# Poisson distribution

The Poisson distribution counts events in a fixed time or space when they happen at a steady average rate lambda and independently. P(X = k) = e^(-lambda) x lambda^k / k!. Mean and variance both equal lambda.

Statistics (university) · Part 03: Random variables and discrete distributions · Card 6 of 8 · Last updated 2026-10-19

## Analogy

Emails arriving in your inbox: you know the hourly average, but you cannot say when each one comes.

## Example

**Calls per hour**

Average lambda = 4 per hour
P(X=2) = e^-4 x 4^2 / 2
= 0.01832 x 8 = 0.1465
P(X=0) = e^-4 = 0.0183

## In practice

Models calls to a help desk, typos per page and cars passing a point per minute.

## Practice

### Flashcards

- **Mean and variance of a Poisson?** Both equal lambda.

## Questions people ask

### When should I use Poisson instead of binomial?

Use Poisson when you count events over time or space and there is no fixed number of trials, only an average rate. Use binomial when there are n trials with a set success chance. For huge n and tiny p, the binomial is well approximated by a Poisson.

### What happens to lambda if I change the time window?

Lambda scales with the window. If calls average 4 per hour, then lambda is 2 for a 30-minute window and 8 for two hours. Always match lambda to the interval in the question before using the formula, because a wrong window gives a confidently wrong answer.

## Related concepts

- [Binomial distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-5)
- [Geometric distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-7)
- [Cumulative distribution function](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-8)

Web page: https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-6
Study this card in the app: https://studyvisually.com/app/learn?category=statistics-university&card=statistics-university-part-3-concept-6