Table of Contents
- What Is a Poisson Distribution?
- Understanding Poisson Distributions
- Formula for the Poisson Distribution
- Fast Fact
- The Poisson Distribution in Finance
- Explain Like I'm 5
- When Should the Poisson Distribution Be Used?
- What Assumptions Does the Poisson Distribution Make?
- Is the Poisson Distribution Discrete or Continuous?
- The Bottom Line
What Is a Poisson Distribution?
Let me tell you about the Poisson distribution—it's a discrete probability distribution in statistics that shows the chances of an event happening a certain number of times in a fixed period. As a count distribution, it relies on the parameter lambda (λ), which is the average number of events in that interval. In finance and investing, we use Poisson distributions to model things like trading events, set price ranges, or forecast market jumps.
Since it's discrete, the variable only takes specific whole number values—0, 1, 2, 3, and so on—with no fractions or decimals allowed. You can't have it covering a continuous range of values.
We often apply Poisson distributions to independent events that happen at a steady rate over a set time frame. It gets its name from the French mathematician Siméon Denis Poisson.
Key Takeaways
- The Poisson distribution, named after Siméon Denis Poisson, helps estimate how many times an event might occur over 'X' periods.
- It's ideal when your variable is a discrete count.
- Economic and financial data often come as counts, like unemployment instances in a year or trades in a session, making them perfect for Poisson analysis.
Understanding Poisson Distributions
You can use a Poisson distribution to figure out the odds of something happening exactly 'X' times. For instance, if a fast-food spot averages 200 cheeseburger sales on a Friday night, it can tell you the probability of selling over 300.
This lets managers set up better scheduling systems that wouldn't fit with something like a normal distribution.
One classic example is estimating Prussian cavalry deaths from horse kicks each year. Today, you might use it for car crashes in a city or neurotransmitter secretions in physiology.
Or consider a video store averaging 400 customers on Friday nights—what's the chance of 600 showing up on a given night?
Formula for the Poisson Distribution
The formula is f(x) = (λ^x / x!) * e^{-λ}, where e is Euler's number (about 2.71828), x is the number of occurrences, x! is the factorial of x, and λ is the expected value of x, which also matches its variance.
If your data follows this, it graphs out showing probabilities for different counts. For example, with a 3% error rate over 100 trials, it plots the likelihood of various error numbers in a day.
Fast Fact
When the mean gets really large, the Poisson distribution starts looking like a normal distribution.
The Poisson Distribution in Finance
In finance, we commonly use the Poisson distribution for small or zero-count data. Take the number of trades an investor makes in a day—it could be 0, 1, 2, etc.
It also predicts market shocks over periods like a decade.
Explain Like I'm 5
Think of the Poisson distribution as the chance of one event happening in a set time. The events have to be independent and come at a steady rate.
When Should the Poisson Distribution Be Used?
Apply it when your variable is a count, like how many times X happens based on inputs—say, defective products from an assembly line with certain settings.
What Assumptions Does the Poisson Distribution Make?
For accuracy, events must be independent, the rate constant over time, and no events happening at the exact same moment. Plus, mean and variance are equal.
Is the Poisson Distribution Discrete or Continuous?
It's discrete because it deals with counts, unlike the continuous normal distribution.
The Bottom Line
The Poisson distribution predicts variation from an average occurrence rate in a time frame. It's discrete, so only whole numbers apply—no fractions. It's a solid tool for evaluating financial and trading scenarios.
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