Distributions
Poisson Distribution Calculator
The probability of k events in an interval, when events arrive independently at a steady average rate.
How to read this result
λ is both the mean and the variance. That is the Poisson's defining property, and it is a testable one: if your counts have a variance much larger than their mean, they are overdispersed and a Poisson model is understating the variability.
Rescale λ with the interval. Three arrivals per hour is λ = 3 for one hour, λ = 0.5 for ten minutes, λ = 72 for a day. The distribution only knows the expected count for the window you are asking about.
The shape changes with λ. At λ = 1 the distribution is strongly right-skewed and the most likely outcome is zero or one. By λ = 20 it is almost symmetric and close to normal.
It is the binomial's limit. As n grows and p shrinks with np held at λ, the binomial becomes the Poisson — which is why it fits rare events among many opportunities: many customers, each unlikely to arrive in this particular minute.
The formula
- the average number of events per interval
- the number of events whose probability you want
Mean λ, variance λ, standard deviation √λ. The cumulative form is computed from the incomplete gamma function rather than by summing, which keeps it exact for large λ.
Worked example
Four calls in an hour when the average is 2.5
A support line takes an average of 2.5 calls an hour. What is the probability of exactly four in the next hour, and of four or fewer?
- λ to the power k.2.5⁴ = 39.0625
- Times e to the minus λ.e^(−2.5) = 0.082085
- Divide by k factorial.4! = 24
- Put it together.P(X = 4) = 39.0625 × 0.082085 / 24 = 0.13360
- For four or fewer, add k = 0 to 4.0.08208 + 0.20521 + 0.25652 + 0.21376 + 0.13360 = 0.89118
Eleven per cent of hours will see five or more calls. Staffing for the average of 2.5 means being overwhelmed roughly one hour in nine — which is the practical reason queueing systems are planned from the distribution rather than the mean.
Checked against R's dpois(4, 2.5) and ppois(4, 2.5).
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
- Events occur independently. One arrival does not make the next more or less likely.
- A constant average rate over the interval. A shop with a lunchtime rush violates this within a day, though it may hold within each hour.
- Two events cannot occur at exactly the same instant.
- The count has no upper bound, at least in principle.
When counts are more variable than the Poisson allows — variance well above the mean — the usual culprit is a rate that varies (clustering, seasonality), and a negative binomial model handles it.
- There is a fixed number of trialsBinomial distributionTen inspections with a pass/fail outcome is binomial, not Poisson.
- λ is large and you want a normal approximationNormal distributionReasonable once λ is above about 10, with SD √λ.
- You want to test observed counts against expectedChi-square goodness of fitTests whether a set of counts matches a Poisson or any other expectation.
Questions people ask
What is the difference between the Poisson and the binomial?
The binomial counts successes in a fixed number of trials; the Poisson counts events in an interval with no fixed number of opportunities. Ten coin flips is binomial; calls arriving in an hour is Poisson.
How do I choose λ?
It is the average count for the interval you are asking about. Estimate it from data — the sample mean of past counts is the maximum-likelihood estimate — and rescale it if your interval differs from the one it was measured over.
Why are the mean and variance equal?
It falls out of the limit that produces the distribution: the binomial's variance np(1 − p) tends to np as p goes to zero, and np is λ. It is also a useful diagnostic — counts whose variance far exceeds their mean are not Poisson.
Can λ be a decimal?
Yes. λ is an average, so 2.5 calls per hour is perfectly sensible even though you can never observe half a call. k must be a whole number; λ need not be.