ThmDex – An index of mathematical definitions, results, and conjectures.
Finite sum of I.I.D. Poisson random natural numbers is Poisson
Formulation 0
Let $X_1, \ldots, X_N \in \text{Poisson}(\theta)$ each be a D2854: Poisson random natural number such that
(i) $X_1, \ldots, X_N$ is an D2713: Independent random collection
Then \begin{equation} \sum_{n = 1}^N X_n \overset{d}{=} \text{Poisson} (N \theta) \end{equation}
Proofs
Proof 0
Let $X_1, \ldots, X_N \in \text{Poisson}(\theta)$ each be a D2854: Poisson random natural number such that
(i) $X_1, \ldots, X_N$ is an D2713: Independent random collection