Let $T_1, T_2, T_3, \, \ldots \overset{d}{=} \text{Exponential}(\theta)$ each be an D214: Exponential random positive real number such that

(i)	$T_1, T_2, T_3, \, \ldots$ is an D2713: Independent random collection

A D6140: Random natural number process $X : [0, \infty) \to \text{Random}(\mathbb{N})$ is a real poisson process with parameter $\theta$ if and only if \begin{equation} \forall \, t \in [0, \infty) : X_t \overset{d}{=} \max \left\{ N \in \mathbb{N} : \sum_{n = 1}^N T_n \leq t \right\} \end{equation}

Let $T_1, T_2, T_3, \, \ldots \overset{d}{=} \text{Exponential}(\theta)$ each be an D214: Exponential random positive real number such that

(i)	$T_1, T_2, T_3, \, \ldots$ is an D2713: Independent random collection

A D6140: Random natural number process $X : [0, \infty) \to \text{Random}(\mathbb{N})$ is a real poisson process with parameter $\theta$ if and only if \begin{equation} \forall \, t \in [0, \infty) : X_t \overset{d}{=} \max \left\{ N \in \mathbb{N} : \sum_{n = 1}^N T_n \in [0, t] \right\} \end{equation}