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Definition D3838
Gamma random positive real number

A D5722: Random positive real number $X \in \text{Random}(0, \infty)$ is a gamma random positive real number with parameters $\alpha, \beta \in (0, \infty)$ if and only if $$\forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = (1 - i t \beta^{-1})^{- \alpha}$$

A D5722: Random positive real number $X \in \text{Random}(0, \infty)$ is a gamma random positive real number with parameters $\alpha, \beta \in (0, \infty)$ if and only if $$\forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = \frac{1}{\left( 1 - \frac{i t}{\beta} \right)^{\alpha}}$$

A D5722: Random positive real number $X \in \text{Random}(0, \infty)$ is a gamma random positive real number with parameters $\alpha, \beta \in (0, \infty)$ if and only if $$\forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = \left( 1 - \frac{i t}{\beta} \right)^{- \alpha}$$

A D5722: Random positive real number $X \in \text{Random}(0, \infty)$ is a gamma random positive real number with parameters $\alpha, \beta \in (0, \infty)$ if and only if $$\forall \, t \in \mathbb{R} : \mathfrak{F}_X (t) = \left( 1 - \frac{i t}{\beta} \right)^{- \alpha}$$
Children
 ▶ D4867: Beta random positive real number
Results
 ▶ R5251: Exponential random positive real number is a gamma random positive real number ▶ R2338: Finite sum of I.I.D. exponential random positive real numbers is a gamma random random positive real number ▶ R5243: Finite sum of uncorrelated identically distributed exponential random positive real numbers is a gamma random random positive real number