Probability Theory and Distributions

  1. Conditional Probability and Independence
    1. Conditional Probability
      1. Definition and Motivation
        1. Conditional Probability Formula
          1. Interpretation and Meaning
            1. Conditions for Well-Definition
            2. Properties of Conditional Probability
              1. Conditional Probability as Probability Measure
                1. Complement Rule for Conditional Probability
                  1. Addition Rule for Conditional Probability
                  2. Multiplication Rules
                    1. General Multiplication Rule
                      1. Chain Rule for Multiple Events
                        1. Sequential Experiments
                        2. Conditional Probability in Practice
                          1. Tree Diagrams
                            1. Sequential Decision Making
                              1. Updating Information
                            2. Law of Total Probability
                              1. Partitions of Sample Space
                                1. Definition of Partition
                                  1. Properties of Partitions
                                    1. Construction of Partitions
                                    2. Statement of Law of Total Probability
                                      1. Formula and Interpretation
                                        1. Geometric Interpretation
                                        2. Applications
                                          1. Probability Calculations
                                            1. Complex Event Analysis
                                              1. Multi-stage Experiments
                                            2. Bayes' Theorem
                                              1. Derivation from Basic Rules
                                                1. From Conditional Probability Definition
                                                  1. From Law of Total Probability
                                                  2. Components of Bayes' Theorem
                                                    1. Prior Probabilities
                                                      1. Likelihood
                                                        1. Posterior Probabilities
                                                          1. Normalizing Constant
                                                          2. Bayes' Theorem for Multiple Hypotheses
                                                            1. General Form
                                                              1. Computational Aspects
                                                              2. Applications
                                                                1. Medical Diagnosis
                                                                  1. Quality Control
                                                                    1. Information Update
                                                                      1. Decision Theory
                                                                    2. Independence of Events
                                                                      1. Definition of Independence
                                                                        1. Multiplicative Definition
                                                                          1. Conditional Probability Definition
                                                                            1. Equivalence of Definitions
                                                                            2. Properties of Independent Events
                                                                              1. Symmetry of Independence
                                                                                1. Independence and Complements
                                                                                  1. Independence and Unions
                                                                                  2. Multiple Event Independence
                                                                                    1. Pairwise Independence
                                                                                      1. Mutual Independence
                                                                                        1. Relationship Between Types
                                                                                        2. Examples and Counterexamples
                                                                                          1. Independent vs Disjoint Events
                                                                                            1. Common Misconceptions
                                                                                              1. Practical Examples

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                                                                                          1. Foundational Concepts of Probability

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                                                                                          3. Random Variables and Their Properties