PoissonProcess 2005-02-09 20:22:19 2006-10-04 09:49:54 Definition simple Poisson process intensity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{simple} A counting process $\lbrace X(t)\mid t\in\mathbb{R}^{+}\cup\lbrace0\rbrace\rbrace$ is called a \emph{simple Poisson}, or simply a \emph{Poisson process} with parameter $\lambda$, also known as the \emph{intensity}, if \begin{enumerate} \item $X(0)=0$, \item $\lbrace X(t)\rbrace$ has stationary independent increments, \item $P(X(t)=1)=\lambda t+o(t)$, \item $P(X(t)>1)=o(t)$, \end{enumerate} where $o(t)$ is the O notation. \textbf{Remarks}. \begin{itemize} \item The intensity $\lambda$ is assumed to be a constant in terms of $t$. \item Condition 3 above says that the \emph{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). \item It can be shown that $X(t)$ has a Poisson distribution (hence the name of the stochastic process) with parameter $\lambda t$: $$P(X(t)=n)=e^{-\lambda t}\frac{{(\lambda t)}^n}{n!}.$$ \item Therefore, $\operatorname{E}[X(t)]=\lambda t$. \end{itemize}