Common Probability Distribution - Poisson

Poisson Distribution

Definition

The Poisson distribution describes the number of events occurring in a fixed interval of time, given a constant average rate of occurrence. It is a discrete probability distribution for count data.

Mathematical Formula

Probability Mass Function

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

Where:

• $\lambda$ = average rate of occurrence, expectation of number of events in a given interval
• $k$ = number of events in the same interval

Mean and Variance

\(\mu = \lambda\)

\[\sigma^2 = \lambda\]

Python Implementation

For detailed Python implementations, examples, and data science applications, see the accompanying Jupyter notebook:

📓 Poisson Distribution - Python Implementation

Notes

• Normal Approximation: When $\lambda > 20$ (varies depending on source), Poisson can be approximated by normal distribution