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
Measurable map
Random variable
Class of random variables
Collection of random variables
Set of random variables
Random collection
Random process
Random real collection process
Random Euclidean real process
Random complex process
Random real process
Random rational process
Random integer process
Random natural number process
Renewal process
Definition D5207
Real poisson process
Formulation 1
Let $T_1, T_2, T_3, \ldots \in \text{Exponential}(\theta)$ 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 $N : [0, \infty) \to \text{Random}(\mathbb{N})$ is a real poisson process with parameter $\theta$ if and only if \begin{equation} \forall \, t \in [0, \infty) : N_t \overset{d}{=} \max \left\{ M \in \mathbb{N} : \sum_{m = 1}^M T_m \leq t \right\} \end{equation}