Poisson Distribution
2026-02-28 10:23 Diff
https://brightchamps.com/en-us/math/data/poisson-distribution

A Poisson distribution is used to predict or estimate the number of times an event might occur within a given period of time. This type of distribution method is specifically used when the variables are discrete count variables. We usually use Poisson distribution when dealing with variables, such as economic and financial data.

The formula used to calculate the probability of an event occurring discreetly over a given period of time is:

\(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\)

Where:


e is approximately 2.718 (Euler’s number),


λ is the average number of events in the interval


k! is the factorial of k


k is the actual number of occurrences.

In a Poisson distribution, both the mean and variance are equal to λ. Here, λ is greater than 0.

Example:


Imagine you are waiting for text messages from your friends. Based on your history, you know that on average, you receive three text messages per hour.
Even though the average is 3, you know that in any specific hour, the actual number could vary. You might get 0, 3, or 5 texts. The text messages arrive independently of one another (receiving one doesn't change the chance of receiving another).


This is a classic Poisson Distribution problem because:

  • We know the average rate (\lambda = 3).
     
  • We are looking at a fixed time interval (1 hour).
     
  • Events are independent.

How the Probability Looks:


Using the Poisson distribution, we can calculate the likelihood of receiving a specific number of texts in the next hour.

  • 0 Texts: There is about a 5% chance you receive absolute silence.
     
  • 1 Text: There is about a 15% chance.
     
  • 2 Texts (The Average): There is about a 22% chance.
     
  • 3 Texts: There is about a 22% chance.
     
  • 5 Texts: There is only about a 10% chance.

Calculation:

\(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\)


Where:

  • \(P(X=k)\) is the probability of getting exactly k texts.
     
  • \(\lambda\) (Lambda) is the average rate (3 texts).
     
  • e is Euler's number (\approx 2.718).
     
  • k! is the factorial of k.

Example Calculation for three texts:


\(P(X=3) = \frac{3^3 \cdot e^{-3}}{3!} = \frac{27 \cdot 0.0498}{6} \approx 0.22\)


Use the Poisson distribution calculator to check.