# Poisson Distribution – Formula, Graphs & Examples

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## Poisson Distribution – In a Nutshell

Used by statisticians to describe particular types of discrete probability distributions, the Poisson distribution is commonly used to describe a series of events occurring within a given time period. Given the average number of times any such event might occur over the time period concerned, Poisson distribution can be utilized for reliability engineering.1 Several conditions must be so if this form of statistical distribution is to be invoked:

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

The Poisson distribution is named after the French mathematician Siméon Denis Poisson who developed this model of probability distribution in the 19th century. In this article, you will learn what distinguishes this form of statistical distribution from other models, the formula that is needed to calculate it and what it looks like when displayed on a graph with numerous examples.

## Definition: Poisson distribution

In 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.2 3 As such, the formula that can be used to describe the Poisson distribution is as follows: It is important to note that the formula can be adapted to describe the average rate at which events occur rather than the average number of events that occur. If this is what is being statistically described, then the formula to use is: Note that ‘λ’ represents the average number of events while ‘r’ is used to describe the rate at which events occur.

## What is a Poisson distribution?

Statistically speaking, a Poisson distribution is used to define a discrete outcome. Simply put, a discrete outcome is something that can occur in a ‘yes’ or ‘no’ way.4

• If something does not occur – within a given time period – then it can be ascribed a zero value.
• If it occurs just once, it will be ascribed 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 barks at night, for instance.

Note that this distribution model does not offer an average figure for such occurrences but takes a sample that allows mathematicians to infer the likelihood – or otherwise – that an event may occur. As such, it tends to be useful for potentially rare events, such as extreme weather modelling, for example.

## 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 mean number of such deaths to be 0.61 per year.5 Note that in this example, under Poisson distribution:

• An event occurrence is defined as a death from a horse kick.
• The mean average, or ‘λ’, for an event is 0.61 per year.
• The measured time period 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 period.

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

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

Example:

Retail outlets can use it to analyze 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, that 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 that will occur in the given time period.

## Poisson distribution mean and variance

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

Alternatives to λ for Poisson distribution mean 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’ is Euler’s number.

Example – 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 that 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 that 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.

## FAQs

#### How can you know if data has a Poisson distribution?

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.

#### Can space be used instead of time in a Poisson distribution?

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.

#### How is this distribution model used in finance?

It can be used to help with anything from manpower planning to the number of expected product returns within a given period.6

#### Is Poisson distribution infinite?

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

## Sources

1 Nassar, Mazen, Sanku Dey and Saralees Nadarajah. “Reliability analysis of exponentiated Poisson-exponential constant stress accelerated life test model.” Wiley Online Library. April 26, 2021. https://onlinelibrary.wiley.com/doi/abs/10.1002/qre.2893

2 MathIsFun. “e (Euler’s Number).“ Accessed January 9, 2023. https://www.mathsisfun.com/numbers/e-eulers-number.html.

3 Factorial. “The factorial function.” Accessed January 9, 2023. https://factorialhr.co.uk/number-function-factorial.

4 Young, Julie. “Discrete Probability Distribution: Overview and Examples.” Investopedia. July 24, 2021. https://www.investopedia.com/terms/d/discrete-distribution.asp.

5 Teo, Bee Guan. “Poisson Distribution – From Horse Kick History Data to Modern Analytic.” Towards Data Science. December 5, 2020. https://towardsdatascience.com/poisson-distribution-from-horse-kick-history-data-to-modern-analytic-5eb49e60fb5f.

6 Hayes, Adam. “Poisson Distribution Formula and Meaning in Finance. Investopedia. May 19, 2022. https://www.investopedia.com/terms/p/poisson-distribution.asp.

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