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}

Let $\alpha, \beta \in (0, \infty)$ each be a D5407: Positive real number such that

(i)	$x \mapsto \lfloor x \rfloor$ is the D764: Real-integer floor function
(ii)	\begin{equation} N := \lfloor \alpha \rfloor \end{equation}
(iii)	$T_1, \, \ldots, \, T_N \overset{d}{=} \text{Exponential}(\beta)$ are each an D214: Exponential random positive real number
(iv)	$A \overset{d}{=} \text{Alpha}(\alpha - N, \beta)$ is an D6332: Alpha random positive real number
(v)	$A, T_1, \, \ldots, \, T_N$ is an D2713: Independent random collection

A D5722: Random positive real number $X \in \text{Random}(0, \infty)$ is a gamma random positive real number with parameter $(\alpha, \beta)$ if and only if \begin{equation} X \overset{d}{=} A + \sum_{n = 1}^N T_n \end{equation}