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MVPstats — What is a Poisson Distribution?

A Poisson Distribution is a discrete Random Sampling Distribution. The sampling is from a given unit of inspection or observation. These units of observation can be properties such as length, area, number of items, volume, or time. The Poisson distribution is generated by counts of the number of occurrences of some phenomenon.

Calculations

The probability of exactly X occurrences in unit of observation may be calculated by the following Poisson formula.

PoissonEq

Where:

Other Properties

The poisson distribution may be approximated with a Normal Distribution with a Mean equal to Lambda, and a standard deviation equal to the square-root of Lambda.

The limit of the Binomial Distribution, as n goes toward infinity, and p goes toward zero, yields a Poisson Distribution, with Lambda equal to np.

The distance or time between occurrence of Poisson phenomenon can be model with an Exponential distribution.

Example

Here is an example Poisson Distribution, with a mean number of occurrences, Lambda, equal to 2. The probability of 3 or more occurrences, when Lambda is 2, is 0.3233.

Poisson Distribution

   Lambda = 2.0000

             Prob     Equal &     Equal &
    X        at X       Above       Below
    0      0.1353      1.0000      0.1353 |------------------
    1      0.2707      0.8647      0.4060 |-----------------------------------
    2      0.2707      0.5940      0.6767 |-----------------------------------
    3      0.1804      0.3233      0.8571 |-----------------------
    4      0.0902      0.1429      0.9473 |------------
    5      0.0361      0.0527      0.9834 |-----
    6      0.0120      0.0166      0.9955 |--
    7      0.0034      0.0045      0.9989 |
    8      0.0009      0.0011      0.9998 |
    9      0.0002      0.0002      1.0000 |
   10      0.0000      0.0000      1.0000 |
   11      0.0000      0.0000      1.0000 |
   12      0.0000      0.0000      1.0000 |
   13      0.0000      0.0000      1.0000 |
   14      0.0000      0.0000      1.0000 |
   15      0.0000      0.0000      1.0000 |
    \          \           \           \
  Inf.     0.0000      0.0000      1.0000 | 
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