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Poisson Cumulative Probability Calculator

Enter a rate λ and a count k for the Poisson distribution.

How it is calculated

The rate λ is the average number of events per interval and must be above 0; k is the count you are asking about and must be a whole number, 0 or more (2.5 or −1 give no result). The engine sums the Poisson probabilities from 0 up to k: P(X ≤ k) = Σ e^(−λ) × λ^i ÷ i! for i = 0…k. Each term is computed in logarithms, ln term = −λ + i × ln λ − ln i!, so large rates such as 500 still give correct values; λ and k above 1,000,000 give no result. Three results are shown to four decimals: the cumulative probability, the exact probability P(X = k), and the mean, which for a Poisson distribution equals λ (shown with more decimals for a very small λ).

Worked example

A help desk gets on average λ = 4 calls per hour. What is the chance of 6 or fewer calls? The probabilities for 0 to 6 calls are 0.0183, 0.0733, 0.1465, 0.1954, 0.1954, 0.1563 and 0.1042. Added together they give a cumulative probability of 0.8893, so there is about an 89% chance of at most 6 calls. The last term, 0.1042, is the chance of exactly 6 calls. The mean is 4.00, and P(more than 6) = 1 − 0.8893 = 0.1107.

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Common mistakes

  • Reading the cumulative result as the chance of exactly k events. That is the second figure; the first is the chance of k or fewer.
  • Using the cumulative value for "at least k". P(X ≥ k) is 1 − P(X ≤ k − 1), so for at least 6 enter k = 5 and subtract the result from 1.
  • Entering the rate for the wrong interval. If calls average 4 per hour and you ask about a 30-minute window, λ must be 2, not 4.
  • Entering a non-integer k, a negative k, or a percentage for the count. The count must be a whole number of 0 or more, and the rate is an average count, not a percentage.
  • Applying the model to events that are not independent or that cluster in time, such as arrivals in rush-hour waves; the result will understate the chance of large counts.

About this calculator

The Poisson distribution models the number of times a rare, independent event happens in a fixed interval of time or space, given a known average rate λ (lambda). The probability of exactly k events is P(X = k) = e^(−λ) λ^k / k!, and the cumulative probability P(X ≤ k) — the probability of seeing k or fewer events — is the running sum of that formula from 0 up to k.

This distribution and its cumulative probability are used wherever counts of rare independent events need modeling: call centers forecasting how many calls will arrive in an hour, reliability engineers estimating how many failures a system will see in a given period, epidemiologists modeling disease case counts, and insurers pricing policies based on the expected number of claims. The cumulative version answers practical questions like ‘what is the chance of getting 10 or fewer calls this hour,’ rather than just the probability of exactly 10.

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