Sums of Gamma Random Variables

           1. Probability Spaces and Random Variables
              1.1  Probability Spaces
              1.2  Random Variables
              1.3  Expected Values of Random Variables
              1.4  Several Random Variables
              1.5  Convolutions
              1.6  Sums of Random Variables

           2. Some Important Probability Distributions
              2.1  The Normal Distribution
              2.2  The Gamma Distribution
              2.3  The Chi-Square Distribution

           3. Sums of Independent Gamma Random Variables
              3.1  Introduction
              3.2  Sums of Gamma Random Variables
              3.3  Integral Representations for A(t)
              3.4  Moschopoulos' Formula for A(t)
              3.5  Hypoexponential Random Variables
              3.6  The Incomplete Gamma Function

           4. The Welch-Satterthwaite Approximation
              4.1  The Approximation for Sums of Gamma Random Variables
              4.2  The Approximation for Sums of Chi-Square Random Variables
              4.3  The Approximation for Hypoexponential Random Variables
              4.4  Examples of the Approximation for Hypoexponential Random Variabless

           5. Partial Orderings for Random Variables
              5.1  Ordinary Ordering
              5.2  The Stochastic Ordering
              5.3  The Hazard Rate Ordering
              5.4  Preservation of the Hazard Rate Ordering Under Log-concave Sums
              5.5  The Likelihood Ratio Ordering
              5.6  Preservation of the Likelihood Ratio Ordering Under Log-concave Sums

           6. Partial Orderings for Hypoexponential Random Variables
              6.1  Majorization


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