A Quick Guide to the Poisson Distribution

Actuarial Concepts
Written By Eryk Faracik

The Poisson distribution is widely used to model insurance claims. It is a discrete probability distribution with its mean equal to its variance. Knowing its formula and properties is essential when preparing for actuarial exams.

Whether you're analyzing insurance claims, call center traffic, or rare events in sports, understanding the Poisson distribution can simplify complex problems.

Here's a brief overview of its key facts and practical insights.

I’ll go through the formula of the Poisson distribution, and its characteristics, concluding with a calculation example.

What is the Poisson Distribution?

The Poisson distribution models the number of events that occur in a fixed interval, given that these events happen independently and at a constant average rate.

It’s a discrete probability distribution, so the function that describes its distribution is called the probability mass function (PMF), not the probability density function (PDF), as in the continuous case.

PMF for the Poisson distribution is given by the formula:

Formula 1. PMF for the Poisson distribution.

Key Characteristics

Discrete Distribution

The Poisson distribution only takes integer values (0, 1, 2,...).

The plot of the Poisson distribution is presented as dots linked with the line.

Image 1. Plot of the Poisson distribution for different values of the parameter λ.

Mean = Variance

The expected value (λ) is also the variance, a defining property.

  • Since the variance equals the mean, a higher expected value implies greater variability. For instance, if λ=2, outcomes will typically cluster closely around 2. However, if λ=100, outcomes will vary significantly, reflecting the broader spread.

  • Unlike distributions with constant variance, the Poisson distribution’s variability grows as the mean increases. This is crucial when modeling rare events that become more frequent over time (e.g., insurance claims in a growing customer base).

  • The Poisson distribution is inherently skewed, especially for small λ values. As λ increases, the distribution shape becomes more symmetric and resembles a normal distribution — a property known as the Poisson limit theorem.

  • Since the Poisson variance scales with the mean, it's particularly useful as an approximation for binomial distributions in cases with a large number of trials and low probability of success.

Events are Independent

One event occurring doesn’t affect the likelihood of another.

Also, the sum of two or more Poisson random variables follows the Poisson distribution, with the expected number of occurrences equal to the sum of the expected number of occurrences.

Formula 2. The sum of the Poisson random variables follows the Poisson distribution.

Example

Imagine an insurance company observes an average of 3 car accidents per day in a certain city. What’s the probability of exactly 5 accidents occurring tomorrow?

The expected number of car accidents in one day is 3.

Therefore, we can say that the number of car accidents in one day follows a Poisson distribution with parameter λ=3. Let’s denote this variable as X.

The probability of exactly 5 accidents can be written as:

Therefore, the desired probability is equal to:

So, there’s about a 10% chance of exactly 5 accidents.

Summary

The Poisson distribution is a simple yet powerful model for random, independent events. Its versatility makes it a cornerstone in actuarial science, statistics, and beyond.

Whether estimating claims or forecasting rare events, mastering this distribution can greatly enhance your analytical toolkit.

After this quick guide, you should have a basic understanding of what the Poisson distribution is. However, if you’d like to know more — which I encourage you to do — I've included a couple of links for reference. Have fun!

Eryk Faracik
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