Probabilities for the number of events in a fixed interval, given an average rate λ.
A help desk receives 24 calls per hour on average. What is the chance of exactly 6 calls in a 10-minute window, and how unusual would 8 or more be?
Enter λ = 4 and k = 6 or k = 8 above to check these values. A quick sanity check on any Poisson model: the mean and variance should be roughly equal in your data. If the counts vary much more than their mean, events are clustered and a negative binomial model fits better.
The Poisson distribution models the number of events that occur in a fixed interval of time or space, when those events happen independently at a constant average rate λ. Classic examples: calls arriving at a switchboard, decay events from a radioactive source, or typos per page.
Use the binomial when you have a fixed number of trials n each with probability p. Use the Poisson when events occur continuously and you only know the average rate — there's no fixed n. The Poisson is also the limit of the binomial when n is large and p is small (with λ = np).
P(X = k) is the probability of exactly k events. P(X ≤ k) is the cumulative probability of k or fewer. P(X ≥ k) = 1 − P(X ≤ k−1) is the probability of at least k events — useful for questions like "what's the chance of 10 or more calls this hour?"
A key property: the mean and variance are both equal to λ. If your data's variance is much larger than its mean (overdispersion), a negative binomial model may fit better.