The **Poisson distribution **is one of the most popular distributions in statistics.

To understand the Poisson distribution, it helps to first understand Poisson experiments.

**Poisson Experiments**

A **Poisson experiment** is an experiment that has the following properties:

- The number of successes in the experiment can be counted.
- The mean number of successes that occurs during a specific interval of time (or space) is known.
- Each outcome is independent.
- The probability that a success will occur is proportional to the size of the interval.

One example of a Poisson experiment is the number of births per hour at a given hospital. For example, suppose a particular hospital experiences an average of 10 births per hour. This is a Poisson experiment because it has the following four properties:

- The number of successes in the experiment can be counted – We can count the number of births.
- The mean number of successes that occurs during a specific interval of time is known – It is known that an average of 10 births per hour occur.
- Each outcome is independent – The probability that one mother gives birth during a given hour is independent of the probability of another mother giving birth.
- The probability that a success will occur is proportional to the size of the interval – the longer the interval of time, the higher the probability that a birth will occur.

We can use the Poisson distribution to answer questions about probabilities regarding this Poisson experiment such as:

- What is the probability that more than 12 births occur in a given hour?
- What is the probability that less than 5 births occur in a given hour?
- What is the probability that between 8 to 11 births occur in a given hour?

**The Poisson Distribution**

The **Poisson distribution** describes the probability of obtaining *k* successes during a given time interval.

If a random variable *X* follows a Poisson distribution, then the probability that *X* = *k* successes can be found by the following formula:

**P(X=k) = λ ^{k} * e^{– λ} / k!**

where:

**λ:**mean number of successes that occur during a specific interval**k:**number of successes**e:**a constant equal to approximately 2.71828

For example, suppose a particular hospital experiences an average of 2 births per hour. We can use the formula above to determine the probability of experiencing 0, 1, 2, 3 births, etc. in a given hour:

**P(X=0) **= 2^{0} * e^{– 2} / 0! = **0.1353**

**P(X=1) **= 2^{1} * e^{– 2} / 1! = **0.2707**

**P(X=2) **= 2^{2} * e^{– 2} / 2! = **0.2707**

**P(X=3) **= 2^{3} * e^{– 2} / 3! = **0.1805**

We can calculate the probability for any number of births up to infinity. We create then create a simple histogram to visualize this probability distribution:

**Calculating Cumulative Poisson Probabilities**

It’s straightforward to calculate a single Poisson probability (e.g. the probability of a hospital experiencing 3 births during a given hour) using the formula above, but to calculate cumulative Poisson probabilities we need to add individual probabilities.

For example, suppose we want to know the probability that the hospital experiences 1 or fewer births in a given hour. We would use the following formula to calculate this probability:

**P(X≤1)** = P(X=0) + P(X=1) = 0.1353 + 0.2707 = **0.406**

This is known as a **cumulative probability** because it involves adding more than one probability. We can calculate the cumulative probability of experiencing *k* or less births in a given hour using a similar formula:

**P(X≤0)** = P(X=0) = **0.1353**

**P(X≤1)** = P(X=0) + P(X=1) = 0.1353 + 0.2707 = **0.406**

**P(X≤2)** = P(X=0) + P(X=1) + P(X=2) =0.1353 + 0.2707 + 0.2707 = **0.6767**

We can calculate these cumulative probabilities for any number of births up to infinity. We can then create a histogram to visualize this cumulative probability distribution:

**Properties of the Poisson Distribution**

The Poisson distribution has the following properties:

The mean of the distribution is** λ**.

The variance of the distribution is also **λ**.

The standard deviation of the distribution is **√λ**.

For example, suppose a hospital experiences an average of 2 births per hour.

The mean number of births we would expect in a given hour is λ = 2 births.

The variance in the number of births we would expect is λ = 2 births.

**Poisson Distribution Practice Problems**

Use the following practice problems to test your knowledge of the Poisson distribution.

**Note: **We will use the Poisson Distribution Calculator to calculate the answers to these questions.

**Problem 1**

**Question: **It is known that a certain website makes 10 sales per hour. In a given hour, what is the probability that the site makes exactly 8 sales?

**Answer:** Using the Poisson Distribution Calculator with λ = 10 and x = 8, we find that P(X=8) = **0.1126**.

**Problem 2**

**Question: **It is known that a certain realtor makes an average of 5 sales per month. In a given month, what is the probability that she makes more than 7 sales?

**Answer:** Using the Poisson Distribution Calculator with λ = 5 and x = 7, we find that P(X>7) = **0.13337**.

**Problem 3**

**Question: **It is known that a certain hospital experience 4 births per hour. In a given hour, what is the probability that 4 or less births occur?

**Answer:** Using the Poisson Distribution Calculator with λ = 4 and x = 4, we find that P(X≤4) = **0.62884**.

**Additional Resources**

The following articles explain how to work with the Poisson distribution in different statistical software:

How to Use the Poisson Distribution in R

How to Use the Poisson Distribution in Excel

How to Calculate Poisson Probabilities on a TI-84 Calculator

Real-Life Examples of the Poisson Distribution

Poisson Distribution Calculator

Really awesome articles. Give proper intuition about distributions rather than limiting them to rigorous maths.