📮 Poisson Distribution Calculator
Enter the average rate λ and a count k to get the exact Poisson probabilities and the distribution's mean, variance and standard deviation.
P(X = 2) — exactly 2 events
0.22404
P(X ≤ 2)
0.42319
P(X ≥ 2)
0.80085
P(X < 2)
0.19915
P(X > 2)
0.57681
For a Poisson process the mean and variance both equal λ = 3, so the standard deviation is √λ ≈ 1.732.
The Poisson distribution models the number of independent events in a fixed interval when they happen at a constant average rate λ — arrivals in a queue, defects per batch, calls per hour. P(X = k) = e^(−λ)·λ^k ÷ k!. Its defining feature is that the mean and variance are both λ. Use it when events are rare and independent; if the variance is much larger than the mean, a different model may fit better. 🔒 In your browser.
How the poisson distribution calculator works
The Poisson distribution gives the probability of a number of independent events in a fixed interval when they occur at a constant average rate λ. The tool computes the exact probability of exactly k events, P(X = k) = e^(−λ)·λ^k ÷ k!, plus the cumulative probabilities (at most k, at least k, fewer than, more than), and reports the distribution's mean and variance — which, distinctively, are both equal to λ.
Use the Poisson model for rare, independent events happening at a steady average rate: customer arrivals, machine failures, typos per page, calls per hour. Its signature property is that the variance equals the mean; if your real data is much more spread out than that (overdispersion), a different model such as the negative binomial may fit better.
Frequently asked questions
What is the Poisson distribution used for?
Modelling the count of independent events in a fixed interval of time or space at a constant average rate — arrivals in a queue, defects per batch, calls per hour, goals per match. It answers "what's the probability of exactly k events?"
How do I calculate a Poisson probability?
P(X = k) = e^(−λ)·λ^k ÷ k!, where λ is the average number of events and k the count you're asking about. For λ = 3 and k = 2 that's e^(−3)·9 ÷ 2 ≈ 0.224. The tool also gives the cumulative probabilities.
What is λ (lambda) in a Poisson distribution?
The average rate — the expected number of events per interval. It's the single parameter of the distribution, and both the mean and the variance equal λ, so the standard deviation is √λ.
What is the difference between the Poisson and binomial distributions?
The binomial counts successes in a fixed number of trials with a fixed success probability; the Poisson counts events over a continuous interval with no fixed number of trials. The Poisson is the limit of the binomial when trials are many and each success is rare.
Why do the mean and variance both equal λ?
It's a defining mathematical property of the Poisson distribution. It's also a useful check: if your observed data has a variance much larger than its mean, the events may not be independent or the rate may not be constant, and Poisson may not fit.