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

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Practice Questions on Poisson Distribution

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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.