Poisson process

A counting process $\{X(t)\mid t\in {ℝ}^{+}\cup \{0\}\}$ 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	2013-03-22 15:01:29
Last modified on	2013-03-22 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