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 \begin{equation} \forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = (1 - i t \beta^{-1})^{- \alpha} \end{equation}

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 \begin{equation} \forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = \frac{1}{\left( 1 - \frac{i t}{\beta} \right)^{\alpha}} \end{equation}

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 \begin{equation} \forall \, t \in \mathbb{R} : \mathbb{E} (e^{i t X}) = \left( 1 - \frac{i t}{\beta} \right)^{- \alpha} \end{equation}

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 \begin{equation} \forall \, t \in \mathbb{R} : \mathfrak{F}_X (t) = \left( 1 - \frac{i t}{\beta} \right)^{- \alpha} \end{equation}