Poisson Distribution

Poisson equation

The Poisson Distribution is a discrete distribution that is often grouped with the Binomial Distribution. The Poisson Distribution is quite useful when you desire to estimate the probabilities of events that occur randomly in some unit of measure (e.g., the number of traffic accidents at a particular intersection per month).

The Poisson constant

The Poisson constant is (2.71828)

The Poisson Equation

Poisson equation

Poisson equation

Poisson and Time Intervals

If the number of occurrences of some event follows a Poisson distribution, the time between successive occurrences will follow an Exponential Distribution.

In probability theory and statistics, the exponential distribution (a.k.a. radical exponential distribution) is the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. I



When to use the Poisson Distribution:

  • Examples of seeing or hearing the word ‘PER’:
  • Used in rates, frequencies, and flows
  • The Poisson Distribution may be applicable under the following conditions: <text>
    • The events occur at random during some interval of measure (e.g., ..per <xyz>)
    • Sample size is at least 16 (e.g., 16 months, 100 feet, megabyte, 180 days)
    • The population size is at least 10 times the sample size.
    • The probability of each occurrence is less than 0.1
    • The occurrence of the events are independent
    • The average number of occurrences per unit of measure is constant

Poisson Distribution Example Questions

When we hear “What is the probability of occurrences?” in a question, we know it’s time to use Poisson. Also, there are examples of seeing or hearing the word ‘PER’.

  • What is the probability of zero occurrences?
  • What is the probability that not one will occur?
  • Defects found in 100 feet of extruded plastic could be reworded to defects PER 100 feet of extruded plastic
  • Lost-time accidents reported a month could be reworded to lost-time accidents PER month
  • Bugs PER megabyte of code
  • Number of ‘sick days’ PER school year


Example 1 (nobody, Or fewer, standard deviation)

The average number of people on hold for technical support during working hours is 4. During working hours on any given day, what is:

A) The probability that there is nobody on hold.
B) That there are 4 or fewer people on hold?
C) What is the standard deviation for the people on hold?

Poisson 1

Example 2 (or more)

There is an average of 2.4 fatal car wrecks per week in a large city. In any given week in a large city, what is the probability that there are 3 or more fatal car accidents?

Poisson 2

Example 3 (Less than)

A farmer has an average of 7 weeds per acre on his farm during the summer months. On any acre in the summer, what is the probability that the farmer has less than 3 weeds?

Poisson 3


Example 4 (More than)

An industrial supply company averages 3.4 minor defects per vehicle sold. For any given vehicle, what is the probability that there are more than 5 minor defects?

Poisson 4

Example 5 (Exactly)

In the Pacific ocean, there is an average of 1.5 fish per gallon of water. If you looked at any given gallon of water in the Pacific Ocean, what is the probability that there would be exactly 3 fish?

Poisson 5

Other good examples here:



Standard Deviation for Poisson

Poisson Distribution Average and Standard Deviation

µ = σ^2  :  σ= Sqrt(µ)  : σ= Sqrt(average)

Six Sigma Black Belt Certification Poisson Distribution Questions:

Question: A process produces nonconformities according to a Poisson distribution. If the mean of the nonconformities is 25, what is the standard deviation?

(A) 2.5
(B) 5.0
(C) 12.5
(D) 25.0

Answer: 5.0  For Poisson distributions, the standard deviation is approximated by the square root of the mean. The square root of 25 is 5.

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