What are the conditions of a Poisson experiment?

A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times [k] within a given interval of time or space.

The Poisson distribution has only one parameter, λ [lambda], which is the number of events. The graph below shows examples of Poisson distributions with different values of λ.

Table of contents

What is a Poisson distribution?

A Poisson distribution is a discrete probability distribution, meaning that it gives the probability of a [i.e., countable] outcome. For Poisson distributions, the discrete outcome is the number of times an event occurs, represented by k.

You can use a Poisson distribution to predict or explain the number of events occurring within a given interval of time or space. “Events” could be anything from disease cases to customer purchases to meteor strikes. The interval can be any specific amount of time or space, such as 10 days or 5 square inches.

You can use a Poisson distribution if:

  1. Individual events happen at random and independently. That is, the probability of one event doesn’t affect the probability of another event.
  2. You know the number of events occurring within a given interval of time or space. This number is called λ [lambda], and it is assumed to be constant.

When events follow a Poisson distribution, λ is the only thing you need to know to calculate the probability of an event occurring a certain number of times.

Examples of Poisson distributions

In general, Poisson distributions are often appropriate for count data. Count data is composed of observations that are non-negative integers [i.e., numbers that are used for counting, such as 0, 1, 2, 3, 4, and so on].

Horse kick deaths

One of the first applications of the Poisson distribution was by statistician Ladislaus Bortkiewicz. In the late 1800s, he investigated accidental deaths by horse kick of soldiers in the Prussian army. He analyzed 20 years of data for 10 army corps, equivalent to 200 years of observations of one corps.

The following histogram shows simulated data that are similar to what Bortkiewicz observed:

He found that a of 0.61 soldiers per corps died from horse kicks each year. However, most years, no soldiers died from horse kicks. On the other end of the spectrum, one tragic year there were four soldiers in the same corps who died from horse kicks.

Using modern terminology:

  • A death by horse kick is an “event.”
  • The time interval is one year.
  • The mean number of events per time interval, λ, is 0.61.
  • The number of deaths by horse kick in a specific year is k.

The army corps that Bortkiewicz observed were a sample of the population of all Prussian army corps. Because of the random nature of sampling, samples rarely follow a probability distribution perfectly. The deaths by horse kick in the sample approximately follow a Poisson distribution, so we can reasonably that the population follows a Poisson distribution.

Other examples of Poisson distributions

Since Bortkiewicz’s time, Poisson distributions have been used to describe many other things. For example, a Poisson distribution could be used to explain or predict:

  • Text messages per hour
  • Male grizzly bears per hectare
  • Machine malfunctions per year
  • Website visitors per month
  • Influenza cases per year

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Probability mass function graphs

A Poisson distribution can be represented visually as a graph of the probability mass function. A probability mass function is a function that describes a discrete probability distribution.

The most probable number of events is represented by the peak of the distribution—the .

  • When λ is a non-integer, the mode is the closest integer smaller than λ.
  • When λ is an integer, there are two modes: λ and  λ−1.

When λ is low, the distribution is much longer on the right side of its peak than its left [i.e., it is strongly ].

As λ increases, the distribution looks more and more similar to a normal distribution. In fact, when λ is 10 or greater, a normal distribution is a good approximation of the Poisson distribution.

Mean and variance of a Poisson distribution

The Poisson distribution has only one parameter, called λ.

  • The of a Poisson distribution is λ.
  • The variance of a Poisson distribution is also λ.

In most distributions, the mean is represented by µ [mu] and the variance is represented by σ² [sigma squared]. Because these two parameters are the same in a Poisson distribution, we use the λ symbol to represent both.

Poisson distribution formula

The probability mass function of the Poisson distribution is:

Where:

  • is a random variable following a Poisson distribution
  • is the number of times an event occurs
  • ] is the probability that an event will occur k times
  • is Euler’s constant [approximately 2.718]
  • is the average number of times an event occurs
  • ! is the factorial function
Example: Applying the Poisson distribution formulaAn average of 0.61 soldiers died by horse kicks per year in each Prussian army corps. You want to calculate the probability that exactly two soldiers died in the VII Army Corps in 1898, assuming that the number of horse kick deaths per year follows a Poisson distribution.

Calculation

The specific army corps [VII Army Corps] and year [1898] don’t matter because the probability is constant.

= 2 deaths by horse kick

= 0.61 deaths by horse kick per year

= 2.718

The probability that exactly two soldiers died in the VII Army Corps in 1898 is 0.101.

Practice questions

Frequently asked questions about Poisson distributions

What does “e” mean in the Poisson distribution formula?

The e in the Poisson distribution formula stands for the number 2.718. This number is called Euler’s constant. You can simply substitute e with 2.718 when you’re calculating a Poisson probability. Euler’s constant is a very useful number and is especially important in calculus.

What does lambda [λ] mean in the Poisson distribution formula?

In the Poisson distribution formula, lambda [λ] is the number of events within a given interval of time or space. For example, λ = 0.748 floods per year.

What is the difference between a normal and a Poisson distribution?

This table summarizes the most important differences between normal distributions and Poisson distributions:

CharacteristicNormalPoissonContinuous or discreteContinuousParameterMean [µ] and standard deviation [σ]Lambda [λ]ShapeBell-shapedDepends on λSymmetrySymmetricalAsymmetrical [right-skewed]. As λ increases, the asymmetry decreases.Range−∞ to ∞0 to ∞

When the of a Poisson distribution is large [>10], it can be approximated by a normal distribution.

What is a normal distribution?

In a normal distribution, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.

The measures of central tendency [mean, mode, and median] are exactly the same in a normal distribution.

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Turney, S. [2022, December 05]. Poisson Distributions | Definition, Formula & Examples. Scribbr. Retrieved January 3, 2023, from //www.scribbr.com/statistics/poisson-distribution/

What are the 3 conditions for a Poisson distribution?

A variable follows a Poisson distribution when the following conditions are true: Data are counts of events. All events are independent. The average rate of occurrence does not change during the period of interest.

What are the properties of a Poisson's experiment?

Properties of Poisson Distribution The events are independent. The average number of successes in the given period of time alone can occur. No two events can occur at the same time. The Poisson distribution is limited when the number of trials n is indefinitely large.

What is the key condition that a random variable must follow to use the Poisson distribution to model this variable?

If an event happens independently and randomly over time and the mean rate of occurrence is constant over time, then the number of occurrences in a fixed amount of time will follow the Poisson distribution.

Under what conditions will you use the Poisson and binomial distributions?

If, on the other hand, an exact probability of an event happening is given and you are asked to calculate the probability of this event happening k times out of n, then the Binomial Distribution must be used. The Poisson distribution can be derived as a limit of the binomial distribution.

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