Theoretical / proof-based
Mathematical statistics · Graduate · Math
Learning objectives from the Mathematical statistics syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Mathematical statistics — outline derived from course README sections and typical US statistics syllabi (OpenIntro / standard OER where applicable).
Probability foundations
- Probability spaces and sigma-algebras — Probability spaces and sigma-algebras
- Measures — Measures
- Random variables — Random variables
- Induced measures — Induced measures
- Expectation via Lebesgue integral — Expectation via Lebesgue integral
- Independence and product measures — Independence and product measures
- Convergence modes: a.s., in probability, Lp, in distribution — Convergence modes: a.s., in probability, Lp, in distribution
Distribution and limit theory
- Characteristic functions — Characteristic functions
- Law of large numbers — Law of large numbers
- Central limit theorems — Central limit theorems
- Multivariate normal — Multivariate normal
- Quadratic forms — Quadratic forms
- Sufficient and complete statistics — Sufficient and complete statistics
- Ancillary statistics — Ancillary statistics
- Exponential families — Exponential families
Statistical inference theory
- Point estimation: UMVU and MLE — Point estimation: UMVU and MLE
- Efficiency — Efficiency
- Hypothesis testing: Neyman–Pearson — Hypothesis testing: Neyman–Pearson
- UMP tests — UMP tests
- Confidence sets — Confidence sets
- Duality with testing — Duality with testing
- Asymptotic theory: delta method — Asymptotic theory: delta method
- Fisher information — Fisher information
- Decision theory — Decision theory
- Admissibility (introduction) — Admissibility (introduction)
Learning objectives
Click an objective for study materials.
Probability foundations
- Probability spaces and sigma-algebras — Probability spaces and sigma-algebras
- Measures — Measures
- Random variables — Random variables
- Induced measures — Induced measures
- Expectation via Lebesgue integral — Expectation via Lebesgue integral
- Independence and product measures — Independence and product measures
- Convergence modes: a.s., in probability, Lp, in distribution — Convergence modes: a.s., in probability, Lp, in distribution
Distribution and limit theory
- Characteristic functions — Characteristic functions
- Law of large numbers — Law of large numbers
- Central limit theorems — Central limit theorems
- Multivariate normal — Multivariate normal
- Quadratic forms — Quadratic forms
- Sufficient and complete statistics — Sufficient and complete statistics
- Ancillary statistics — Ancillary statistics
- Exponential families — Exponential families
Statistical inference theory
- Point estimation: UMVU and MLE — Point estimation: UMVU and MLE
- Efficiency — Efficiency
- Hypothesis testing: Neyman–Pearson — Hypothesis testing: Neyman–Pearson
- UMP tests — UMP tests
- Confidence sets — Confidence sets
- Duality with testing — Duality with testing
- Asymptotic theory: delta method — Asymptotic theory: delta method
- Fisher information — Fisher information
Definitions and structure
- Probability spaces and sigma-algebras — Probability spaces and sigma-algebras
- Measures — Measures
- Random variables — Random variables
- Induced measures — Induced measures
- Expectation via Lebesgue integral — Expectation via Lebesgue integral
Proofs and reasoning
- Independence and product measures — Independence and product measures
- Convergence modes: a.s., in probability, Lp, in distribution — Convergence modes: a.s., in probability, Lp, in distribution
- Characteristic functions — Characteristic functions
- Law of large numbers — Law of large numbers
- Central limit theorems — Central limit theorems
Abstraction and generalization
- Multivariate normal — Multivariate normal
- Quadratic forms — Quadratic forms
- Sufficient and complete statistics — Sufficient and complete statistics
- Ancillary statistics — Ancillary statistics
- Exponential families — Exponential families
Modeling and computation
- Apply probability spaces and sigma-algebras in engineering contexts — Apply probability spaces and sigma-algebras in engineering contexts
- Apply measures in engineering contexts — Apply measures in engineering contexts
- Apply random variables in engineering contexts — Apply random variables in engineering contexts
- Apply induced measures in engineering contexts — Apply induced measures in engineering contexts
- Apply expectation via lebesgue integral in engineering contexts — Apply expectation via lebesgue integral in engineering contexts
Data and technology
- Use software to explore mathematical statistics problems numerically — Use software to explore mathematical statistics problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Independence and product measures — Independence and product measures
- Convergence modes: a.s., in probability, Lp, in distribution — Convergence modes: a.s., in probability, Lp, in distribution
- Characteristic functions — Characteristic functions
- Law of large numbers — Law of large numbers
- Central limit theorems — Central limit theorems
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Probability foundations
Probability spaces and sigma-algebras
Coming soon - Distribution and limit theory
Characteristic functions
Coming soon - Statistical inference theory
Point estimation: UMVU and MLE
Coming soon
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