ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Subset ▼ Power set ▼ Hyperpower set sequence ▼ Hyperpower set ▼ Hypersubset ▼ Subset algebra ▼ Subset structure ▼ Measurable space ▼ Measure space ▼ Probability space ▼ Filtered probability space ▼ Random time ▼ Stopping time ▼ Negative binomial random number ▼ Geometric random positive integer ▼ Standard exponential random positive real number ▼ Exponential random positive real number ▼ Erlang random positive real number ▼ Real poisson process
Definition D5211
Standard real poisson process

Let $T_1, T_2, T_3, \, \ldots \overset{d}{=} \text{Exponential}(1)$ each be a D4000: Standard exponential random positive real number such that
 (i) $T_1, T_2, T_3, \, \ldots$ is an D2713: Independent random collection
A D6135: Random unsigned real process $X : [0, \infty) \to \text{Random}(\Omega \to [0, \infty))$ is a standard real poisson process if and only if $$\forall \, t \in [0, \infty) : X_t \overset{d}{=} \max \left\{ N \in \mathbb{N} : \sum_{n = 1}^N T_n \leq t \right\}$$