# 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. 1.

$X(0)=0$,

2. 2.

$\{X(t)\}$ has stationary independent increments,

3. 3.

$P(X(t)=1)=\lambda t+o(t)$,

4. 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, $\operatorname{E}[X(t)]=\lambda t$.

Title Poisson process PoissonProcess 2013-03-22 15:01:29 2013-03-22 15:01:29 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 60G51 homogeneous Poisson process simple Poisson process intensity