Poisson Distribution Calculator
The Poisson distribution calculator computes the probability of exactly k events occurring in a fixed time or space period when events happen at a constant average rate (lambda). The formula is P(X=k) = (lambda^k * e^(-lambda)) / k!. For example, if a web server receives an average of 10 requests per second, the Poisson distribution can calculate the probability of receiving exactly 8 requests in a given second. This calculator also shows the cumulative probability P(X <= k), which is the probability of k or fewer events. The distribution parameters (mean, variance, standard deviation) are shown below the outputs.
Formula
P(X = k) = (lambda^k * e^(-lambda)) / k!
where e ≈ 2.71828 and k! is the factorial of k
Distribution parameters
Poisson distribution calculator: frequently asked questions
What is the Poisson distribution?
The Poisson distribution models the number of events occurring in a fixed time or space when events occur at a constant average rate, independently of time since the last event. Examples include phone calls arriving at a switchboard, defects in manufacturing, or traffic accidents.
What is lambda?
Lambda (lambda) is the mean rate: the average number of events expected in the time or space period. For example, if a call center receives an average of 5 calls per hour, lambda = 5.
When should I use Poisson instead of binomial?
Use Poisson when the number of trials is very large, the probability of success is very small, and you know the average number of successes (lambda). Use binomial when you have a fixed number of trials and constant probability per trial.
What is e in the formula?
e is Euler's number, approximately 2.71828. It is the base of the natural logarithm and appears in many areas of mathematics and science.
What does cumulative probability mean in Poisson?
Cumulative probability P(X <= k) is the probability of getting k or fewer events. For example, P(X <= 3) is the probability of 0, 1, 2, or 3 events occurring.
Official sources
- Wolfram MathWorld: Poisson Distribution.
- Wikipedia: Poisson Distribution.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.