The formula
P(X = k) = e^(−λ) · λᵏ / k!
What it means
The Poisson counts rare events across a continuous interval of time or space: there is no fixed number of trials, only an average rate. Its defining feature is that the mean and the variance are the same number, and that is also its check: if your data show a variance far above the mean, the events are not independent and the Poisson does not describe what is happening.
How to work it out by hand
- Note the average number of events per interval, which is lambda
- Raise lambda to the power k and divide by the factorial of k
- Multiply by e to the power minus lambda
- For the cumulative figures, add the cases from zero up to k
What is worth knowing
Lambda has to match the interval you are asking about: if the rate is twelve calls an hour and you ask about fifteen minutes, lambda is three, not twelve. That is the commonest mistake with this distribution. And independence matters: calls to a support line are not independent when there is a general outage, and on those days the Poisson badly underestimates the chance of a spike.