📊Poisson Distribution Calculator
Enter the average rate (λ) and a value k to find Poisson probabilities.
Free Poisson Distribution Calculator
This Poisson distribution calculator finds the probability of observing exactly k events, at most k events, or at least k events, given an average rate λ — using P(X=k) = (λ^k × e^−λ) / k!. The Poisson distribution models the number of independent events occurring in a fixed interval, given a known average rate.
Common examples include the number of customers arriving per hour, defects per batch, or calls received per minute — anywhere events happen independently at a constant average rate. The bar chart shows the full probability distribution around your value of k.
Free, instant, and runs entirely in your browser — no sign-up required.
About the Poisson Distribution Calculator
Features
Three probability types
Exact, at-most, and at-least probabilities together.
Distribution chart
Visualize the shape of the distribution around k.
Mean and variance
Both equal λ, shown alongside the probabilities.
Instant, private
Runs entirely in your browser. No data is sent anywhere.
Frequently Asked Questions
What is the Poisson distribution?
The Poisson distribution models the probability of a given number of independent events occurring in a fixed interval of time or space, given a known average rate (λ).
What is the Poisson probability formula?
P(X = k) = (λ^k × e^−λ) / k!, where λ is the average rate, k is the number of events, and e is Euler's number (≈2.71828).
What does λ (lambda) represent?
λ is the average number of events expected in the interval — for example, if a call center receives 4 calls per hour on average, λ = 4.
What's the difference between P(X=k), P(X≤k), and P(X≥k)?
P(X=k) is the probability of exactly k events; P(X≤k) is the cumulative probability of k or fewer events; P(X≥k) is the probability of k or more events.
What are the mean and variance of a Poisson distribution?
Both the mean and the variance equal λ — this is a defining property of the Poisson distribution.
When is the Poisson distribution appropriate?
It applies when events happen independently of each other at a constant average rate, such as customer arrivals, website hits per minute, or manufacturing defects per batch.
Can k be a decimal?
No — k must be a whole number, since it represents a count of discrete events (you can't observe 'half an event').
Can λ be a decimal?
Yes, λ is typically a decimal, since it represents an average rate that doesn't need to be a whole number, like 2.5 events per hour.
How does the Poisson distribution relate to the binomial distribution?
The Poisson distribution is the limiting case of the binomial distribution when the number of trials is very large and the probability of success per trial is very small, with np converging to λ.
Is this calculator free to use?
Yes, completely free with no sign-up, and it runs entirely in your browser.