Simulating the Poisson Process

This Demonstration shows simulated paths of the Poisson process. You can see how the cumulative number of events increases as time lapses. You can adjust the mean number of events per unit time. The Demonstration also shows the mean of the process (the blue line) and approximate confidence intervals (the green curves). The confidence intervals are based on the normal approximation to the Poisson distribution. The Poisson process is a special case of a continuous-time Markov chain.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: unjoined paths
Snapshot 2: joined paths
Snapshot 3: one of the paths goes outside of the 99.9% confidence interval
Snapshots 1 and 2 are the same except that in Snapshot 2 the paths are joined. These snapshots show that when you show several paths, joining the paths with vertical lines makes the paths clearer.
Recall that in a Poisson process events occur randomly in time. If time starts at 0, then the number of events occurring up to time is a random variable that has a Poisson distribution with mean . Here, is the mean number of events that occur in one unit of time. The time between the events has an exponential distribution with mean .
For the Poisson process, see [1, pp. 204–208]. For simulation of the Poisson process and other stochastic processes with Mathematica, see [2, pp. 987–1002]. Andrzej Kozlowski has also created a Demonstration, The Poisson Process, that shows simulated paths of the Poisson process. That Demonstration also shows so-called compensated Poisson processes but does not show the mean or confidence intervals.
[1] A. O. Allen, Probability, Statistics, and Queueing Theory with Computer Science Applications, 2nd ed., Boston: Academic Press, 1990.
[2] H. Ruskeepää, Mathematica Navigator: Mathematics, Statistics, and Graphics, 3rd ed., San Diego, CA: Elsevier Academic Press, 2009.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+