## What Is a Poisson Distribution?

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution, the variable can only take whole number values (0, 1, 2, 3, etc.), with no fractions or decimals.

### Key Takeaways

- A Poisson distribution, named after French mathematician Siméon Denis Poisson, can be used to estimate how many times an event is likely to occur within "X" periods of time.
- Poisson distributions are used when the variable of interest is a discrete count variable.
- Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

## Understanding Poisson Distributions

A Poisson distribution can be used to estimate how likely it is that something will happen "X" number of times. For example, if the average number of people who buy cheeseburgers from a fast-food chain on a Friday night at a single restaurant location is 200, a Poisson distribution can answer questions such as, "What is the probability that more than 300 people will buy burgers?" The application of the Poisson distribution thereby enables managers to introduce optimal scheduling systems that would not work with, say, a normal distribution.

One of the most famous historical, practical uses of the Poisson distribution was estimating the annual number of Prussian cavalry soldiers killed due to horse-kicks. Modern examples include estimating the number of car crashes in a city of a given size; in physiology, this distribution is often used to calculate the probabilistic frequencies of different types of neurotransmitter secretions. Or, if a video store averaged 400 customers every Friday night, what would have been the probability that 600 customers would come in on any given Friday night?

## The Formula for the Poisson Distribution Is

Where:

*e*is Euler's number (*e*= 2.71828...)*x*is the number of occurrences*x*! is the factorial of*x*- λ is equal to the expected value (EV) of
*x*when that is also equal to its variance

Given data that follows a Poisson distribution, it appears graphically as:

In the example depicted in the graph above, assume that some operational process has an error rate of 3%. If we further assume 100 random trials, the Poisson distribution describes the likelihood of getting a certain number of errors over some period of time, such as a single day.

If the mean is very large, then the Poisson distribution is approximately a normal distribution.

## The Poisson Distribution in Finance

The Poisson distribution is also commonly used to model financial count data where the tally is small and is often zero. As one example in finance, it can be used to model the number of trades that a typical investor will make in a given day, which can be 0 (often), or 1, or 2, etc.

As another example, this model can be used to predict the number of "shocks" to the market that will occur in a given time period, say, over a decade.

## When Should the Poisson Distribution Be Used?

The Poisson distribution is best applied to statistical analysis when the variable in question is a count variable. For instance, how many times X occurs based on one or more explanatory variables. For instance, to estimate how many defective products will come off an assembly line given different inputs.

## What Assumptions Does the Poisson Distribution Make?

In order for the Poisson distribution to be accurate, all events are independent of each other, the rate of events through time is constant, and events cannot occur simultaneously. Moreover, the mean and the variance will be equal to one another.

## Is the Poisson Distribution Discrete or Continuous?

Because it measures discrete counts, the Poisson distribution is also a discrete distribution. This can be contrasted with the normal distribution, which is continuous.