Math · Statistics
Poisson distribution calculator.
Enter the average number of events in an interval and the count you are asking about. Get exact, cumulative and between-two-numbers probabilities.
If events happen on average 4 times per interval (λ = 4), the chance of exactly 2 events is 0.1465. The chance of 2 or fewer is 0.2381, of 2 or more is 0.9084, and of between 2 and 5 events is 0.6936. The mean is 4 and the standard deviation is 2.
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The formula
The Poisson distribution models the number of events in a fixed interval when λ is the average number of events: P(x) = e−λ λx / x! for x = 0, 1, 2, … The cumulative probability adds P(0) through P(x). The mean is λ, the variance is λ and the standard deviation is √λ. The calculator evaluates cumulative sums with the incomplete gamma function, which stays accurate for far tails.
Worked example
λ = 4 and k = 2: P(2) = e−4 × 4² / 2! = 0.018316 × 8 = 0.1465. P(X ≤ 2) = e−4(1 + 4 + 8) = 0.2381. P(X ≥ 2) = 1 − P(X ≤ 1) = 1 − 5e−4 = 0.9084. P(2 ≤ X ≤ 5) = 0.1465 + 0.1954 + 0.1954 + 0.1563 = 0.6936.
Limits
Poisson counts assume events arrive independently at a steady average rate. Clustered events (for example, calls that come in bursts) have more variance than λ and do not fit. Make sure λ matches the interval you ask about: a rate of 12 per hour is 3 per 15 minutes.
Source, opened 8 October 2026: NIST/SEMATECH e-Handbook, 1.3.6.6.19 Poisson Distribution (probability mass function, cumulative function, mean λ, standard deviation √λ, λ as the average number of events in the interval).
Frequently asked questions
How do you calculate a Poisson probability?
Use P(x) = e^(-lambda) x lambda^x / x!. For lambda = 4 and x = 2 that is e^(-4) x 16 / 2 = 0.1465.
What does lambda mean?
Lambda is the average number of events in the interval you are looking at. If you average 12 events an hour and ask about 15 minutes, use lambda = 3.
How do I find "at least" a number of events?
Use P(X >= k), which equals 1 - P(X <= k - 1). The calculator shows it directly. For lambda = 4 and k = 2 it is 0.9084.
How do I find the probability between two numbers?
Add the exact probabilities from the lower to the upper number, inclusive. Enter the lower number as k and the upper in the between box. For 2 to 5 events at lambda = 4 the answer is 0.6936.
What are the mean and standard deviation of a Poisson distribution?
The mean equals lambda and the standard deviation is the square root of lambda. For lambda = 4 they are 4 and 2.
When should I use binomial instead of Poisson?
Use binomial when you have a fixed number of trials with a fixed success chance. Use Poisson when you count events in an interval with no fixed number of trials and only know the average rate.
Learning aid, not for decisions
Educational calculator. Results assume the conditions listed on this page, which real data often only roughly meet. Do not use it for clinical, regulatory, legal or financial decisions.