Poisson Probability Calculator
The Poisson distribution models the number of times a random event occurs in a fixed interval when events happen independently at a constant average rate (lambda). It applies to situations such as the number of customers arriving at a bank per hour, the number of defects per meter of wire, or the number of calls received by a switchboard per minute. Enter lambda (the average number of events per interval) and k (the specific count of interest) to get the exact probability P(X = k), the cumulative P(X at most k), and the tail probability P(X at least k).
Poisson probability formula
P(X = k) = (λ^k × e^(-λ)) / k!
Where lambda is the average number of events per interval, k is the specific count (a non-negative integer), e is Euler's number (approximately 2.71828), and k! is k factorial. The calculation uses the log-space formula to avoid overflow for large values.
Poisson distribution properties
- Mean and variance both equal lambda. The standard deviation equals the square root of lambda.
- For small lambda, the distribution is right-skewed. As lambda increases, the distribution approaches the normal distribution.
- The Poisson approximates binomial(n, p) when n is large and p is small, with lambda = n times p.
- The sum of independent Poisson random variables is also Poisson: Poisson(lambda1) plus Poisson(lambda2) = Poisson(lambda1 plus lambda2).
Frequently asked questions
What is the Poisson distribution?
The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate (lambda). It is used for rare events: customer arrivals, defects per unit, calls per hour, radioactive decays.
What is the formula for Poisson probability?
P(X = k) = (lambda^k times e^(-lambda)) divided by k!, where lambda is the average rate (mean number of events per interval), k is the specific count you want the probability for, e is Euler's number (approximately 2.71828), and k! is k factorial.
What does lambda represent?
Lambda (also written as the Greek letter, or just as the mean) is the average number of events in the interval. For example, if a call center receives on average 10 calls per hour, lambda = 10 per hour. You then compute probabilities for specific values of k.
When is the Poisson distribution a good model?
The Poisson distribution fits well when: events occur independently; the average rate is constant; two events cannot occur simultaneously; and probabilities are small relative to the observation interval. It approximates the binomial distribution when n is large and p is small.
What is the cumulative Poisson probability?
The cumulative probability P(X at most k) is the sum of P(X = 0) plus P(X = 1) plus ... plus P(X = k). It tells you the probability of observing k or fewer events. The complement P(X at least k) equals 1 minus P(X at most k-1).
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Poisson Distribution.
- NIST Digital Library of Mathematical Functions: Chapter 26: Combinatorial Analysis.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.