Poisson Distribution: Decision-Making with Uncertainty

Key takeaways

• The Poisson distribution is a discrete probability distribution used to model the number of events in a fixed interval.

• The Poisson distribution formula is P(X = k) = (λ^k * e^(−λ)) / k!, hinging on the number of expected events (k) and the average number of expected events (λ).

• One of the differences between the Poisson distribution and the normal distribution is that in the Poisson distribution, the parameter lambda (λ) represents both the mean and the variance of the distribution, while the mean and variance vary independently in normal distributions.

It’s often appropriate to use a Poisson distribution when your events are independent, occur at a relatively consistent average rate, and you are counting how many events happen. Explore what the Poisson distribution is, when you should use it, real-world examples, and more.

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What is the Poisson distribution?

The Poisson distribution is a probability distribution that represents how many times a certain event is likely to happen within a fixed interval. For example, it might represent how many calls you’d expect a call center to receive in an hour, how many visits you’d predict a website to get in one week, how many people you’d expect to have a certain disease in one year, and so on [1].

What is lambda in Poisson distribution?

One of the central values of a Poisson distribution is the average rate, represented as lambda (λ), at which events occur. Based on this average, you can measure how likely different counts are around this average. For example, if the average number of website visits per week is 2,000, you can use a Poisson distribution to predict the likelihood of the website receiving 3,000 in a given week [2].

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

A Poisson distribution measures the count of events, meaning its outcomes are whole numbers, like the count of sales made, while normal distributions model continuous outcomes like height or temperature. Poisson distributions have an equal mean and variance, represented by λ, while normal distributions have means and variances that vary independently.

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Poisson distribution formula

The probability of observing “k” events is modeled by the Poisson probability mass function (PMF):

P(X = k) = (λ^k * e^(−λ)) / k!

In this formula, you have:

• k: the exact number of events

• λ: the average number of events

So, for example, if you expect to make 15 sales in a day, the λ would equal 15. If you then want to model the probability of making 20 sales in a day, k would equal 20, and you would model the equation as: P(X = 20) = (15^20 * e^(−15)) / 20! = 0.042 = 4.2 percent

This tells you that the probability of making 20 sales in a day in this scenario is 4.2 percent.

Properties to consider

When thinking about Poisson distributions, consider the following properties:

• Poisson distributions typically describe and analyze events that are relatively rare.

• λ represents both the mean and the variance and is the only parameter.

• The smaller the λ, the more biased the distribution will be.

• At λ=20, the distribution tends to approach normality.

When should you use Poisson distribution?

In general, you can choose to use a Poisson distribution to model how many times something happens in a fixed interval when events are reasonably independent and steady. It is typically best chosen for scenarios where events are relatively rare compared to the number of chances for them to happen, and where you don’t have a fixed upper limit on the count.

Assumptions

The main assumptions to keep in mind for this distribution are:

1) Events occur independently

2) Two events cannot happen at the same instant

3) Probability is proportional to interval length

Limitations

Because Poisson distributions assume the mean and variance are equal, they can become less accurate in situations where the counts are more spread than the model predicts, the events cluster together or depend on one another, or the event rate shifts over time.

In some cases, you might opt for alternative models. For example, if your counts are more spread than a Poisson model can account for, this is “overdispersion.” In this case, alternatives like the negative binomial distribution may fit better.

Real-world examples

To start seeing where a Poisson distribution might come up in everyday life, consider places where it could be helpful to measure and predict how many events happen in a given amount of time or space.

For example, professionals in related industries might find it helpful to model different types of event counts. A health professional might model the number of patients diagnosed with a certain condition in a year. A marketing professional might model the number of website visits in an hour. A business owner might predict the number of customer support tickets received in a day to make hiring decisions. Other professionals might track the:

• Number of defects found in a production run

• Number of insurance claims made in a year

• Number of beaver habitats in one mile of river

• Number of flowering plants in one square mile

• Number of arrivals at a clinic, store, or service center

Binomial distribution vs. Poisson distribution

A binomial distribution counts successes across a fixed number of trials that each have the same probability (p), while a Poisson distribution counts events that occur over an interval with no fixed number of trials or upper limit. For example, a binomial distribution might model the likelihood of getting six heads when you flip a coin 10 times, while a Poisson distribution might measure the likelihood that 10 people walk into a clinic in one hour if the average number of people per hour is six [3].

How to start learning the Poisson distribution

One good way to learn to use the Poisson distribution is to practice with real data. You can come up with your own practice problems by picking an average value, like 10 emails per day or 20 people arriving at a clinic. That is your λ.

Based on this, practice using the equation for different scenarios. How likely would it be to then receive eight emails in a day? What about 15? How likely would it be that two people arrived at the clinic? What about 22?

From there, you can start expanding to a broader set of probability distributions, such as binomial or Bernoulli, to understand how they relate to each other or differ.

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