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
2.
Part III: Bayesian Methods
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2.
Part III: Bayesian Methods
2.1.
Overview
This part covers bayesian methods.
Chapters
2.1.1. Bayesian Philosophy
Introduction
Key Concepts
Mathematical Framework
Python Implementation
Examples
Summary