Probability Theory and Distributions

  1. Random Variables and Their Properties
    1. Introduction to Random Variables
      1. Definition as Functions
        1. Mapping from Sample Space to Real Line
          1. Measurability Conditions
            1. Notation and Conventions
            2. Motivation and Interpretation
              1. Numerical Representation of Outcomes
                1. Facilitating Mathematical Analysis
                2. Types of Random Variables
                  1. Discrete Random Variables
                    1. Continuous Random Variables
                      1. Mixed Random Variables
                        1. Singular Random Variables
                      2. Discrete Random Variables
                        1. Characterization
                          1. Countable Range
                            1. Point Mass Distributions
                            2. Probability Mass Function
                              1. Definition and Properties
                                1. Validity Conditions
                                  1. Graphical Representation
                                  2. Cumulative Distribution Function
                                    1. Definition for Discrete Variables
                                      1. Step Function Properties
                                        1. Right Continuity
                                          1. Relationship to PMF
                                          2. Functions of Discrete Random Variables
                                            1. Transformation of PMF
                                              1. One-to-One Transformations
                                                1. Many-to-One Transformations
                                              2. Continuous Random Variables
                                                1. Characterization
                                                  1. Uncountable Range
                                                    1. Absolutely Continuous Distributions
                                                    2. Probability Density Function
                                                      1. Definition and Properties
                                                        1. Validity Conditions
                                                          1. Interpretation as Density
                                                            1. Graphical Representation
                                                            2. Cumulative Distribution Function
                                                              1. Definition for Continuous Variables
                                                                1. Continuity Properties
                                                                  1. Relationship to PDF
                                                                    1. Fundamental Theorem of Calculus Application
                                                                    2. Functions of Continuous Random Variables
                                                                      1. Change of Variables Technique
                                                                        1. Jacobian Method
                                                                          1. Inverse Transform Method