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Poisson process
A counting process $\{X(t)\mid t\in\mathbb{R}^{{+}}\cup\{0\}\}$ is called a simple Poisson, or simply a Poisson process with parameter $\lambda$, also known as the intensity, if
1. $X(0)=0$,
2. $\{X(t)\}$ has stationary independent increments,
3. $P(X(t)=1)=\lambda t+o(t)$,
4. $P(X(t)>1)=o(t)$,
where $o(t)$ is the O notation.
Remarks.

It can be shown that $X(t)$ has a Poisson distribution (hence the name of the stochastic process) with parameter $\lambda t$:
$P(X(t)=n)=e^{{\lambda t}}\frac{{(\lambda t)}^{n}}{n!}.$ 
Therefore, $\operatorname{E}[X(t)]=\lambda t$.
Defines:
simple Poisson process, intensity
Synonym:
homogeneous Poisson process
Type of Math Object:
Definition
Major Section:
Reference
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