Poisson process

A counting process $\{X(t)\mid t\in {ℝ}^{+}\cup \{0\}\}$ is called asimple Poisson, or simply a Poisson process withparameter $\lambda$, 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, $\mathrm{E}[X(t)]=\lambda t$.