What Is a Poisson Distribution?

In statistics, a Poisson distribution is a discrete probability distribution that tells how many times an event is likely to occur over a specified period. It is a count distribution, the parameter of which is lamb🦋da (λ); the mean num♎ber of events in the specific interval.

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, etc.), 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 mathem🅰atician Siméon Denis Poisson.

Understanding Poisson Distributions

A Poisson 澳洲幸运5开奖号码历史查询: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 che𓆏eseburgers 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 澳洲幸运5开奖号码历史查询:normal distribution.

One of the most famous historical, practical uses of the P♈oisson 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 physiolo▨gy, 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

$\begin{aligned}&f(x)=\frac{\lambda^x}{x!}e^{-\lambda}\\&\textbf{where:}\\&e=\text{Euler's number } (e=2.71828\dots)\\&x=\text{Number of occurrences}\\&x!=\text{Factorial of }x\\&\lambda=\text{Equal to the expected value (EV) of }x\text{ when that is also equal to its variance}\end{aligned}$​f(x)=x!λx​e−λwhere:e=Euler’s number (e=2.71828…)x=Number of occurrencesx!=Factorial of xλ=Equal to the 🍃;expected value&♒nbsp;(EV) of x when that is also equal&n💖bsp;to its variance​

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

In the example depicted in the graphꦕ 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.

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 mo✤del the number of trades that a typical investor will make in a given day, which can be 0 (often), 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.

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 b🐼e a helpfu𓆉l tool to evaluate and predict financial and trading operations.