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Poisson Distribution Calculator

This calculator finds the probability of an event occurring a certain number of times (

in a fixed interval of time or space when the mean number of events

is known. This type of probability distribution is known as a Poisson distribution.

Calculation

Inputs



λ
:7.00​



k
:4.00​



Output



P (X=k)
:0.09​


P(X=k)=e−λλkk!P (X = k) = \dfrac{e^{-\lambda} \lambda^k}{k!}
Where:
  1. 
    
    is a random variable following a Poisson distribution (unitless)
  2. 
    
    is the number of times an event occurs (unitless)
  3. 
    
    is the mean number of times an event occurs (unitless)
  4. 
    
    is Euler's constant (approximately 2.718)
  5. 
    
    is the factorial function
  1. 
    
    is the probability mass function, the probability that an event will occur
    
    times

Explanation

The Poisson distribution is a discrete probability distribution that describes the probability of an event occurring a certain number of times (

) in a fixed interval of time or space when the mean number of events (

) is known. It applies to problems in which random events occur independently of each other at a known average rate.
A Poisson distribution can be represented visually as a graph of the probability mass function. A probability mass function is a function that describes a discrete probability distribution.
Poisson distributions of three random independent events, based on their λ values

The most probable number of events is represented by the peak of the distribution, called the mode. When

is:
  1. a non-integer, the mode is the closest integer smaller than
    
    
  2. an integer, there are two modes:
    
    and
    
    
When

is low, then the distribution is strongly rightly skewed. As

increases, the distribution looks more balanced. In fact as

, Poisson distribution tends to a normal distribution.

Related Resources

  1. https://www.scribbr.com/statistics/poisson-distribution/
  1. https://www.geeksforgeeks.org/poisson-distribution/
  2. https://www.statology.org/poisson-distribution-real-life-examples/