The Poisson distribution
The number of events in a fixed interval when they happen independently at an average rate λ. There is no upper limit, and the mean and the variance are both λ. Move λ, then ask a question: the bars it counts are shaded.
Parameters
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Compare with
P(X ≤ 3) = 0.4335
P(X ≤ 3) = F(3) = 0.4335
Summary
| Measure | Poisson(λ = 4) |
|---|---|
| mean | 4.0000 |
| variance | 4.0000 |
| standard deviation | 2.0000 |
| skewness | 0.5000 |
The same in code
- Python (scipy.stats)
poisson.cdf(3, 4)- Excel
=POISSON.DIST(3, 4, TRUE)
Formulas
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Mean equals variance
Move λ anywhere and compare the mean with the variance.
Look for: They stay equal. If your counts have a variance well above their mean, the Poisson is the wrong model for them.
The binomial's limit
Compare with the binomial and raise its number of trials from 20 to 500, keeping λ = 4.
Look for: The binomial's p = λ/n shrinks and its points close in on the Poisson's bars: many trials, each unlikely, with a steady mean.
Skewed for small λ
Set λ = 1 with the normal approximation on, then raise λ to 30.
Look for: At λ = 1 the bars pile up at 0 and 1 with a long right tail. As λ grows the skewness 1/√λ falls and the normal curve fits.
A bar you could miss
Open the preset for more than 15 calls, then switch to P(X ≥ 15).
Look for: P(X > 15) = 1 − F(15) = 0.1556, while P(X ≥ 15) = 1 − F(14) is larger by the bar at 15.