Poisson Distribution Formula

Poisson Distribution Formula

The Poisson distribution, commonly referred to as the Poisson distribution probability mass function, is a theoretical discrete probability. It is used to determine the likelihood of an independent event occurring at a certain rate over a fixed period of time. Other fixed intervals, like volume, area, distance, etc., can also be employed with the Poisson distribution probability mass function. If there are few successes over many attempts, a Poisson random variable will reasonably reflect the occurrence. When the trials are large indefinitely, the Poisson distribution is utilised as a limiting instance of the binomial distribution. The same binomial phenomena is modelled by a Poisson distribution, where np takes the place of λ . Denis Poisson, a French mathematician, is honoured by the name Poisson distribution.

What is Poisson Distribution?

The definition of a poisson distribution is used to simulate a discrete probability of an event when independent occurrences happen at a known constant mean rate over a set period of time. In other words, the Poisson distribution is used to calculate the likelihood that an event will happen a certain number of times during a certain time frame. The Poisson rate parameter, which is denoted by the symbol λ, shows the anticipated value of the typical number of events within the set time interval. Both the business and biological areas make extensive use of the poisson distribution.

Use an example to try to better grasp this: a customer service centre receives 100 calls each hour, eight hours per day. The calls are independent of one another, as we can see. A Poisson probability distribution describes the likelihood of the number of calls per minute. No matter how many calls were received in the minute before, any number of calls can be made per minute. Based on an understanding of the Poisson distribution, the call centre may raise its standards for customer care by adding more services and attending to the demands of its clients if they were to determine the likelihood that more than 150 calls could be received every hour.

Poisson Distribution Formula

When the mean rate of occurrence is constant throughout time, the Poisson Distribution Formula is used to determine the likelihood of an event occurring independently, discreetly, over a certain time period. When many alternative outcomes exist, the Poisson Distribution Formula is used. When x is the average rate of value and X is a discrete random variable with a Poisson distribution, the probability of x is given by: f(x) = P(X=x) = (e-x)/x!


The Euler number is e, which equals 2.718.

is the predicted value’s average rate, = variance, and >0.

Poisson Distribution Mean and Variance

The mean of the Poisson Distribution Formula and the variance value will be the same for a Poisson distribution with the average rate over a defined period of time. It can therefore be claimed that both the mean and the variance of the distribution for X having a Poisson distribution.

Consequently, E(X) = V(X) = λ


The anticipated mean is E(X).Variance V(λ ) is greater than zero.

Properties of Poisson Distribution

When an event has a lot of unusual and independent alternative outcomes, the Poisson distribution might be used. The Poisson Distribution has the following characteristics. The Poisson distribution has

The occurrences are separate.

Only the typical number of successes in the allotted time can happen. Events cannot take place simultaneously.

When n trials are infinitely big, the Poisson distribution is constrained.

Mean = variance = λ

np =, where is a constant, is a finite equation.

The square root of the mean is always equal to the standard deviation.

 P(X= a) = μa / a! e -μ- gives the precise probability that the random variable X has a mean of a.

The Poisson distribution resembles a normal distribution when the mean is high.

Poisson Distribution Table

One can rapidly determine the probability mass function of an event that follows the Poisson distribution by using a Poisson distribution table, which is similar to the binomial distribution. For various values of, where >0, the Poisson distribution table displays various Poisson distribution values. The probability mass function has a value of 0.6065, or 60.65%, as shown in the table below for P(X = 0) and = 0.5.

Applications of Poisson Distribution

The Poisson distribution has a wide range of uses. The following random variables have a Poisson distribution:

To determine the quantity of flaws in a finished product

To determine the total number of fatalities caused by any sickness or disaster in a nation

To count the amount of diseased plants in the field, the amount of bacteria in living things, or the amount of radioactive decay in atoms

To figure out how long the events will be apart.

Poisson Distribution Examples

Extramarks is an online learning platform that concentrates on the segments of pre-school, K–12, higher education, and Test Prep Through the expert fusion of education and technology. Extramarks’ support students’ academic development in every manner. It offers all of the tools students need to succeed on any in-person, board, or competitive exam. In addition to the solved examples on the Poisson Distribution Formula, it offers students a comprehensive range of study resources, such as K12 preparation materials, NCERT Textbook Solutions, solved sample papers, and more. The two most crucial study materials for examination preparation are solved examples  and practice papers. 

Students can organise their key ideas and avoid chapter problems by using Extramarks’ provided solved examples on the Poisson Distribution Formula. Students who have online accounts can access it whenever they want to study from wherever. Students can benefit greatly from Extramarks’ solved examples. The Poisson Distribution Formula solved examples are available on Extramarks, a trusted website, so they do not need to go elsewhere.

The Poisson Distribution Formula solved examples are available on the Extramarks website for students to use in exam preparation. Students can improve their preparation by frequently using the Extramarks Poisson Distribution Formula solved examples. If students want to do well on their exams, they must fully understand the solved examples. Students will not have any issue understanding the chapter thanks to the thorough Poisson Distribution Formula solved examples on the Extramarks website. To perform well in the exam, students should be more familiar with the concepts covered in the Poisson Distribution Formula solved examples.

The Extramarks provide visual and interactive learning modules that include the  solved examples. By offering assignments, projects, essays, quizzes, or other tasks, Extramarks also improves the effectiveness of home study. The Poisson Distribution Formula solved examples can be found on the Extramarks website. Many students find the Poisson Distribution Formula to be difficult but with the help of the  solved examples created by subject matter experts at Extramarks, students can comprehend them easily. Reading the  solved examples is the first and most crucial thing that students need to do in order to get ready for the Poisson Distribution Formula.

Practice Questions on Poisson Distribution

Students can learn at their own pace using the self-learning platform, Extramarks. The textbooks do not include all of the solutions that are present in them, so Extramarks is providing the solved examples. They also provide students practice questions on the Poisson Distribution Formula so that they can get an idea of how questions may appear in the examination. The Extramarks website’s Poisson Distribution Formula practice questions helps students fully practice each concept and deepen their grasp of it. Students may solve problems more quickly and accurately with the help of the Poisson Distribution Formula practice questions. If students consistently and thoroughly practice the Poisson Distribution Formula practice questions, they can attain outstanding examination results.

If they wish to swiftly learn each topic and perform well, students should thoroughly practice all of the Poisson Distribution Formula practice questions, including the questions from the exercises. Extramarks’ subject-matter specialists have created all of the practice questions to help students pass with distinction. Keeping in mind the school marking systems, the Extramarks website provides the best practice questions. It offers the Poisson Distribution Formula practice questions with solutions so that students do not face any difficulty. Students can get all of the practice questions as needed. Because they may use the practice questions to thoroughly understand all of the answers without investing a lot of time, students should pay great attention to them.

Students can also get the Poisson Distribution Formula practice questions in the PDF format on the Extramarks’ website. The Poisson Distribution Formula practice questions are included in this PDF along with thorough solutions. The purpose of the solution resources is to eliminate all of the students’ uncertainties. In addition to the Poisson Distribution Formula practice questions, Extramarks also includes additional problems with the NCERT solutions in the PDF to ensure that the students do not overlook anything when studying for the examinations.

Using the Poisson Distribution Formula practice questions offered by the Extramarks website helps students learn more swiftly and efficiently. Students can access chapter-by-chapter worksheets and past years’ papers from Extramarks in addition to the practice questions. It also monitors how students are learning and assists them if necessary. 

Maths Related Formulas
Z Score Formula Slope Intercept Form Formula
Unit Vector Formula Antiderivative Formula
Surface Area Of A Cone Formula Cosec Cot Formula
Area Of A Pentagon Formula Decay Formula
Bayes Theorem Formula Implicit Differentiation Formula
Definite Integral Formula Linear Correlation Coefficient Formula
Ellipse Formula Point Slope Form Formula
Half Life Formula Side Angle Side Formula
Inverse Matrix Formula Volume Of An Ellipsoid Formula
Lcm Formula Surface Area Of A Square Pyramid Formula
FAQs (Frequently Asked Questions)

1. What circumstances apply the Poisson distribution?

When an interesting variable is a discrete count variable, poisson distributions are used. Many financial and economic data are presented as count variables, such as the frequency of unemployment in a given year, which makes them amenable to study using a Poisson distribution.

2. Poisson is it continuous or discrete?

A discrete probability distribution is a Poisson distribution. It provides the likelihood that an event will occur a specific number of times (k) over a predetermined period of time or area.