Poisson Cumulative Probability Calculator

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).

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.