STAT 418: Computational Methods in Data Science

Course Content

  • Part I: Foundations of Probability and Computation
    • Chapter 1: Statistical Paradigms and Core Concepts
      • Section 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
        • Practice Problems
    • Chapter 2: Monte Carlo Simulation
      • Section 2.1 Monte Carlo Fundamentals
        • The Historical Development of Monte Carlo Methods
        • The Core Principle: Expectation as Integration
        • Theoretical Foundations
        • Variance Estimation and Confidence Intervals
        • Worked Examples
        • Comparison with Deterministic Methods
        • Sample Size Determination
        • Convergence Diagnostics and Monitoring
        • Practical Considerations
        • Chapter 2.1 Exercises: Monte Carlo Fundamentals Mastery
        • Bringing It All Together
        • Transition to What Follows
      • Section 2.2 Uniform Random Variates (Optional)
        • Why Uniform? The Universal Currency of Randomness
        • The Paradox of Computational Randomness
        • Chaotic Dynamical Systems: An Instructive Failure
        • Linear Congruential Generators
        • Shift-Register Generators
        • The KISS Generator: Combining Strategies
        • Modern Generators: Mersenne Twister and PCG
        • Statistical Testing of Random Number Generators
        • Practical Considerations
        • Chapter 2.2 Exercises: Uniform Random Variates Mastery
        • Bringing It All Together
        • Transition to What Follows
      • Section 2.3 Inverse CDF Method
        • Mathematical Foundations
        • Continuous Distributions with Closed-Form Inverses
        • Numerical Inversion
        • Discrete Distributions
        • Mixed Distributions
        • Practical Considerations
        • Chapter 2.3 Exercises: Inverse CDF Method Mastery
        • Lessons from the Exercises
        • Bringing It All Together
        • Transition to What Follows
      • Section 2.4 Transformation Methods (Optional)
        • Why Transformation Methods?
        • The Box–Muller Transform
        • The Polar (Marsaglia) Method
        • Method Comparison: Box–Muller vs Polar vs Ziggurat
        • The Ziggurat Algorithm
        • The CLT Approximation (Historical)
        • Distributions Derived from the Normal
        • Multivariate Normal Generation
        • Implementation Guidance
        • Chapter 2.4 Exercises: Transformation Methods Mastery
        • Bringing It All Together
      • Section 2.5 Rejection Sampling
        • The Dartboard Intuition
        • The Accept-Reject Algorithm
        • Efficiency Analysis
        • Choosing the Proposal Distribution
        • Python Implementation
        • The Squeeze Principle
        • Geometric Example: Sampling from the Unit Disk
        • Worked Examples
        • Limitations and the Curse of Dimensionality
        • Connections to Other Methods
        • Practical Considerations
        • Chapter 2.5 Exercises: Rejection Sampling Mastery
        • Bringing It All Together
        • References
      • Section 2.6 Variance Reduction Methods
        • The Variance Reduction Paradigm
        • Importance Sampling
        • Control Variates
        • Antithetic Variates
        • Stratified Sampling
        • Common Random Numbers
        • Conditional Monte Carlo (Rao–Blackwellization)
        • Combining Variance Reduction Techniques
        • Practical Considerations
        • Bringing It All Together
        • Chapter 2.6 Exercises: Variance Reduction Mastery
      • Section 2.7 Chapter 2 Summary
        • The Complete Monte Carlo Workflow
        • Method Selection Guide
        • Quick Reference Tables
        • Common Pitfalls Checklist
        • Connections to Later Chapters
        • Learning Outcomes Checklist
        • Further Reading: Optimization and Missing Data
        • Final Perspective
  • Part II: Frequentist Inference
    • Chapter 3: Parametric Inference and Likelihood Methods
      • Section 3.1 Exponential Families
        • Historical Origins: From Scattered Results to Unified Theory
        • The Canonical Exponential Family
        • Converting Familiar Distributions
        • The Log-Partition Function: A Moment-Generating Machine
        • Sufficiency: Capturing All Parameter Information
        • Minimal Sufficiency and Completeness
        • Conjugate Priors and Bayesian Inference
        • Exponential Dispersion Models and GLMs
        • Python Implementation
        • Practical Considerations
        • Chapter 3.1 Exercises: Exponential Families Mastery
        • Bringing It All Together
      • Section 3.2 Maximum Likelihood Estimation
        • The Likelihood Function
        • The Score Function
        • Fisher Information
        • Closed-Form Maximum Likelihood Estimators
        • Numerical Optimization for MLE
        • Asymptotic Properties of MLEs
        • The Cramér-Rao Lower Bound
        • The Invariance Property
        • Likelihood-Based Hypothesis Testing
        • Confidence Intervals from Likelihood
        • Practical Considerations
        • Connection to Bayesian Inference
        • Chapter 3.2 Exercises: Maximum Likelihood Estimation Mastery
        • Bringing It All Together
      • Section 3.3 Sampling Variability and Variance Estimation
        • Statistical Estimators and Their Properties
        • Sampling Distributions
        • The Delta Method
        • The Plug-in Principle
        • Variance Estimation Methods
        • Applications and Worked Examples
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 3.4 Linear Models
        • Matrix Calculus Foundations
        • The Linear Model
        • Ordinary Least Squares: The Calculus Approach
        • Ordinary Least Squares: The Geometric Approach
        • Properties of the OLS Estimator
        • The Gauss-Markov Theorem
        • Estimating the Error Variance
        • Distributional Results Under Normality
        • Diagnostics and Model Checking
        • Bringing It All Together
        • Numerical Stability: QR Decomposition
        • Model Selection and Information Criteria
        • Regularization: Ridge and LASSO
        • Chapter 3.4 Exercises: Linear Models Mastery
      • Section 3.5 Generalized Linear Models
        • Historical Context: Unification of Regression Methods
        • The GLM Framework: Three Components
        • Score Equations and Fisher Information
        • Iteratively Reweighted Least Squares
        • Logistic Regression: Binary Outcomes
        • Poisson Regression: Count Data
        • Gamma Regression: Positive Continuous Data
        • Inference in GLMs: The Testing Triad
        • Model Diagnostics
        • Model Comparison and Selection
        • Quasi-Likelihood and Robust Inference
        • Practical Considerations
        • Bringing It All Together
        • Further Reading
        • Chapter 3.5 Exercises: Generalized Linear Models Mastery
      • Section 3.6 Chapter 3 Summary
        • The Parametric Inference Pipeline
        • The Five Pillars of Chapter 3
        • How the Pillars Connect
        • Method Selection Guide
        • Quick Reference: Core Formulas
        • Connections to Future Material
        • Practical Guidance
        • Final Perspective
    • Chapter 4: Resampling Methods
      • Section 4.1 The Sampling Distribution Problem
        • The Fundamental Target: Sampling Distributions
        • Historical Development: The Quest for Sampling Distributions
        • Three Routes to the Sampling Distribution
        • When Asymptotics Fail: Motivating the Bootstrap
        • The Plug-In Principle: Theoretical Foundation
        • Computational Perspective: Bootstrap as Monte Carlo
        • Practical Considerations
        • Bringing It All Together
        • Chapter 4.1 Exercises
      • Section 4.2 The Empirical Distribution and Plug-in Principle
        • The Empirical Cumulative Distribution Function
        • Convergence of the Empirical CDF
        • Parameters as Statistical Functionals
        • The Plug-in Principle
        • When the Plug-in Principle Fails
        • The Bootstrap Idea in One Sentence
        • Computational Implementation
        • Bringing It All Together
        • Section 4.2 Exercises: ECDF and Plug-in Mastery
      • Section 4.3 The Nonparametric Bootstrap
        • The Bootstrap Principle
        • Bootstrap Standard Errors
        • Bootstrap Bias Estimation
        • Bootstrap Confidence Intervals
        • Bootstrap for Regression
        • Bootstrap Diagnostics
        • When Bootstrap Fails
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 4.4: The Parametric Bootstrap
        • The Parametric Bootstrap Principle
        • Location-Scale Families
        • Parametric Bootstrap for Regression
        • Confidence Intervals
        • Model Checking and Validation
        • When Parametric Bootstrap Fails
        • Parametric vs. Nonparametric: A Decision Framework
        • Practical Considerations
        • Bringing It All Together
      • Section 4.5: Jackknife Methods (Optional)
        • Historical Context and Motivation
        • The Delete-1 Jackknife
        • Jackknife Bias Estimation
        • The Delete-\(d\) Jackknife
        • Jackknife versus Bootstrap
        • The Infinitesimal Jackknife
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 4.6 Bootstrap Hypothesis Testing and Permutation Tests
        • From Confidence Intervals to Hypothesis Tests
        • The Bootstrap Hypothesis Testing Framework
        • Permutation Tests: Exact Tests Under Exchangeability
        • Testing Equality of Distributions
        • Bootstrap Tests for Regression
        • Bootstrap vs Classical Tests
        • Permutation vs Bootstrap: Choosing the Right Approach
        • Multiple Testing with Bootstrap
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 4.7 Bootstrap Confidence Intervals: Advanced Methods (Optional)
        • Why Advanced Methods?
        • The Studentized (Bootstrap-t) Interval
        • Bias-Corrected (BC) Intervals
        • Bias-Corrected and Accelerated (BCa) Intervals
        • Choosing B and Assessing Monte Carlo Error
        • Diagnostics for Advanced Bootstrap Methods
        • Method Selection Guide
        • Bringing It All Together
        • Chapter 4.7 Exercises: Bootstrap Confidence Interval Mastery
      • Section 4.8: Cross-Validation Methods (Optional)
        • Historical Context: From Jackknife to Cross-Validation
        • Leave-One-Out Cross-Validation
        • K-Fold Cross-Validation
        • Generalized Cross-Validation
        • Nested Cross-Validation
        • Bootstrap Prediction Error Estimation
        • Connection to Information Criteria
        • The Variance Estimation Problem
        • Cross-Validation for Structured Data
        • Computational Considerations
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 4.9 Chapter 4 Summary
        • The Resampling Philosophy
        • The Complete Resampling Workflow
        • The Eight Pillars of Chapter 4
        • Method Selection Guide
        • Quick Reference Tables
        • Common Pitfalls Checklist
        • Connections to Other Chapters
        • Learning Outcomes Checklist
        • Practical Guidance
        • Further Reading: Advanced Resampling Topics
        • Final Perspective
  • Part III: Bayesian Inference
    • Chapter 5: Bayesian Inference
      • Section 5.1 Foundations of Bayesian Inference
        • Historical Development: From Bayes’ Essay to the MCMC Revolution
        • The Bayesian Workflow
        • Bayes’ Theorem for Parameters
        • Discrete Parameter Spaces
        • Continuous Parameters: Grid Approximation
        • The Posterior as Complete Inference
        • The Likelihood Principle Revisited
        • Practical Considerations
        • Bringing It All Together
        • Chapter 5.1 Exercises
      • Section 5.2 Prior Specification and Conjugate Analysis
        • The Role of the Prior
        • Beta-Binomial Conjugate Analysis
        • Normal-Normal Model: Known Variance
        • Normal-Inverse-Gamma: Unknown Mean and Variance
        • Poisson-Gamma Model
        • Multinomial-Dirichlet Model
        • Beyond Conjugacy: The Full Landscape of Prior Specification
        • Bayesian vs. Frequentist Synthesis and the Limits of Conjugacy
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 5.3 Posterior Inference: Credible Intervals and Hypothesis Assessment
        • Credible Intervals: Probability Statements About Parameters
        • Posterior Probability and Directional Hypothesis Assessment
        • Posterior Predictive Intervals
        • Communicating Bayesian Results
        • Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 5.4 Markov Chains: The Mathematical Foundation of MCMC
        • From Grid Approximation to Markov Chains
        • Markov Chains
        • Stationary Distributions and Detailed Balance
        • The Ergodic Theorem: Why MCMC Averages Converge
        • The MCMC Estimator, Effective Sample Size, and \(\hat{R}\)
        • Python: Simulating Convergence and Diagnosing Chains
        • Mixing, Thinning, and Practical Considerations
        • Bringing It All Together
        • Exercises
      • Section 5.5 MCMC Algorithms: Metropolis-Hastings and Gibbs Sampling
        • The Metropolis-Hastings Algorithm
        • The Gibbs Sampler
        • Hamiltonian Monte Carlo: A Brief Introduction
        • Practical Workflow
        • Bringing It All Together
        • Exercises
      • Section 5.6 Probabilistic Programming with PyMC
        • How PyMC Works: From Model Spec to Gradient
        • The Diagnostic Toolkit
        • Full Worked Example 1: Logistic Regression
        • Full Worked Example 2: Poisson Regression
        • Scale Parameters: Half-Normal and Half-Cauchy Priors
        • Derived Quantities with pm.Deterministic
        • Practical Workflow Checklist
        • Bringing It All Together
        • Exercises
      • Section 5.7 Bayesian Model Comparison
        • The Predictive Accuracy Target
        • WAIC
        • PSIS-LOO Cross-Validation
        • Implementation: Comparing Two Models
        • Bayes Factors
        • Bringing It All Together
        • Exercises
      • Section 5.8 Hierarchical Models and Partial Pooling (Optional)
        • The Pooling Problem
        • The Mathematical Structure
        • Worked Example 1: Eight Schools (Normal Likelihood)
        • Worked Example 2: NBA Team Scoring – Does Shrinkage Pay Off?
        • Worked Example 3: Dirichlet-Multinomial (Categorical Likelihood)
        • LOO Model Comparison: Three Pooling Strategies
        • When to Use Hierarchical Models
        • Practical Considerations
        • Bringing It All Together
        • Exercises
        • References
      • Section 5.9 Chapter 5 Summary
        • The Bayesian Workflow
        • Section-by-Section Synthesis
        • Decision Guide: Which Bayesian Tool When
        • Quick Reference: Core Formulas
        • Common Pitfalls
        • Connections Across the Chapter
        • Connections to Earlier Chapters
  • Part IV: Large Language Models in Data Science
    • Chapter 6: LLMs in Data Science Workflows
      • Section 6.1 LLM Foundations: Architecture, Training, and Deployment
        • From Statistical Models to Neural Language Models
        • The Transformer Architecture
        • Pre-Training, Fine-Tuning, and In-Context Learning
        • Model Families and the Landscape
        • Open vs. Closed Models: Trade-offs for Data Scientists
        • Deployment Options
        • Getting Started with GenAI Studio
        • Chapter 6.1 Exercises: LLM Foundations
        • Transition to What Follows
      • Section 6.2 Embeddings and Feature Extraction
        • From Sparse to Dense Representations
        • Generating Embeddings with GenAI Studio
        • Similarity Search
        • What an Embedding Encodes
        • Classification with Embeddings
        • Integration with Statistical Models
        • Chapter 6.2 Exercises: Embeddings and Feature Extraction
        • Transition to What Follows
      • Section 6.3 Text Preprocessing for LLM Pipelines
        • Tokenization: How Models See Text
        • Context Windows and Token Limits
        • Chunking Strategies
        • Text Normalization and Cleaning
        • Building a Complete Preprocessing Pipeline
        • Chapter 6.3 Exercises: Text Preprocessing
        • Transition to What Follows
      • Section 6.4 LLM-Assisted Data Annotation
        • The Annotation Bottleneck
        • LLM as Annotator
        • Annotation Tasks for Data Science
        • Quality Control and Validation
        • When LLM Annotation Works and When It Does Not
        • Chapter 6.4 Exercises: Data Annotation
        • Transition to What Follows
      • Section 6.5 Retrieval-Augmented Generation
        • Why RAG?
        • Building RAG with GenAI Studio
        • Building RAG from Scratch
        • Chunking Strategies for RAG
        • Evaluating RAG Systems
        • RAG Failure Modes and Mitigations
        • Chapter 6.5 Exercises: Retrieval-Augmented Generation
        • Transition to What Follows
      • Section 6.6 Prompt Engineering for Data Science
        • Prompts as Code
        • Systematic Prompt Design
        • Few-Shot Prompting
        • Chain-of-Thought Reasoning
        • Advanced Techniques
        • Debugging and Iterating on Prompts
        • Data Science Prompt Patterns
        • Chapter 6.6 Exercises: Prompt Engineering
        • Transition to What Follows
      • Section 6.7 Tool Use
        • Why Tool Use?
        • What Tool Use Is
        • Declaring a Tool with @tool
        • A Single Tool Call, End to End
        • When to Use Tool Use
        • Risks and Safeguards
        • Where Tool Use Ends and Agents Begin
        • Chapter 6.7 Exercises: Tool Use
        • Transition to What Follows
      • Section 6.8 Reliability and Evaluation
        • The Reliability Challenge
        • Consistency Assessment
        • Can You Trust a Confidence Number?
        • LLM-as-Judge
        • Uncertainty Quantification
        • Evaluation Protocols
        • Chapter 6.8 Exercises: Reliability and Evaluation
        • Transition to What Follows
      • Section 6.9 Responsible AI Practices
        • Privacy and Data Protection
        • Bias in LLM Outputs
        • Transparency and Disclosure
        • Ethical Frameworks
        • Appropriate Use in Data Science
        • Building a Personal AI Use Policy
        • Chapter 6.9 Exercises: Responsible AI
        • Transition to What Follows
      • Section 6.10 Chapter Summary
        • Section-by-Section Recap
        • GenAI Studio Quick Reference
        • Connections to Earlier Chapters
        • Purdue AI Working Competency Mapping
        • Common Pitfalls Checklist
        • End-to-End Example
        • Learning Outcomes Checklist
        • Further Reading
        • Final Perspective

Appendices

  • Appendices
    • Appendix A: Calculus Review
      • From Formulas to Statistical Reasoning
      • Univariate Differentiation
        • Basic Rules
        • Logarithmic Differentiation
        • Higher Derivatives and the Second Derivative Test
        • Common Derivatives Reference
      • Integration
        • Definite Integrals as Expectations
        • Integration by Parts
        • Change of Variables — Univariate
        • Improper Integrals and Convergence
      • Taylor Series and Approximation
        • Taylor’s Theorem
        • First-Order Taylor: The Delta Method Connection
        • Second-Order Taylor: Quadratic Likelihood Approximation
        • Convergence of Taylor Series
      • Leibniz Rule and Differentiation Under the Integral
        • Statement and Regularity Conditions
        • Application: The Score Has Mean Zero
        • Application: Fisher Information Equivalence
      • Multivariate Differential Calculus
        • Partial Derivatives and the Gradient
        • The Hessian Matrix
        • The Multivariate Chain Rule
        • Second-Order Conditions
      • Convexity and Optimization
        • Convex Sets and Functions
        • Hessian Characterization of Convexity
        • Convexity and Exponential Families
        • Strict Convexity and Unique Optimization
      • Multivariate Taylor Expansion
        • The Multivariate Taylor Formula
        • The Wald Statistic
      • Multiple Integrals and Change of Variables
        • Multiple Integrals
        • The Jacobian Determinant
        • Worked Example: Polar Coordinates
      • Connections: From Calculus to Computation
      • Practice Problems
    • Appendix B: Linear Algebra for Data Science
      • From Notation to Intuition
      • Vectors and Inner Products
        • Inner Product and Norm
        • Orthogonality
      • Matrix Operations and Properties
        • Multiplication, Transpose, and Inverse
        • Trace
        • Determinant
        • Rank
      • Special Matrix Structures
        • Symmetric Matrices
        • Diagonal and Identity Matrices
        • Idempotent Matrices
        • Positive Definite and Positive Semi-Definite Matrices
        • Stochastic Matrices
      • Block Matrices
        • Block Partitioning
        • Block Multiplication
        • Block Transpose and Block Diagonal
        • The Schur Complement
        • Sherman-Morrison-Woodbury Formula
      • Column Space, Projections, and Least Squares
        • Column Space and Null Space
        • Orthogonal Projection
        • The Hat Matrix
        • Leverage
      • Matrix Decompositions
        • Eigendecomposition (Spectral Decomposition)
        • Cholesky Decomposition
        • QR Decomposition
        • Singular Value Decomposition (SVD)
        • Decomposition Summary
      • Quadratic Forms and Covariance
        • Quadratic Forms
        • Covariance Matrices
      • Matrix Calculus
        • Gradients of Scalar Functions
        • The Hessian Matrix
        • Derivative of Log-Determinant
        • Jacobian and Change of Variables
      • Numerical Considerations
        • Condition Number and Stability
        • Avoiding \(\mathbf{X}^\top\mathbf{X}\)
        • Near-Singular Matrices
      • Key Takeaways
      • Looking Ahead
      • Practice Problems
    • Appendix C: Numerical Analysis Review
      • From Exact to Approximate
      • Floating-Point Arithmetic
        • IEEE 754 Basics
        • Catastrophic Cancellation
        • Log-Space Arithmetic
      • Root Finding
        • Fixed-Point Iteration and Contraction Mapping
        • Bisection
        • Newton’s Method for Root Finding
      • Numerical Optimization
        • Optimization as Fixed-Point Iteration
        • Gradient Descent
        • Newton’s Method for Optimization
        • Quasi-Newton Methods
      • Numerical Differentiation
        • Forward and Central Differences
        • The Step Size Dilemma
        • The Complex-Step Method
      • Numerical Integration
        • Error Analysis for Quadrature
        • Adaptive Quadrature
        • Monte Carlo Error Rate
      • Convergence and Stopping Criteria
        • Asymptotic (Big-O) Notation
        • Convergence Orders
        • Stopping Criteria
        • Diagnosing Non-Convergence
      • Connections: From Numerical Analysis to Computation
      • Practice Problems
    • Appendix D: Probability Distributions — Theory and Computation
      • From Abstract Foundations to Concrete Tools
        • The Moment of Discovery: De Moivre’s Insight
        • This Appendix’s Philosophy: Theory Meets Computation
        • Structure of This Appendix
      • The Python Ecosystem for Probability
        • Practical Example: Complete Distribution Analysis
        • Why Study Named Distributions?
      • Discrete Distributions
        • Bernoulli Distribution
        • Binomial Distribution
        • Poisson Distribution
        • Geometric Distribution
        • Negative Binomial Distribution
      • Continuous Distributions
        • Uniform Distribution
        • Normal (Gaussian) Distribution
        • Exponential Distribution
        • Gamma Distribution
        • Beta Distribution
      • Additional Important Distributions
        • Student’s t-Distribution
        • Chi-Square Distribution
        • F-Distribution
      • Summary and Practical Guidelines
        • Choosing the Right Distribution
        • Key Relationships Between Distributions
        • Python Tools Summary
        • Parameterization Notes
        • Common Pitfalls and Best Practices
      • Conclusion
      • Practice Problems
    • Appendix E: Statistical Inference Review
      • Point Estimation
        • Estimators and Their Properties
        • Uniformly Minimum Variance Unbiased Estimators
        • Common Estimators
        • Method of Moments
        • Sampling Distributions
      • Confidence Intervals
        • Constructing Intervals from Pivotal Quantities
        • Margin of Error and Sample Size Determination
        • Duality of Confidence Intervals and Hypothesis Tests
        • Common Misinterpretations
        • Computational Verification: Coverage
        • When Coverage Breaks Down: The Wald Proportion Interval
      • Hypothesis Testing
        • The Neyman-Pearson Framework
        • Error Types and Power
        • P-Values
        • Power Analysis
        • The Neyman-Pearson Lemma
        • Uniformly Most Powerful Tests
        • Common Hypothesis Tests
        • Multiple Testing
      • Sufficiency and Information
        • Sufficient Statistics
        • The Fisher-Neyman Factorization Theorem
        • Examples of Sufficient Statistics
        • Minimal Sufficiency
        • Completeness
        • Ancillary Statistics and Basu’s Theorem
        • Fisher Information
        • Two Equivalent Forms of Fisher Information
        • The Cramér-Rao Lower Bound
        • Efficiency of Estimators
      • The Likelihood Function
        • Likelihood versus Probability
        • Maximum Likelihood Estimation
        • The Score Function and Information
        • MLE Properties
        • Profile Likelihood
        • Computational Illustration: Likelihood Anatomy
      • Asymptotic Theory
        • Modes of Convergence
        • The Weak Law of Large Numbers
        • The Central Limit Theorem
        • Slutsky’s Theorem
        • The Continuous Mapping Theorem
        • The Delta Method
        • Asymptotic Theory of the MLE
      • Connections: From Review to Computation
      • Practice Problems
    • Appendix F: Python Random Generation
      • From Mathematical Distributions to Computational Samples
      • The Python Ecosystem at a Glance
      • Understanding Pseudo-Random Number Generation
        • The Nature of Pseudo-Randomness
        • What Makes a Good PRNG?
        • The Mersenne Twister
        • NumPy’s PCG64
      • The Standard Library: random Module
        • Generating Random Numbers
        • Random Operations on Sequences
        • Distribution Generators
        • Controlling Randomness: Seeds and State
      • NumPy: Fast Vectorized Random Sampling
        • Why NumPy Is the Default for Scientific Work
        • The Modern Generator API
        • Performance Comparison
        • Univariate Distributions
        • Multivariate Distributions
        • NumPy Sampling Utilities
        • Parallel Random Number Generation
      • SciPy Stats: The Complete Statistical Toolkit
        • Why SciPy Is the “Next Stop” After NumPy
        • The Frozen Distribution Pattern
        • The Unified Interface
        • Parameterization: The Most Common Error Source
        • Complete Analysis Example
      • Bringing It All Together: Library Selection Guide
      • Looking Ahead: From Random Numbers to Monte Carlo Methods
      • Practice Problem
STAT 418: Computational Methods in Data Science
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