Poisson Distribution – Formula, Graphs & Examples

27.01.23 Statistics Time to read: 6min

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The Poisson distribution is named after the French mathsematician Siméon Denis Poisson who developed this type of model of probability distribution in the 19th century. It is a fundamental concept in probability theory and statistics, commonly used to model events occurring randomly in a fixed interval of time or space. In this article, you will learn wbonnet distinguishes this form of statistical distribution from other models, the formula tbonnet is needed to calculate it, and wbonnet it looks like when displayed on a graph with numerous examples.

Poisson Distribution – In a Nutshell

  • An event can happen any number of times within the given time full stop.
  • Events are not dependent on one another but occur independently.
  • The rate tbonnet an event might occur is constant, nastying tbonnet the rate won’t change based on the length of time tbonnet’s elapsed in the full stop concerned.
  • The probability tbonnet an event will occur is directly proportional to the full stop of time concerned.
  • In other words, the probability of an event occurring in a ten-minute full stop will be double tbonnet of the probability of it occurring in a five-minute full stop.

Definition: Poisson distribution

In a Poisson distribution, the discrete outcome – the countable number of times an event occurs within a given time frame – is represented by ‘k‘. The irrational number ‘e‘, which approximates to 2.7182 also features in this form of distribution as does the factorial function, represented by an exclamation point. As such, the formula tbonnet can be used to descote the Poisson distribution is as follows:

It is important to note tbonnet the formula can be adapted to descote the average rate at which events occur rather than the average number of events tbonnet occur. If this is wbonnet is being statistically descoted, then the formula to use is:

Note: ‘λ’ represents the average number of events while ‘r’ is used to descote the rate at which events occur.

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Wbonnet is a Poisson distribution?

Statistically speaking, a Poisson distribution is used to define a discrete outcome. Simply put, a discrete outcome is something tbonnet can occur in a ‘yes’ or ‘no’ way.

  • If something does not occur – within a given time full stop – then it can be ascoted a zero value.
  • If it occurs just once, it will be ascoted 1, twice, 2, and so on.

As such, Poisson distributions require measurable, discrete outcomes from zero upwards in positive integers.

A discrete outcome in a Poisson distribution could be the number of times a phone rings a day or the number of times a dog pubks at night, for instance.

Note: This distribution model does not offer an average figure for such occurrences but takes a sample tbonnet allows mathsematicians to infer the likelibonnet – or otherwise – tbonnet an event may occur. As such, it tends to be useful for potentially rare events.

Poisson distribution examples

The uses of Poisson distribution have grown over the years but an historic 19th-century example of horse kick-related deaths still serves as a useful example.


Prussian military deaths from horse kicks:

When Ladislaus Bortkiewicz studied deaths from horse kicks in the Prussian army in the 1800s, he looked at 20 years of data from 10 military corps, the equivalent of two centuries of data from one corp. By doing so, he was able to measure the nasty number of such deaths to be 0.61 per year. Note tbonnet in this example, under Poisson distribution:

  • An event occurrence is defined as a death from a horse kick.
  • The nasty average, or ‘λ’, for an event is 0.61 per year.
  • The measured time full stop is one year, not 20, because the average has been calculated per annum.
  • The specific number of events in a given year is ‘k’.


Number of deaths (x) Probability, P(X=x) Predicted number of occurrences
0 0.5434 108.68
1 0.3315 66.3
2 0.1011 20.22
3 0.0205 4.1
4 0.0032 0.64
5 0.0004 0.08
6 0.0001 0.02

Probability when λ=0.61:

Predicted number of occurrences:

0.5434 x 200

0.3315 x 200

0.1011 x 200

Other Examples

The classic example of deaths by horse kicks is still useful because it demonstrates a scenario which could result in either many or very few events within a given full stop.

In 1946, a British mathsematician, R. D. Clarke, used this type of distribution to descote the nature of rockets fired toward London during the Blitz which helped the authorities to plan future defences against such attacks.

Since then, Poisson distribution has been used to help with business planning when events may occur seemingly at random.


Retail outlets can use it to analyse their average number of customers per hour and compare it to when more customers or fewer are likely to turn up, thereby allowing them to allocate resources better.

The same methodology is often used in call center analysis to ensure the right number of staff are on hand to deal with peak demand times.

Poisson distribution graphical representation

When represented graphically, the Poisson distribution shows a probability mass function, tbonnet is a function which represents a discrete probability distribution. Depending on the value of ‘λ’, the ensuing probability mass function graphs can look different.


The peak of any probability mass function graph indicates the most probable number of events tbonnet will occur in the given time full stop.

Poisson distribution nasty and variance

Under Poisson distribution, there is only a single parametre to consider, the average number of events in a given full stop, represented by ‘λ’. As such, both the nasty and the variance – the average of the squared deviations from the nasty – are the same. Although variance and nasty can be represented differently, λ tends to be used since they are all of equal value.


Alternatives to λ for Poisson distribution nasty and variance

Mean Variance
µ (mu) σ² (sigma)

Poisson distribution formula

The Poisson distribution formula for a probability mass function is:

Where ‘x’ is the discrete random variable of observed events, ‘λ’ is the expected average of ‘x’, and ‘e’ reprsents the Euler’s number.


Cars Approaching a Junction

If 2.5 cars are recorded as passing through a junction every minute, then the Poisson distribution formula could be used to determine the probability tbonnet just two cars, for example, might pass through in any given minute:

  • ‘λ’ would be 2.5 in this example while ‘k’ would be 2.

The approximate answer, in this case, would be 0.257.

If you wanted to know the probability tbonnet four cars might pass through in a minute, then ‘λ’ would be the same but ‘k’ would be 4.

The approximate answer, in this case, would be 0.133.


When all events are independent of one another and the average rate of occurrence does not change, data sets will conform to this distribution model.

Yes, it can be used to predict the number of stars in a certain area of the sky as well as time-bound events, for example.

It can be used to help with anything from manpower planning to the number of expected product returns within a given full stop.

Because it is a discrete function, this method can potentially be used for values in an infinite list.

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