What is statistics? – Poisson Probability Distribution

Poisson Probability Distribution: A little History
The Poisson Probability Distribution has taken its name after Siméon Denis Poisson (1781-1840). Mr Poisson was exploring the probability of the trial convictions that were wrongful. Moreover, Mr. Ladislaus Bortkiewicz (1898) found that “the number of Prussian soldiers that have died because they were kicked by a horse” follows also a Poisson distribution!

Simeon_Denis_Poisson_Ladislaus_Bortkiewicz_probability_distribution

Poisson Probability Distribution: Theoretical Definition
The Poisson Probability Distribution is a Discrete Probability distribution which represents random events which can happen in integral points of time or space (volume, area, or distance), and its occurrence is known. Also, a Poisson Probability Distribution can be a Binomial Probability Distribution when is counted the probability of non occurrence of the same event in the same time or space intervals. A such example can be the number of pink bicycles that can pass by outside your house (or not) in relation to the total number of bicycles that can pass by outside your house in e.g. one hour. Therefore, it is suitable to describe rare events that have a probability of occurrence enough that can be counted in standard intervals. Therefore, it is also called the “Law of Rare events”.

poisson_distribution_example_bicycles

Poisson Probability Distribution: Applications
The Poisson Probability Distribution is applied to numerous scientific fields such as Telecommunications, Astronomy (the rate of incoming photons in telescopes), in Biology (the number of mutations per DNA piece/length), in Insurances, in Seismography and many other scientific fields.

poisson_distribution_examples_biology_seismography_insurances

Poisson Probability Distribution: Statistical Definition
If we define a random Discrete variable X which follows a Poisson Probability Distribution, then the Probability Mass Function (pmf) for the Poisson Probability Distribution is the following one:

f(k;\lambda )=Pr(X=k)=\frac{\lambda^ke^{-1}}{k!}

The parameter E(X)=\lambda>0 takes always positive values, and it is the expected value of X, that is, the mean of the rate that X events occur.
The k=0,1,2,3...n takes always integral vales, as well the zero value.
The e=2.718 is the base of the natural logarithm, and it is also called Euler’s number.
The k!=k_1*k_2*...*k_n is factorial, that is, its members are multiplied e.g. 3!=1*2*3=6

Poisson Probability Distribution: Statistical Example
Let’s assume that you have visited the same beach for 100 days. Also, let’s assume that you have been stung by a urchin 20 times in these 100 days. Note that 5 random days, you have been stung by the urchin multiple times e.g. 2 times/day. What is the probability that a random day between these 100 days, the urchin has stung you only once ?

poisson_distribution_beach_example_urchin

In order to calculate the mean of times that you have been stung by a urchin these 100 days, we must multiple the times that you have been stung by the number of days that these stung-events happened. That is: 80 days you have not been stung by a urchin, 10 days you have been stung 1 time by the urchin, and 5 days you have been stung 2 times by the urchin. Then, we divide this result by the total number of frequency which is 100 (days):

E(X)=\mu=\lambda=0*80+1*10+2*5=0+10+10=20/100=0.2

And thus, \lambda=0.2. Because we are searching the probability the urchin has stung you only once in these 100 days: X=k=1. Then, by replacing formula’s symbols by the corresponding values, as a result, we have:

f(k;\lambda )=Pr(X=k)=\frac{0.2^1e^{-1}}{1!}=\frac{0.2*0.37}{1}=0.074 ή 7.4\%.

poisson_distribution_beach_example_urchin_2

Poisson Probability Distribution: Cumulative Probability Distribution
The Cumulative Probability Distribution for the Poisson Probability Distribution (cdf) is the following one:

cdf=e^{-\lambda} \sum_{i=0}^{\left \lfloor k \right \rfloor}\frac{\lambda^i}{i!}

Symbol Explanation
\sum: Indicate the summation of all the results that will be produced when the math operations related to every nth member (i) inside \sum have finished.
\left \lfloor k \right \rfloor: The maximum integral value of time or space, in relation to the event of interest
i: it takes integral values e.g. 0,1,2,3…n.

The pmf and cdf of the Poisson Probability Distribution for λ=2. It must be noted that the dashed lines exist for illustration reasons. It is a Discrete Probability Distribution, and thus, only integral values exist such as 1 and 2, no 1.3 or 1.6.
poisson_cdf_pmf Cumulative distribution function Probability mass function

Resources
Siméon Denis Poisson (1781-1840)
Ladislaus Bortkiewicz