Simulating the Simple Random Walk

This Demonstration shows simulated paths of the simple random walk. Thus, you can see how the path evolves with time. The Demonstration also shows approximate confidence intervals (the green curves), which are based on the normal approximation.


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


Snapshot 1: some of the 10 paths go outside of the 95% confidence interval
Snapshot 2: all 10 paths stay within the 99.9% confidence interval
Snapshot 3: 10 paths, each of 10,000 steps, 99.9% confidence interval
The simple random walk starts at 0. At each time step , 1 is added or subtracted from the current value. Addition and subtraction are done with equal probabilities. In the plots, the values are plotted on the vertical axis and the time axis is horizontal.
The confidence intervals can be obtained from the following result. Let be the position of the walk at step . The probability that is greater than approaches, as approaches infinity, the probability that the standard normal variable is greater than ; see [1], p. 76.
For the simple random walk, see [1], pp. 67–97. For simulation of the simple random walk and other stochastic processes with Mathematica, see [2], pp. 987–1002.
[1] W. Feller, An Introduction to Probability and Its Applications, vol. 1, 3rd ed., revised printing, New York: Wiley, 1968.
[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+