Probability Theory and Distributions

  1. Limit Theorems and Convergence
    1. Types of Convergence
      1. Convergence in Distribution
        1. Definition and Characterization
          1. Weak Convergence
          2. Convergence in Probability
            1. Definition and Properties
              1. Relationship to Other Types
              2. Almost Sure Convergence
                1. Definition and Interpretation
                  1. Strong Convergence
                  2. Convergence in Mean Square
                    1. Definition and Properties
                      1. L² Convergence
                      2. Relationships Between Convergence Types
                        1. Hierarchy of Convergence
                          1. Implications and Counterexamples
                        2. Law of Large Numbers
                          1. Weak Law of Large Numbers
                            1. Statement for IID Variables
                              1. Proof Using Chebyshev's Inequality
                                1. General Conditions
                                2. Strong Law of Large Numbers
                                  1. Statement and Interpretation
                                    1. Almost Sure Convergence
                                      1. Conditions for Validity
                                      2. Applications
                                        1. Empirical Probability
                                          1. Monte Carlo Methods
                                            1. Statistical Estimation
                                          2. Central Limit Theorem
                                            1. Classical Central Limit Theorem
                                              1. Statement for IID Variables
                                                1. Standardization Process
                                                  1. Rate of Convergence
                                                  2. Conditions and Assumptions
                                                    1. Finite Variance Requirement
                                                      1. Independence Assumption
                                                        1. Identical Distribution
                                                        2. Extensions and Generalizations
                                                          1. Lindeberg-Feller Theorem
                                                            1. Lyapunov Theorem
                                                              1. Non-identical Distributions
                                                              2. Applications
                                                                1. Normal Approximations
                                                                  1. Binomial Approximation
                                                                    1. Poisson Approximation
                                                                      1. Sample Mean Distribution
                                                                      2. Confidence Intervals
                                                                        1. Hypothesis Testing
                                                                          1. Quality Control
                                                                        2. Probability Inequalities
                                                                          1. Markov's Inequality
                                                                            1. Statement and Proof
                                                                              1. Applications and Examples
                                                                                1. Limitations
                                                                                2. Chebyshev's Inequality
                                                                                  1. Statement and Proof
                                                                                    1. One-sided Version
                                                                                      1. Applications to Probability Bounds
                                                                                      2. Hoeffding's Inequality
                                                                                        1. Statement for Bounded Variables
                                                                                          1. Applications to Concentration
                                                                                          2. Jensen's Inequality
                                                                                            1. Statement for Convex Functions
                                                                                              1. Applications to Expectation
                                                                                              2. Applications of Inequalities
                                                                                                1. Probability Bounds
                                                                                                  1. Convergence Proofs
                                                                                                    1. Statistical Applications

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                                                                                                8. Transformations and Functions of Random Variables

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                                                                                                1. Foundational Concepts of Probability