Poisson Distribution


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A Poisson distribution, named for French mathematician Siméon Denis Poisson, predicts the occurrence of certain events in an interval (e.g., time, distance, volume) if you know how often the event has occurred. Most applications of the distribution are used to model the number of events happening in a fixed interval of time [1].

Poisson Distribution PMF and CDF

The Poisson distribution is discrete, which means it’s probability mass function (PMF) can only take on integer values of x (1, 2, 3, …).

graph poisson distribution PMF
Poisson distribution graph of PMF, valid only for integers on the horizontal axis. λ (also written as μ) is the expected number of event occurrences.

The Poisson Distribution Probability Mass Function (PMF) is

Where:

  • k = number of occurrences.
  • e = Euler’s number (   2.71828).
  • ! = the factorial function.
  • λ  (the shape parameter) = average (expected) number of events. Sometimes written as μ, this is a positive real number and is equal to:
    • The expected value of X and
    • The variance of X [2].

The cumulative distribution function is [2]

Properties

  • Mean: λ
  • Range 0 to ∞
  • Standard Deviation: √(λ)
  • Coefficient of Variation: 1/√(λ)
  • Skewness: 1/√(λ)
  • Kurtosis: 3 + (1/λ)

Poisson distribution vs. Binomial

It can be challenging to figure out if you should use a binomial distribution or a Poisson distribution. A rule of thumb: [3]

  • If you know the average probability of an event happening per unit (e.g., per unit of time, cycle, event) and you want to find probability of a certain number of events happening in a period of time (or number of events), then use the Poisson Distribution.
  • If you know the exact probability and you want to find the probability of the event happening a certain number of times out of x (e.g., 10 times out of 100, or 99 times out of 1000), use the Binomial Distribution formula.

References

[1] Haight, Frank A. (1967), Handbook of the Poisson Distribution, New York, NY, USA: John Wiley & Sons, ISBN 978-0-471-33932-8

[2] Engineering Statistics Handbook. Poisson Distribution. Online: https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm

[3] Stephanie Glen. “Poisson Curve: Simple Definition” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/probability-distribution/poisson-distribution/