Poisson Distribution Calculator
Compute Poisson probabilities — exactly k, at most k and at least k occurrences for a given average rate λ — with the mean and standard deviation.
How it works
P(X = k) = e^(−λ) · λ^k ÷ k! · P(X ≤ k) = Σ · mean = λ · σ = √λ
The Poisson distribution models the count of independent events in a fixed interval of time or space when they occur at a constant average rate λ. It answers questions like how many calls a help desk gets in an hour or how many typos appear on a page. A defining feature is that its mean and variance are both equal to λ, so the spread grows with the rate. This calculator sums the mass function to give cumulative probabilities and evaluates the factorial through a log-gamma function for numerical stability.
Worked example
If a call centre averages λ = 4 calls per hour, the chance of exactly 2 calls in an hour is e⁻⁴ · 4² ÷ 2! = e⁻⁴ · 8 ≈ 0.1465, or 14.65%. Summing k = 0, 1, 2 gives P(at most 2) ≈ 23.81%, and the standard deviation is √4 = 2.
Sources: Poisson distribution — Wikipedia
Frequently asked questions
When should I use a Poisson distribution?
Use it to count independent events that happen at a steady average rate over a fixed interval — arrivals, defects, accidents, decay events — where each occurrence does not affect the next. It is the natural model when you know an average count but not a fixed number of trials.
How is the Poisson related to the binomial?
The Poisson is the limit of the binomial when the number of trials n is large and the success probability p is small, with λ = np. That makes it a convenient approximation for rare events across many opportunities, where the binomial would be unwieldy.
Why do the mean and variance both equal λ?
It is a mathematical property of the Poisson distribution: both its mean and its variance equal the rate parameter λ, so its standard deviation is √λ. If real data has a variance much larger than its mean, it is overdispersed and a different model may fit better.
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For education — check assumptions (sample size, distribution) before using statistics in real decisions.