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.

•
The intensity $\lambda $ is assumed to be a constant in terms of $t$.

•
Condition 3 above says that the rate in which the an event occurs once in time interval $t$, as $t$ approaches 0, is $\lambda $. Condition 4 says that the event occurs more than once is very unlikely (the rate approaches zero as the time interval shrinks to zero).

•
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, $\mathrm{E}[X(t)]=\lambda t$.
Title  Poisson process 

Canonical name  PoissonProcess 
Date of creation  20130322 15:01:29 
Last modified on  20130322 15:01:29 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60G51 
Synonym  homogeneous Poisson process 
Defines  simple Poisson process 
Defines  intensity 