Poisson Distribution Calculator
Find the probability of a given number of rare, independent events occurring in a fixed interval using the Poisson distribution.
Mode
Poisson probability for the selected mode
- 1
e^(−λ)
e^(−3) = 0.049787 - 2
λ^k
3^2 = 9 - 3
k! (factorial)
2! = 2 - 4
P(X = k) × 100%
(e^(−λ) × λ^k ÷ k!) × 100 = 22.4042Probability expressed as a percentage to match the headline.
How does this calculator work?
The Poisson distribution gives the probability of k independent events in a fixed interval when the average rate is λ, using P(X=k)=e^(-λ)·λ^k/k!. Choose exactly, at most, or at least k. Both the mean and variance equal λ, so the standard deviation is √λ.
Formula
How this is calculated
The Poisson distribution models the count of independent events that occur at a constant average rate λ over a fixed interval of time, space, or volume. Enter λ (the mean number of events) and k (the event count of interest). The exact probability mass is P(X = k) = e^(-λ) · λ^k / k!, where k! is the factorial of k.
The mode dropdown selects which probability to report. "Exactly" returns the single PMF value P(X = k). "At most" returns the cumulative probability P(X ≤ k) = Σ from i=0 to k of e^(-λ)·λ^i/i!. "At least" returns P(X ≥ k) = 1 − P(X ≤ k−1). Factorials are computed via the log-gamma (lgamma) function so the math stays stable for large k, and probabilities are evaluated in log space before exponentiating.
For a Poisson random variable the mean and variance are both equal to λ, so the standard deviation is √λ. The model assumes events are independent, occur one at a time, and have a constant rate; k must be a non-negative integer and λ must be non-negative. The PMF bars chart the distribution around the mean, highlighting the bar at your chosen k.
Frequently asked questions
λ (lambda) is the expected average number of events per interval. It equals both the mean and the variance of the distribution, so the standard deviation is √λ.
Use it for counts of rare, independent events at a constant average rate over a fixed interval — for example calls per hour, typos per page, or decays per second.
P(X ≥ k) is calculated as 1 − P(X ≤ k−1), the complement of the cumulative probability up to k−1 events.
Also known as
TG we-Calculate Editorial Team. (2026). Poisson Distribution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/poisson-distribution-calculator
TG we-Calculate Editorial Team. "Poisson Distribution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/poisson-distribution-calculator.
TG we-Calculate Editorial Team, "Poisson Distribution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/poisson-distribution-calculator
@misc{wecalculate_poisson_distribution_calculator, title = {Poisson Distribution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/poisson-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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