# Poisson Distribution: Formula and Meaning in Finance

![Poisson Distribution](https://www.investopedia.com/thmb/WveF1P9uWSYlDLb2TRCIxS8mC9U=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/poisson-distribution-4191210-98022969865b4b73b8088142d5aac65c.jpg)

*— Investopedia / Joules Garcia*

> **Definition**
>
> The Poisson distribution is a probability distribution that is used by investment analysts in stochastic modeling.

## What Is a Poisson Distribution?

In statistics, a Poisson distribution is a discrete [probability distribution](https://www.investopedia.com/terms/p/probabilitydistribution.asp) that indicates the likelihood of an event occurring a specified number of times over a given period. It is a count distribution, the parameter of which is lambda (λ); the mean number of events in a specific interval. In finance, especially investing, Poisson distributions are created to explain the possibility of trading events, create price ranges, or predict market jumps, among others.

As the Poisson distribution is a discrete function, 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, and so on), with no fractions or decimals.

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.

### 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.
*   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, or how many trades are made in one session, thus lending themselves to analysis using a Poisson distribution.

## Understanding Poisson Distributions

A Poisson [distribution](https://www.investopedia.com/articles/investing/102014/lognormal-and-normal-distribution.asp) 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](https://www.investopedia.com/terms/n/normaldistribution.asp).

One of the most famous historical and 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?

## Formula for the Poisson Distribution

$\begin{aligned}&f(x)=\frac{\lambda^x}{x!}e^{-\lambda}\\&\textbf{where:}\\&e=\text{Euler's number } (e=2.71828\dots)\\&x=\text{Number of occurrences}\\&x!=\text{Factorial of }x\\&\lambda=\text{Equal to the expected value (EV) of }x\text{ when that is also equal to its variance}\end{aligned}$

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

![Poisson Distribution Example](https://www.investopedia.com/thmb/3KhZ7WaJzh-NT75VPVfShDsvsSE=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Clipboard01-23850f297c19425b86aecdc1d560ad40.jpg)

*Poisson Distribution Example.*

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.

> **Fast Fact**
>
> 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), 1, 2, and so on.

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

## Explain Like I'm 5

A Poisson distribution represents the likelihood of a single event occurring within a specific period. The events must be independent and must occur at a constant rate.

## 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, when asking how many times X occurs based on one or more explanatory variables, such as estimating how many defective products will come off an assembly line given different inputs.

## What Assumptions Does the Poisson Distribution Make?

For the Poisson distribution to be accurate, all events must be independent of each other, the rate of events over time must be constant, and events cannot occur simultaneously. Moreover, the mean and the variance will be equal.

## Is the Poisson Distribution Discrete or Continuous?

Because it measures discrete counts, the Poisson distribution is also a [discrete distribution](https://www.investopedia.com/terms/d/discrete-distribution.asp). This can be contrasted with the normal distribution, which is continuous.

## The Bottom Line

A Poisson distribution is a probability distribution that is used to predict the amount of variation from an average rate of occurrence in a given time frame. It's also a discrete function, meaning that the variable can only take whole number values and no fractions or decimals.

The Poisson distribution can be a helpful tool to evaluate and predict financial and trading operations.