Poisson probability distribution

by Stephen R. Schmitt

Contents

1. About
2. The source code
3. Discussion

1. About

This Java Script calculator generates a listing of Poisson probabilities. To operate the calculator, enter the value of λ, the average number of events per unit. Press the Compute button to generate the list. On invalid entries, a popup window will display an error message.

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2. The source code

The Java Script source code for this program can be viewed by using the View|Source command of your web browser.

You may use or modify this source code in any way you find useful, provided that you agree that the author has no warranty, obligations or liability. You must determine the suitability of this source code for your use.

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3. Discussion

The Poisson probability distribution is named for the French mathematician Simeon Denise Poisson (1871-1840). It is used to describe a number of events in an interval given an average number of events. The mean number of events in each unit will be denoted by the Greek letter λ. The Poisson probability distribution gives the number of times a particular event occurs in a given unit interval. Applications include modeling the distribution of telephone calls going through a switchboard system, the demand of patients for service at a health institution, the arrivals of trucks and cars at a tollbooth, the number of accidents at an intersection, and counting nuclear decay events. Most commonly, unit intervals are time intervals; unit intervals may be in units of area, volume, etc.

Restrictions:

• The probability that an event occurs in a given unit interval is the same for all unit intervals.
• The number of events that occur in one unit interval is independent of the number that occur in other unit intervals.

The Poisson probability mass function is:

p(x, λ) =	λxe−λ	x = 0, 1, 2, . . .
	x!

where λ = mean number of events in a given unit, e = 2.718282 . . . (base of natural logarithms).

Its mean is: λ; its variance is: λ.

As an example, for λ = 2; the probability of 3 events in an interval is 0.180 and the probablity of up to 3 events in an interval is 0.857.

n:      p(x = n, 2)            P(x ≤ n, 2)
0:      0.135335283236613      0.135335283236613
1:      0.270670566473225      0.406005849709838
2:      0.270670566473225      0.676676416183064
3:      0.180447044315484      0.857123460498547 ←
4:      0.090223522157742      0.947346982656289
5:      0.036089408863097      0.983436391519386

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