Poisson Distribution Calculator

This calculator finds the probability of an event occurring a certain number of times (﻿k
 in a fixed interval of time or space when the mean number of events﻿(λ)
 is known. This type of probability distribution is known as a Poisson distribution.
CalculationInputs﻿﻿λ
:7.00​
﻿
﻿﻿k
:4.00​
﻿
﻿
Output﻿﻿P (X=k)
:0.09​
﻿
﻿
P(X=k)=e−λλkk!P (X = k) = \dfrac{e^{-\lambda} \lambda^k}{k!}P(X=k)=k!e−λλk​
Where:
﻿﻿X
 is a random variable following a Poisson distribution (unitless)
﻿﻿k
 is the number of times an event occurs (unitless)
﻿﻿λ
 is the mean number of times an event occurs (unitless)
﻿﻿e
 is Euler's constant (approximately 2.718)
﻿﻿!
 is the factorial function 
﻿﻿P(X=k)
 is the probability mass function, the probability that an event will occur ﻿k
 times 
ExplanationThe Poisson distribution is a discrete probability distribution that describes the probability of an event occurring a certain number of times (﻿k
) in a fixed interval of time or space when the mean number of events (﻿λ
) 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 ﻿λ
 is:
a non-integer, the mode is the closest integer smaller than ﻿λ
﻿
an integer, there are two modes: ﻿λ
 and ﻿λ−1
﻿
When ﻿λ
 is low, then the distribution is strongly rightly skewed. As ﻿λ
 increases, the distribution looks more balanced. In fact as  ﻿λ→∞
, Poisson distribution tends to a normal distribution. 
Related Resourceshttps://www.scribbr.com/statistics/poisson-distribution/
https://www.geeksforgeeks.org/poisson-distribution/
https://www.statology.org/poisson-distribution-real-life-examples/