Probability Theory and Distributions

  1. Transformations and Functions of Random Variables
    1. Single Variable Transformations
      1. Discrete Variable Transformations
        1. One-to-One Transformations
          1. Many-to-One Transformations
            1. PMF Transformation Rules
            2. Continuous Variable Transformations
              1. Monotonic Transformations
                1. Inverse Function Method
                  1. Jacobian Calculation
                  2. Non-monotonic Transformations
                    1. Multiple Inverse Method
                      1. Probability Density Transformation
                    2. Common Transformations
                      1. Linear Transformations
                        1. Quadratic Transformations
                          1. Logarithmic Transformations
                            1. Exponential Transformations
                          2. Multiple Variable Transformations
                            1. Bivariate Transformations
                              1. Jacobian Matrix Method
                                1. Change of Variables Formula
                                  1. Examples and Applications
                                  2. Multivariate Transformations
                                    1. General Jacobian Method
                                      1. Coordinate Transformations
                                        1. Polar Coordinates
                                          1. Spherical Coordinates
                                        2. Order Statistics
                                          1. Definition and Motivation
                                            1. Distribution of Order Statistics
                                              1. Single Order Statistic
                                                1. Joint Distribution of Order Statistics
                                                2. Extreme Value Distributions
                                                  1. Minimum and Maximum
                                                    1. Range Distribution
                                                    2. Applications
                                                      1. Reliability Analysis
                                                        1. Quality Control
                                                          1. Risk Assessment
                                                        2. Sums of Random Variables
                                                          1. Sums of Independent Discrete Variables
                                                            1. Convolution Formula
                                                              1. Probability Generating Functions
                                                              2. Sums of Independent Continuous Variables
                                                                1. Convolution Integral
                                                                  1. Moment Generating Function Method
                                                                  2. Special Cases
                                                                    1. Sums of Normal Variables
                                                                      1. Sums of Poisson Variables
                                                                        1. Sums of Gamma Variables
                                                                        2. Central Limit Theorem Applications
                                                                          1. Approximate Distributions
                                                                            1. Normal Approximations

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                                                                        7. Joint Distributions and Dependence

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                                                                        9. Limit Theorems and Convergence