STAT 350: Introduction to Statistics

Course Content

  • 1. Part I: Foundations of Probability and Computation
    • 1.1. Chapter 1: Statistical Paradigms and Core Concepts
      • 1.1.1. Paradigms of Probability and Statistical Inference
        • The Mathematical Foundation: Kolmogorov’s Axioms
        • Interpretations of Probability
        • Statistical Inference Paradigms
        • Historical and Philosophical Debates
        • Bringing It All Together
        • Looking Ahead: Our Course Focus
        • References and Further Reading
      • 1.1.2. Probability Distributions: Theory and Computation
        • From Abstract Foundations to Concrete Tools
        • The Python Ecosystem for Probability
        • Introduction: Why Probability Distributions Matter
        • The Python Ecosystem for Probability
        • Discrete Distributions
        • Continuous Distributions
        • Additional Important Distributions
        • Summary and Practical Guidelines
        • Conclusion
      • 1.1.3. Python Random Generation
        • From Mathematical Distributions to Computational Samples
        • The Python Ecosystem at a Glance
        • Understanding Pseudo-Random Number Generation
        • The Standard Library: random Module
        • NumPy: Fast Vectorized Random Sampling
        • SciPy Stats: The Complete Statistical Toolkit
        • Bringing It All Together: Library Selection Guide
        • Looking Ahead: From Random Numbers to Monte Carlo Methods
  • Part II: Simulation-Based Methods
    • 1. Chapter 2: Monte Carlo Simulation
      • 1.1. Monte Carlo Fundamentals
        • 1.1.1. The Historical Development of Monte Carlo Methods
        • 1.1.2. The Core Principle: Expectation as Integration
        • 1.1.3. Theoretical Foundations
        • 1.1.4. Variance Estimation and Confidence Intervals
        • 1.1.5. Worked Examples
        • 1.1.6. Comparison with Deterministic Methods
        • 1.1.7. Sample Size Determination
        • 1.1.8. Convergence Diagnostics and Monitoring
        • 1.1.9. Practical Considerations
        • 1.1.10. Bringing It All Together
        • 1.1.11. Transition to What Follows
      • 1.2. Uniform Random Variates
        • 1.2.1. The Probability Integral Transform
        • 1.2.2. The Paradox of Computational Randomness
        • 1.2.3. Chaotic Dynamical Systems: An Instructive Failure
        • 1.2.4. Linear Congruential Generators
        • 1.2.5. Shift-Register Generators
        • 1.2.6. The KISS Generator: Combining Strategies
        • 1.2.7. Modern Generators: Mersenne Twister and PCG
        • 1.2.8. Statistical Testing of Random Number Generators
        • 1.2.9. Practical Considerations
        • 1.2.10. Bringing It All Together
        • 1.2.11. Transition to What Follows
      • 1.3. Inverse CDF Method
        • 1.3.1. Mathematical Foundations
        • 1.3.2. Continuous Distributions with Closed-Form Inverses
        • 1.3.3. Numerical Inversion
        • 1.3.4. Discrete Distributions
        • 1.3.5. Mixed Distributions
        • 1.3.6. Practical Considerations
        • 1.3.7. Bringing It All Together
        • 1.3.8. Transition to What Follows
    • 2. Chapter 3: Frequentist Statistical Inference
      • 2.1. Sampling Variability
        • 2.1.1. Introduction
        • 2.1.2. Key Concepts
        • 2.1.3. Mathematical Framework
        • 2.1.4. Python Implementation
        • 2.1.5. Examples
        • 2.1.6. Summary
      • 2.2. Statistical Estimators
        • 2.2.1. Introduction
        • 2.2.2. Key Concepts
        • 2.2.3. Mathematical Framework
        • 2.2.4. Python Implementation
        • 2.2.5. Examples
        • 2.2.6. Summary
      • 2.3. Plugin Methods
        • 2.3.1. Introduction
        • 2.3.2. Key Concepts
        • 2.3.3. Mathematical Framework
        • 2.3.4. Python Implementation
        • 2.3.5. Examples
        • 2.3.6. Summary
      • 2.4. Parametric Inference
        • 2.4.1. Introduction
        • 2.4.2. Key Concepts
        • 2.4.3. Mathematical Framework
        • 2.4.4. Python Implementation
        • 2.4.5. Examples
        • 2.4.6. Summary
      • 2.5. Exponential Families
        • 2.5.1. Introduction
        • 2.5.2. Key Concepts
        • 2.5.3. Mathematical Framework
        • 2.5.4. Python Implementation
        • 2.5.5. Examples
        • 2.5.6. Summary
      • 2.6. Maximum Likelihood
        • 2.6.1. Introduction
        • 2.6.2. Key Concepts
        • 2.6.3. Mathematical Framework
        • 2.6.4. Python Implementation
        • 2.6.5. Examples
        • 2.6.6. Summary
      • 2.7. Linear Models
        • 2.7.1. Introduction
        • 2.7.2. Key Concepts
        • 2.7.3. Mathematical Framework
        • 2.7.4. Python Implementation
        • 2.7.5. Examples
        • 2.7.6. Summary
      • 2.8. Generalized Linear Models
        • 2.8.1. Introduction
        • 2.8.2. Key Concepts
        • 2.8.3. Mathematical Framework
        • 2.8.4. Python Implementation
        • 2.8.5. Examples
        • 2.8.6. Summary
    • 3. Chapter 4: Resampling Methods
      • 3.1. Jackknife Introduction
        • 3.1.1. Introduction
        • 3.1.2. Key Concepts
        • 3.1.3. Mathematical Framework
        • 3.1.4. Python Implementation
        • 3.1.5. Examples
        • 3.1.6. Summary
      • 3.2. Bootstrap Fundamentals
        • 3.2.1. Introduction
        • 3.2.2. Key Concepts
        • 3.2.3. Mathematical Framework
        • 3.2.4. Python Implementation
        • 3.2.5. Examples
        • 3.2.6. Summary
      • 3.3. Nonparametric Bootstrap
        • 3.3.1. Introduction
        • 3.3.2. Key Concepts
        • 3.3.3. Mathematical Framework
        • 3.3.4. Python Implementation
        • 3.3.5. Examples
        • 3.3.6. Summary
      • 3.4. Parametric Bootstrap
        • 3.4.1. Introduction
        • 3.4.2. Key Concepts
        • 3.4.3. Mathematical Framework
        • 3.4.4. Python Implementation
        • 3.4.5. Examples
        • 3.4.6. Summary
      • 3.5. Confidence Intervals
        • 3.5.1. Introduction
        • 3.5.2. Key Concepts
        • 3.5.3. Mathematical Framework
        • 3.5.4. Python Implementation
        • 3.5.5. Examples
        • 3.5.6. Summary
      • 3.6. Bias Correction
        • 3.6.1. Introduction
        • 3.6.2. Key Concepts
        • 3.6.3. Mathematical Framework
        • 3.6.4. Python Implementation
        • 3.6.5. Examples
        • 3.6.6. Summary
      • 3.7. Cross Validation Loo
        • 3.7.1. Introduction
        • 3.7.2. Key Concepts
        • 3.7.3. Mathematical Framework
        • 3.7.4. Python Implementation
        • 3.7.5. Examples
        • 3.7.6. Summary
      • 3.8. Cross Validation K Fold
        • 3.8.1. Introduction
        • 3.8.2. Key Concepts
        • 3.8.3. Mathematical Framework
        • 3.8.4. Python Implementation
        • 3.8.5. Examples
        • 3.8.6. Summary
      • 3.9. Model Selection
        • 3.9.1. Introduction
        • 3.9.2. Key Concepts
        • 3.9.3. Mathematical Framework
        • 3.9.4. Python Implementation
        • 3.9.5. Examples
        • 3.9.6. Summary
  • 2. Part III: Bayesian Methods
    • 2.1. Overview
      • 2.1.1. Bayesian Philosophy
        • Introduction
        • Key Concepts
        • Mathematical Framework
        • Python Implementation
        • Examples
        • Summary
STAT 350: Introduction to Statistics
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