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Definition D2854
Poisson random natural number

Let $\xi = \{ \{ \xi_{n, m} \}_{1 \leq m \leq n} \}_{n \geq 1}$ be a D5163: Random real triangular array such that
 (i) $\lambda \in (0, \infty)$ is a D5407: Positive real number (ii) $\theta_1, \theta_2, \theta_3, \ldots \in (0, 1]$ are each a D5407: Positive real number (iii) $$\theta_n : = \min \left( \frac{\lambda}{n}, 1 \right)$$ (iv) $$\forall \, n \in 1, 2, 3, \ldots : \forall \, m \in 1, \ldots, n : \xi_{n, m} \overset{d}{=} \text{Bernoulli}(\theta_n)$$ (v) $\xi_{n, 1}, \ldots, \xi_{n, n}$ is an D2713: Independent random collection for each $n \geq 1$
A D5216: Random natural number $X \in \text{Random}(\mathbb{N})$ is a Poisson random natural number with parameter $\lambda$ if and only if $$X \overset{d}{=} \lim_{n \to \infty} \sum_{m = 1}^n \xi_{n, m}$$
Children
 ▶ Standard Poisson random natural number
Results
 ▶ Expectation of a Poisson random natural number ▶ Finite sum of I.I.D. Poisson random natural numbers is Poisson ▶ Finite sum of independent Poisson random natural numbers is Poisson ▶ Finite sum of uncorrelated identically distributed Poisson random natural numbers is Poisson ▶ Gaussian approximation to standard Poisson distribution