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 infinity, the probability that the standard normal variable is greater than
; see , p. 76.
For the simple random walk, see , pp. 67–97. For simulation of the simple random walk and other stochastic processes with Mathematica
, see , pp. 987–1002.
 W. Feller, An Introduction to Probability and Its Applications
, vol. 1, 3rd ed., revised printing, New York: Wiley, 1968.
 H. Ruskeepää, Mathematica Navigator: Mathematics, Statistics, and Graphics,
3rd ed., San Diego, CA: Elsevier Academic Press, 2009.