Key Idea: The Poisson distribution counts random events in a fixed interval, such as calls per hour or flaws per kilometre. You write X ~ Po(m), where m is the mean.
| Fact | Rule |
|---|---|
| Formula | P(X = x) = e⁻ᵐ·mˣ/x! |
| Mean and variance | Both equal m, so σ = √m |
| GDC | poissonpdf (= x), poissoncdf (≤ x) |
| ≥ or > | Use the complement |
| Longer interval | Scale the rate; independent Poissons add |
When is Poisson suitable?
- Events occur independently.
- Constant average rate over the interval.
- Events happen singly, with no upper limit on the count.
- Quick check: mean ≈ variance in the data.
- A fixed number of trials n? Use the binomial instead.
Poisson counts events over time or space when you only know an average rate.
Flaws occur at 0.8 per km. What is m for 5 km? 4.
Find P(no flaws) in those 5 km. e⁻⁴ = 0.0183.
Data have mean 3.1 and variance 7.9. Is Poisson a good model? Probably not. Mean and variance should be about equal.
A ~ Po(2) and B ~ Po(3) are independent. What models A + B? Po(5).