,6717,6719],{"className":6718,"style":6083},[6082],[5678,6720,6721],{"xmlns":5680,"width":6086,"height":6087,"viewBox":6088,"preserveAspectRatio":6089},[5685,6722],{"d":6092},[1804,6724,2160],{"className":6725},[2159],[1804,6727,6729],{"className":6728},[2000],[1804,6730,6732],{"className":6731,"style":6654},[2004],[1804,6733],{},[1804,6735,2090],{"className":6736},[1921],[1804,6738],{"className":6739,"style":1968},[1913],[1804,6741,1940],{"className":6742},[1972],[1804,6744],{"className":6745,"style":1968},[1913],[1804,6747,6749,6752,6796],{"className":6748},[1838],[1804,6750],{"className":6751,"style":6600},[1842],[1804,6753,6755],{"className":6754},[1847,6050],[1804,6756,6758,6788],{"className":6757},[1996,2136],[1804,6759,6761,6785],{"className":6760},[2000],[1804,6762,6764,6773],{"className":6763,"style":6622},[2004],[1804,6765,6767,6770],{"className":6766,"style":6065},[6064],[1804,6768],{"className":6769,"style":4771},[2011],[1804,6771,4122],{"className":6772,"style":6072},[1847,1848],[1804,6774,6775,6778],{"style":6634},[1804,6776],{"className":6777,"style":4771},[2011],[1804,6779,6781],{"className":6780,"style":6083},[6082],[5678,6782,6783],{"xmlns":5680,"width":6086,"height":6087,"viewBox":6088,"preserveAspectRatio":6089},[5685,6784],{"d":6092},[1804,6786,2160],{"className":6787},[2159],[1804,6789,6791],{"className":6790},[2000],[1804,6792,6794],{"className":6793,"style":6654},[2004],[1804,6795],{},[1804,6797,2110],{"className":6798},[1847],[1796,6800,6801,6802,6977],{},"因此 ",[1804,6803,6805,6846],{"className":6804},[1807],[1804,6806,6808],{"className":6807},[1811],[1813,6809,6810],{"xmlns":1815},[1817,6811,6812,6843],{},[1820,6813,6814,6820,6822,6824,6826,6828,6830,6833,6835,6837,6841],{},[2075,6815,6816,6818],{},[1823,6817,2820],{},[1823,6819,1937],{},[1864,6821,2084],{"stretchy":1866},[1823,6823,4122],{},[1864,6825,2090],{"stretchy":1866},[1864,6827,1940],{},[1872,6829,1874],{},[1823,6831,6832],{"mathvariant":2109},"\u002F",[1864,6834,2084],{"stretchy":1866},[1872,6836,1948],{},[6011,6838,6839],{},[1823,6840,4122],{},[1864,6842,2090],{"stretchy":1866},[1827,6844,6845],{"encoding":1829},"f_Y(y)=1\u002F(2\\sqrt y)",[1804,6847,6849,6913],{"className":6848,"ariaHidden":1834},[1833],[1804,6850,6852,6855,6895,6898,6901,6904,6907,6910],{"className":6851},[1838],[1804,6853],{"className":6854,"style":1895},[1842],[1804,6856,6858,6861],{"className":6857},[1847],[1804,6859,2820],{"className":6860,"style":3039},[1847,1848],[1804,6862,6864],{"className":6863},[1992],[1804,6865,6867,6887],{"className":6866},[1996,2136],[1804,6868,6870,6884],{"className":6869},[2000],[1804,6871,6873],{"className":6872,"style":2143},[2004],[1804,6874,6875,6878],{"style":3054},[1804,6876],{"className":6877,"style":2012},[2011],[1804,6879,6881],{"className":6880},[2016,2017,2018,2019],[1804,6882,1937],{"className":6883,"style":1964},[1847,1848,2019],[1804,6885,2160],{"className":6886},[2159],[1804,6888,6890],{"className":6889},[2000],[1804,6891,6893],{"className":6892,"style":2167},[2004],[1804,6894],{},[1804,6896,2084],{"className":6897},[1899],[1804,6899,4122],{"className":6900,"style":3377},[1847,1848],[1804,6902,2090],{"className":6903},[1921],[1804,6905],{"className":6906,"style":1968},[1913],[1804,6908,1940],{"className":6909},[1972],[1804,6911],{"className":6912,"style":1968},[1913],[1804,6914,6916,6920,6924,6927,6930,6974],{"className":6915},[1838],[1804,6917],{"className":6918,"style":6919},[1842],"height:1.0869em;vertical-align:-0.3369em;",[1804,6921,6923],{"className":6922},[1847],"1\u002F",[1804,6925,2084],{"className":6926},[1899],[1804,6928,1948],{"className":6929},[1847],[1804,6931,6933],{"className":6932},[1847,6050],[1804,6934,6936,6966],{"className":6935},[1996,2136],[1804,6937,6939,6963],{"className":6938},[2000],[1804,6940,6942,6951],{"className":6941,"style":6060},[2004],[1804,6943,6945,6948],{"className":6944,"style":6065},[6064],[1804,6946],{"className":6947,"style":4771},[2011],[1804,6949,4122],{"className":6950,"style":6072},[1847,1848],[1804,6952,6953,6956],{"style":6075},[1804,6954],{"className":6955,"style":4771},[2011],[1804,6957,6959],{"className":6958,"style":6083},[6082],[5678,6960,6961],{"xmlns":5680,"width":6086,"height":6087,"viewBox":6088,"preserveAspectRatio":6089},[5685,6962],{"d":6092},[1804,6964,2160],{"className":6965},[2159],[1804,6967,6969],{"className":6968},[2000],[1804,6970,6972],{"className":6971,"style":6102},[2004],[1804,6973],{},[1804,6975,2090],{"className":6976},[1921],"，概率更集中在 0 附近。",[6979,6980],"pyodide",{"code64":6981,"layout":6982,"locale":7,"packages":6983,"title":6984},"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","vertical","numpy","Python：多对一变量变换与 CDF",[1796,6986,6987,6988,7076],{},"直方图的高峰不是概率在 0 点堆积；",[1804,6989,6991,7019],{"className":6990},[1807],[1804,6992,6994],{"className":6993},[1811],[1813,6995,6996],{"xmlns":1815},[1817,6997,6998,7016],{},[1820,6999,7000,7002,7004,7006,7008,7010,7012,7014],{},[1823,7001,2095],{},[1864,7003,2084],{"stretchy":1866},[1823,7005,1937],{},[1864,7007,1940],{},[1872,7009,2272],{},[1864,7011,2090],{"stretchy":1866},[1864,7013,1940],{},[1872,7015,2272],{},[1827,7017,7018],{"encoding":1829},"P(Y=0)=0",[1804,7020,7022,7046,7067],{"className":7021,"ariaHidden":1834},[1833],[1804,7023,7025,7028,7031,7034,7037,7040,7043],{"className":7024},[1838],[1804,7026],{"className":7027,"style":1895},[1842],[1804,7029,2095],{"className":7030,"style":2129},[1847,1848],[1804,7032,2084],{"className":7033},[1899],[1804,7035,1937],{"className":7036,"style":1964},[1847,1848],[1804,7038],{"className":7039,"style":1968},[1913],[1804,7041,1940],{"className":7042},[1972],[1804,7044],{"className":7045,"style":1968},[1913],[1804,7047,7049,7052,7055,7058,7061,7064],{"className":7048},[1838],[1804,7050],{"className":7051,"style":1895},[1842],[1804,7053,2272],{"className":7054},[1847],[1804,7056,2090],{"className":7057},[1921],[1804,7059],{"className":7060,"style":1968},[1913],[1804,7062,1940],{"className":7063},[1972],[1804,7065],{"className":7066,"style":1968},[1913],[1804,7068,7070,7073],{"className":7069},[1838],[1804,7071],{"className":7072,"style":2450},[1842],[1804,7074,2272],{"className":7075},[1847],"。它表示靠近 0 的短区间拥有较高概率密度。",[2028,7078,7080],{"id":7079},"_6-变换题的稳定步骤","6. 变换题的稳定步骤",[2035,7082,7083,7115,7185,7217,7220,7223],{},[2038,7084,7085,7086,7114],{},"写出 ",[1804,7087,7089,7102],{"className":7088},[1807],[1804,7090,7092],{"className":7091},[1811],[1813,7093,7094],{"xmlns":1815},[1817,7095,7096,7100],{},[1820,7097,7098],{},[1823,7099,1937],{},[1827,7101,1937],{"encoding":1829},[1804,7103,7105],{"className":7104,"ariaHidden":1834},[1833],[1804,7106,7108,7111],{"className":7107},[1838],[1804,7109],{"className":7110,"style":1843},[1842],[1804,7112,1937],{"className":7113,"style":1964},[1847,1848]," 的支持集；",[2038,7116,7117,7118,7184],{},"用事件 ",[1804,7119,7121,7145],{"className":7120},[1807],[1804,7122,7124],{"className":7123},[1811],[1813,7125,7126],{"xmlns":1815},[1817,7127,7128,7142],{},[1820,7129,7130,7132,7134,7136,7138,7140],{},[1823,7131,5336],{},[1864,7133,2084],{"stretchy":1866},[1823,7135,1825],{},[1864,7137,2090],{"stretchy":1866},[1864,7139,2102],{},[1823,7141,4122],{},[1827,7143,7144],{"encoding":1829},"g(X)\\le y",[1804,7146,7148,7175],{"className":7147,"ariaHidden":1834},[1833],[1804,7149,7151,7154,7157,7160,7163,7166,7169,7172],{"className":7150},[1838],[1804,7152],{"className":7153,"style":1895},[1842],[1804,7155,5336],{"className":7156,"style":3377},[1847,1848],[1804,7158,2084],{"className":7159},[1899],[1804,7161,1825],{"className":7162,"style":1849},[1847,1848],[1804,7164,2090],{"className":7165},[1921],[1804,7167],{"className":7168,"style":1968},[1913],[1804,7170,2102],{"className":7171},[1972],[1804,7173],{"className":7174,"style":1968},[1913],[1804,7176,7178,7181],{"className":7177},[1838],[1804,7179],{"className":7180,"style":3373},[1842],[1804,7182,4122],{"className":7183,"style":3377},[1847,1848]," 求 CDF；",[2038,7186,7187,7188,7216],{},"分段处理不同 ",[1804,7189,7191,7204],{"className":7190},[1807],[1804,7192,7194],{"className":7193},[1811],[1813,7195,7196],{"xmlns":1815},[1817,7197,7198,7202],{},[1820,7199,7200],{},[1823,7201,4122],{},[1827,7203,4122],{"encoding":1829},[1804,7205,7207],{"className":7206,"ariaHidden":1834},[1833],[1804,7208,7210,7213],{"className":7209},[1838],[1804,7211],{"className":7212,"style":3373},[1842],[1804,7214,4122],{"className":7215,"style":3377},[1847,1848],"；",[2038,7218,7219],{},"若可微，再求导得到 PDF；",[2038,7221,7222],{},"检查非负与积分为 1；",[2038,7224,7225],{},"用模拟检查分位数或矩。",[1796,7227,7228],{},"CDF 法对多对一、非单调和离散—连续混合变换最稳健。",[2028,7230,7232],{"id":7231},"_7-常见误区","7. 常见误区",[2231,7234,7235,7238,7241,7244,7247,7250],{},[2038,7236,7237],{},"把 PDF 的高度当作单点概率；",[2038,7239,7240],{},"忘记支持集，导致密度积分不为 1；",[2038,7242,7243],{},"变量变换漏掉雅可比绝对值；",[2038,7245,7246],{},"多对一映射只保留一个根；",[2038,7248,7249],{},"用零相关推断独立；",[2038,7251,7252,7253,3946,7282,7312],{},"对离散变量把 ",[1804,7254,7256,7269],{"className":7255},[1807],[1804,7257,7259],{"className":7258},[1811],[1813,7260,7261],{"xmlns":1815},[1817,7262,7263,7267],{},[1820,7264,7265],{},[1864,7266,3723],{},[1827,7268,3723],{"encoding":1829},[1804,7270,7272],{"className":7271,"ariaHidden":1834},[1833],[1804,7273,7275,7279],{"className":7274},[1838],[1804,7276],{"className":7277,"style":7278},[1842],"height:0.5782em;vertical-align:-0.0391em;",[1804,7280,3723],{"className":7281},[1972],[1804,7283,7285,7299],{"className":7284},[1807],[1804,7286,7288],{"className":7287},[1811],[1813,7289,7290],{"xmlns":1815},[1817,7291,7292,7296],{},[1820,7293,7294],{},[1864,7295,3433],{},[1827,7297,7298],{"encoding":1829},"\\ge",[1804,7300,7302],{"className":7301,"ariaHidden":1834},[1833],[1804,7303,7305,7309],{"className":7304},[1838],[1804,7306],{"className":7307,"style":7308},[1842],"height:0.7719em;vertical-align:-0.136em;",[1804,7310,3433],{"className":7311},[1972]," 混用。",[2028,7314,7315],{"id":7315},"课堂任务",[1796,7317,7318,7319,7398],{},"令 ",[1804,7320,7322,7350],{"className":7321},[1807],[1804,7323,7325],{"className":7324},[1811],[1813,7326,7327],{"xmlns":1815},[1817,7328,7329,7347],{},[1820,7330,7331,7333,7335,7337,7339,7341,7343,7345],{},[1823,7332,1825],{},[1864,7334,6227],{},[1823,7336,6230],{},[1864,7338,2084],{"stretchy":1866},[1872,7340,2272],{},[1864,7342,1877],{"separator":1834},[1872,7344,1874],{},[1864,7346,2090],{"stretchy":1866},[1827,7348,7349],{"encoding":1829},"X\\sim U(0,1)",[1804,7351,7353,7371],{"className":7352,"ariaHidden":1834},[1833],[1804,7354,7356,7359,7362,7365,7368],{"className":7355},[1838],[1804,7357],{"className":7358,"style":1843},[1842],[1804,7360,1825],{"className":7361,"style":1849},[1847,1848],[1804,7363],{"className":7364,"style":1968},[1913],[1804,7366,6227],{"className":7367},[1972],[1804,7369],{"className":7370,"style":1968},[1913],[1804,7372,7374,7377,7380,7383,7386,7389,7392,7395],{"className":7373},[1838],[1804,7375],{"className":7376,"style":1895},[1842],[1804,7378,6230],{"className":7379,"style":6276},[1847,1848],[1804,7381,2084],{"className":7382},[1899],[1804,7384,2272],{"className":7385},[1847],[1804,7387,1877],{"className":7388},[1909],[1804,7390],{"className":7391,"style":1914},[1913],[1804,7393,1874],{"className":7394},[1847],[1804,7396,2090],{"className":7397},[1921],"，分别研究：",[2035,7400,7401,7479,7587,7619],{},[2038,7402,7403,7478],{},[1804,7404,7406,7431],{"className":7405},[1807],[1804,7407,7409],{"className":7408},[1811],[1813,7410,7411],{"xmlns":1815},[1817,7412,7413,7428],{},[1820,7414,7415,7417,7419,7421,7424,7426],{},[1823,7416,1937],{},[1864,7418,1940],{},[1864,7420,1870],{},[1823,7422,7423],{},"log",[1864,7425,3410],{},[1823,7427,1825],{},[1827,7429,7430],{"encoding":1829},"Y=-\\log X",[1804,7432,7434,7452],{"className":7433,"ariaHidden":1834},[1833],[1804,7435,7437,7440,7443,7446,7449],{"className":7436},[1838],[1804,7438],{"className":7439,"style":1843},[1842],[1804,7441,1937],{"className":7442,"style":1964},[1847,1848],[1804,7444],{"className":7445,"style":1968},[1913],[1804,7447,1940],{"className":7448},[1972],[1804,7450],{"className":7451,"style":1968},[1913],[1804,7453,7455,7459,7462,7465,7472,7475],{"className":7454},[1838],[1804,7456],{"className":7457,"style":7458},[1842],"height:0.8889em;vertical-align:-0.1944em;",[1804,7460,1870],{"className":7461},[1847],[1804,7463],{"className":7464,"style":1914},[1913],[1804,7466,7468,7469],{"className":7467},[2650],"lo",[1804,7470,5336],{"style":7471},"margin-right:0.0139em;",[1804,7473],{"className":7474,"style":1914},[1913],[1804,7476,1825],{"className":7477,"style":1849},[1847,1848]," 的 CDF 与 PDF；",[2038,7480,7481,7114],{},[1804,7482,7484,7520],{"className":7483},[1807],[1804,7485,7487],{"className":7486},[1811],[1813,7488,7489],{"xmlns":1815},[1817,7490,7491,7517],{},[1820,7492,7493,7496,7498,7501,7503,7505,7507,7509,7511,7513,7515],{},[1823,7494,7495],{},"Z",[1864,7497,1940],{},[1823,7499,7500],{},"min",[1864,7502,3410],{},[1864,7504,2084],{"stretchy":1866},[1823,7506,1825],{},[1864,7508,1877],{"separator":1834},[1872,7510,1874],{},[1864,7512,1870],{},[1823,7514,1825],{},[1864,7516,2090],{"stretchy":1866},[1827,7518,7519],{"encoding":1829},"Z=\\min(X,1-X)",[1804,7521,7523,7542,7575],{"className":7522,"ariaHidden":1834},[1833],[1804,7524,7526,7529,7533,7536,7539],{"className":7525},[1838],[1804,7527],{"className":7528,"style":1843},[1842],[1804,7530,7495],{"className":7531,"style":7532},[1847,1848],"margin-right:0.0715em;",[1804,7534],{"className":7535,"style":1968},[1913],[1804,7537,1940],{"className":7538},[1972],[1804,7540],{"className":7541,"style":1968},[1913],[1804,7543,7545,7548,7551,7554,7557,7560,7563,7566,7569,7572],{"className":7544},[1838],[1804,7546],{"className":7547,"style":1895},[1842],[1804,7549,7500],{"className":7550},[2650],[1804,7552,2084],{"className":7553},[1899],[1804,7555,1825],{"className":7556,"style":1849},[1847,1848],[1804,7558,1877],{"className":7559},[1909],[1804,7561],{"className":7562,"style":1914},[1913],[1804,7564,1874],{"className":7565},[1847],[1804,7567],{"className":7568,"style":1964},[1913],[1804,7570,1870],{"className":7571},[3815],[1804,7573],{"className":7574,"style":1964},[1913],[1804,7576,7578,7581,7584],{"className":7577},[1838],[1804,7579],{"className":7580,"style":1895},[1842],[1804,7582,1825],{"className":7583,"style":1849},[1847,1848],[1804,7585,2090],{"className":7586},[1921],[2038,7588,7589,7590,7618],{},"用模拟检查 ",[1804,7591,7593,7606],{"className":7592},[1807],[1804,7594,7596],{"className":7595},[1811],[1813,7597,7598],{"xmlns":1815},[1817,7599,7600,7604],{},[1820,7601,7602],{},[1823,7603,1937],{},[1827,7605,1937],{"encoding":1829},[1804,7607,7609],{"className":7608,"ariaHidden":1834},[1833],[1804,7610,7612,7615],{"className":7611},[1838],[1804,7613],{"className":7614,"style":1843},[1842],[1804,7616,1937],{"className":7617,"style":1964},[1847,1848]," 是否服从指数分布；",[2038,7620,7621,7622,3946,7650,7709],{},"解释 ",[1804,7623,7625,7638],{"className":7624},[1807],[1804,7626,7628],{"className":7627},[1811],[1813,7629,7630],{"xmlns":1815},[1817,7631,7632,7636],{},[1820,7633,7634],{},[1823,7635,1825],{},[1827,7637,1825],{"encoding":1829},[1804,7639,7641],{"className":7640,"ariaHidden":1834},[1833],[1804,7642,7644,7647],{"className":7643},[1838],[1804,7645],{"className":7646,"style":1843},[1842],[1804,7648,1825],{"className":7649,"style":1849},[1847,1848],[1804,7651,7653,7671],{"className":7652},[1807],[1804,7654,7656],{"className":7655},[1811],[1813,7657,7658],{"xmlns":1815},[1817,7659,7660,7668],{},[1820,7661,7662],{},[1942,7663,7664,7666],{},[1823,7665,1825],{},[1872,7667,1948],{},[1827,7669,7670],{"encoding":1829},"X^2",[1804,7672,7674],{"className":7673,"ariaHidden":1834},[1833],[1804,7675,7677,7680],{"className":7676},[1838],[1804,7678],{"className":7679,"style":1982},[1842],[1804,7681,7683,7686],{"className":7682},[1847],[1804,7684,1825],{"className":7685,"style":1849},[1847,1848],[1804,7687,7689],{"className":7688},[1992],[1804,7690,7692],{"className":7691},[1996],[1804,7693,7695],{"className":7694},[2000],[1804,7696,7698],{"className":7697,"style":1982},[2004],[1804,7699,7700,7703],{"style":2007},[1804,7701],{"className":7702,"style":2012},[2011],[1804,7704,7706],{"className":7705},[2016,2017,2018,2019],[1804,7707,1948],{"className":7708},[1847,2019]," 为何显然不独立。",[2028,7711,7712],{"id":7712},"核心阅读",[2231,7714,7715,7723,7730],{},[2038,7716,7717,7718,7722],{},"Casella & Berger, ",[7719,7720,7721],"em",{},"Statistical Inference","，随机变量与变换章节。",[2038,7724,7725,7726,7729],{},"Ross, ",[7719,7727,7728],{},"A First Course in Probability","，随机变量章节。",[2038,7731,7732,7733,2454],{},"Blitzstein & Hwang, ",[2848,7734,7738],{"href":7735,"rel":7736},"https:\u002F\u002Fprojects.iq.harvard.edu\u002Fstat110\u002Fyoutube",[7737],"nofollow","Stat 110 Notes",[1796,7740,7741,7742,7746,7747,2454],{},"上一章：",[2848,7743,7745],{"href":7744},"..\u002F01-prob-theory\u002F","概率论基础","｜下一章：",[2848,7748,7750],{"href":7749},"..\u002F03-moment\u002F","数字特征与矩",{"title":10,"searchDepth":7752,"depth":7752,"links":7753},2,[7754,7755,7756,7757,7758,7759,7760,7761,7762,7763],{"id":2030,"depth":7752,"text":2030},{"id":2055,"depth":7752,"text":2056},{"id":3342,"depth":7752,"text":3343},{"id":4082,"depth":7752,"text":4083},{"id":5313,"depth":7752,"text":5314},{"id":6207,"depth":7752,"text":6208},{"id":7079,"depth":7752,"text":7080},{"id":7231,"depth":7752,"text":7232},{"id":7315,"depth":7752,"text":7315},{"id":7712,"depth":7752,"text":7712},"统一理解 PMF、PDF、CDF、分位数、联合分布、条件分布与变量变换。","md",{"sidebar":7767},{"order":7768},3,true,{"title":1657,"description":7764},"ztA-A03n1fp_Plb-MJZ7YNkGRbEXUXuAlLquHD_5KLk",[7773,7775],{"title":1651,"path":1652,"stem":1653,"description":7774,"children":-1},"从样本空间与概率公理进入条件概率、独立性、全概率和贝叶斯公式。",{"title":1663,"path":1664,"stem":1665,"description":7776,"children":-1},"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。",1785754756393]