What Is a Poisson Distribution?

In statistics, a Poisson distribution is a discrete probability distribution that tells how many times an event 𒆙is likely to occur over a specified period. It is a count distribution, the parameter of which is lambda (λ); the mean number of events in the specific inteꦫrval.

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 eve💛nts that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

Understanding Poisson Distributions

A Poisson 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网: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 numbe📖r 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, ဣif a video store averaged 400 customers every Friday night, what would have been the probability that 600 customers wou🦂ld 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 (💟EV)&ღnbsp;of x when that is also equal&nbs🎶p;to its variance​

Given data that follows a Poi🌳sson distribution, it appears graphically as:🌊

In the example depicted in the grꦺaph 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 of🐼ten 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, 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 be a helpful tool to evaluate and predict financial𒁏 an﷽d trading operations.