Poisson Distribution Formula

Poisson Distribution Formula (Table of Contents)

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What is the Poisson Distribution Formula?

In Probability and Statistics, there are three types of distributions based on continuous and discrete data – Normal, Binomial and Poisson Distributions. Normal Distribution is often as a Bell Curve. Poisson distribution often referred to as Distribution of rare events. This is predominantly used to predict the probability of events that will occur based on how often the event had happened in the past. It gives the possibility of a given number of events occurring in a set of period. It is used in many real-life situations.

Formula to find Poisson distribution is given below:

P(x) = (e^-λ * λ^x) / x!

For x=0, 1, 2, 3…

This experiment generally counts the number of events happened in the area, distance or volume. Along with this, one can find the Chain of events which is nothing but the chain of occurrences of the same event over the particular period of time. The Poisson distribution has the following common characteristics.

An event can happen any number of time at any time.
The event can consider any measures like volume, area, distance and time.
However, the probability of an event happening in any measures specified above is the same.
Each event is not dependent on all other events which mean the probability of an event happening does not affect other event happening at the same time.

Examples of Poisson Distribution Formula

Let’s take an example to understand the calculation of the Poisson Distribution in a better manner.

You can download this Poisson Distribution Formula Excel Template here – Poisson Distribution Formula Excel Template

Poisson Distribution Formula – Example #1

The average number of yearly accidents happen at a Railway station platform during train movement is 7. To identify the probability that there are exactly 4 incidents at the same platform this year, Poisson distribution formula can be used.

Solution:

Poisson Distribution is calculated using the formula given below

P(x) = (e^-λ * λ^x) / x!

P (4) = (2.718^-7* 7⁴⁾ / 4!
P (4) = 9.13%

For the given example, there are 9.13% chances that there will be exactly the same number of accidents that can happen this year.

Poisson Distribution Formula – Example #2

The number of typing mistakes made by a typist has a Poisson distribution. The mistakes are made independently at an average rate of 2 per page. Find the probability that a three-page letter contains no mistakes.

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Here average rate per page = 2 and average rate for 3 pages (λ) = 6

Solution:

Poisson Distribution is calculated using the formula given below

P(x) = (e^-λ * λ^x) / x!

P (0) = (2.718^-6* 6⁰) / 0!
P (0) = 0.25%

Hence there is 0.25% chances that there will be no mistakes for 3 pages.

Note: x^{0 =} 1 (any value power 0 will always be 1); 0! = 1 (zero factorial will always be 1)

Explanation

Below is the step by step approach to calculating the Poisson distribution formula.

Step 1: e is the Euler’s constant which is a mathematical constant. Generally, the value of e is 2.718.

Step 2: X is the number of actual events occurred. It can have values like the following. x = 0,1,2,3…

Step 3: λ is the mean (average) number of events (also known as “Parameter of Poisson Distribution). If you take the simple example for calculating λ => 1, 2,3,4,5. If you apply the same set of data in the above formula, n = 5, hence mean = (1+2+3+4+5)/5=3. For a large number of data, finding median manually is not possible. So it is essential to use the formula for a large number of data sets. Here in calculating Poisson distribution, usually we will get the average number directly. Based on the value of the λ, the Poisson graph can be unimodal or bimodal like below.

Step 4: x! is the Factorial of actual events happened x. Below is an example of how to calculate factorial for the given number.

If you take the simple example for calculating Factorial of the real data set => 1, 2,3,4,5.

x! = x * (x-1) * (x-2) * (x-3) *…… 3*2*1
5! = 5 * (5-1) * (5-2) * (5-3) * (5-4)
5! = 5*4*3*2*1
5! = 120

Relevance and Uses of Poisson Distribution Formula

Poisson distribution can work if the data set is a discrete distribution, each and every occurrence is independent of the other occurrences happened, describes discrete events over an interval, events in each interval can range from zero to infinity and mean a number of occurrences must be constant throughout the process. Depending on the value of Parameter (λ), the distribution may be unimodal or bimodal. The Poisson distribution is a discrete distribution, means the event can only be stated as happening or not as happening, meaning the number can only be stated in whole numbers. Fractional occurrences of the event are not part of this model. The outcome results can be classified as success or failure. This is widely used in the world of:

Data Analytics for Predictive Analysis of Data
Stock Market predictions
Sales Market predictions
Supply and Demand chain predictions
Readily available in Amazon Web Services (AWS) platforms
Review and evaluating business insurance coverage

Other applications of the Poisson distribution are from more open-ended problems. For example, it may be used to help determine the minimum amount of resourcing needed in a call center based on average calls received and calls on hold. In short, the list of applications can be added more and more, as it is used worldwide practical statistical purpose.