Probability Theory and Distributions

  1. Expectation and Moments
    1. Expected Value
      1. Definition for Discrete Random Variables
        1. Weighted Average Interpretation
          1. Convergence Conditions
          2. Definition for Continuous Random Variables
            1. Integral Representation
              1. Existence Conditions
              2. Properties of Expectation
                1. Linearity
                  1. Monotonicity
                    1. Expectation of Constants
                    2. Expectation of Functions of Random Variables
                      1. Law of the Unconscious Statistician
                        1. Discrete Case
                          1. Continuous Case
                        2. Variance and Standard Deviation
                          1. Definition of Variance
                            1. As Second Central Moment
                              1. Alternative Computational Formula
                              2. Properties of Variance
                                1. Non-negativity
                                  1. Effect of Linear Transformations
                                    1. Variance of Constants
                                    2. Standard Deviation
                                      1. Definition and Interpretation
                                        1. Units and Scale
                                        2. Standardization
                                          1. Z-score Transformation
                                            1. Properties of Standardized Variables
                                          2. Higher Order Moments
                                            1. Raw Moments
                                              1. Definition and Notation
                                                1. Relationship to Expectation
                                                2. Central Moments
                                                  1. Definition and Properties
                                                    1. Relationship to Variance
                                                    2. Skewness
                                                      1. Third Standardized Moment
                                                        1. Interpretation and Meaning
                                                          1. Measures of Asymmetry
                                                          2. Kurtosis
                                                            1. Fourth Standardized Moment
                                                              1. Interpretation and Meaning
                                                                1. Excess Kurtosis
                                                              2. Moment Generating Functions
                                                                1. Definition and Existence
                                                                  1. Mathematical Definition
                                                                    1. Domain of Convergence
                                                                      1. Existence Conditions
                                                                      2. Properties of MGFs
                                                                        1. Uniqueness Property
                                                                          1. Moment Generation Property
                                                                            1. Linear Transformation Property
                                                                            2. MGFs of Common Distributions
                                                                              1. Discrete Distributions
                                                                                1. Continuous Distributions
                                                                                2. Applications
                                                                                  1. Finding Moments
                                                                                    1. Proving Distribution Results
                                                                                      1. Sum of Independent Variables

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                                                                                  3. Random Variables and Their Properties

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                                                                                  5. Common Discrete Distributions