Poisson process

A counting process $\lbrace X(t)\mid t\in\mathbb{R}^{+}\cup\lbrace0\rbrace\rbrace$ is called a simple Poisson, or simply a Poisson process with parameter $\lambda$ , also known as the intensity, if

1. $X(0)=0$ ,
2. $\lbrace X(t)\rbrace$ has stationary independent increments,
3. $P(X(t)=1)=\lambda t+o(t)$ ,
4. $P(X(t)>1)=o(t)$ ,

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