Snapshot 1: unjoined 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.