Frequently Asked Questions
Everything you need to know about the Poisson Distribution Calculator.
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.