Poisson distribution calculator
For counts of events in a fixed stretch of time or space: calls per hour, typos per page. Set the average rate and see the whole spread of counts.
Example: a help desk averages 4 calls per hour. What is the chance of 7 or more in the next hour?
P(X = 3) = e^(−4) × 4^3 / 3! = 0.1954
Full probability table
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.0183 | 0.0183 |
| 1 | 0.0733 | 0.0916 |
| 2 | 0.1465 | 0.2381 |
| 3 | 0.1954 | 0.4335 |
| 4 | 0.1954 | 0.6288 |
| 5 | 0.1563 | 0.7851 |
| 6 | 0.1042 | 0.8893 |
| 7 | 0.0595 | 0.9489 |
| 8 | 0.0298 | 0.9786 |
| 9 | 0.0132 | 0.9919 |
| 10 | 0.0053 | 0.9972 |
| 11 | 0.0019 | 0.9991 |
| 12 | 0.0006 | 0.9997 |
| 13 | 0.0002 | 0.9999 |
| 14 | 5.640e-5 | 1.0000 |
| 15 | 1.504e-5 | 1.0000 |
| 16 | 3.760e-6 | 1.0000 |
| 17 | 8.847e-7 | 1.0000 |
What the Poisson distribution counts
The Poisson distribution gives the probability of seeing k events in a fixed interval when events occur independently at a steady average rate λ. It is named after Siméon Denis Poisson (1837). A famous early use was Ladislaus Bortkiewicz's 1898 study of Prussian cavalrymen killed by horse kicks, whose yearly counts fitted a Poisson curve closely.
P(X = k) = e^(−λ) · λᵏ / k!
Scaling the rate
λ must match the interval you ask about. If calls average 4 per hour, then over 30 minutes λ = 2 and over a full 8-hour shift λ = 32. Slide λ upwards and watch the distribution become more symmetric; for large λ it is close to normal with mean λ and SD √λ.
Worked example
With λ = 4 calls per hour: P(X = 0) = e⁻⁴ ≈ 0.0183, so a completely quiet hour happens about 1.8% of the time. P(X ≥ 7) ≈ 0.1107, roughly one hour in nine.
The waiting time between Poisson events follows the exponential distribution.
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Questions students ask
What is the Poisson formula?
P(X = k) = e^(−λ) λ^k / k!, where λ is the average number of events in the interval.
When should I use a Poisson distribution?
When events happen independently, one at a time, at a constant average rate, and you are counting them in a fixed interval. Classic examples are arrivals, defects and radioactive decays.
What are the mean and variance of a Poisson distribution?
Both equal λ. If your count data have a variance much larger than the mean, a Poisson model is probably too simple.
How is Poisson related to binomial?
When n is large and p is small, Binomial(n, p) is close to Poisson(λ = np). For example, 1,000 trials with p = 0.002 behaves like Poisson with λ = 2.