Poisson Distribution Calculator

What does the calculator compute and how?

For a rate λ and a count k it gives P(X = k) = λᵏe⁻λ ÷ k!, the cumulative P(X ≤ k) as the sum of those terms from 0 to k, and P(X > k). Terms are evaluated through logarithms of the factorial to avoid overflow. For λ = 3 and k = 4 the results are 0.168031, 0.815263 and 0.184737.

How do I get P(X ≥ k) or the probability between two counts?

The tool shows P(X ≤ k) and P(X > k) only, so shift k. P(X ≥ 4) equals P(X > 3), which you get by entering k = 3: for λ = 3 that is 1 − 0.647232 = 0.352768. A range such as P(2 ≤ X ≤ 5) is P(X ≤ 5) minus P(X ≤ 1), two runs with the same λ.

What are the limits on λ and k?

λ must be positive, k must be 0 or greater, and k above 170 is rejected with an error. A non-integer k is rounded to the nearest whole number (2.6 is treated as 3), because the Poisson counts whole events. λ itself can be fractional, such as 2.5 calls per hour.

Why do some results appear in scientific notation?

Probabilities below 0.000001 are shown as, for example, 7.961e-17 for P(X = 30) at λ = 4, so that a tiny but nonzero chance is not displayed as 0.000000. P(X > k) in the far tail is summed directly instead of subtracting from 1, which keeps small values such as 1.179e-11 (λ = 3, k = 20) accurate instead of losing them to rounding.

How do I match λ to the interval I care about?

λ must be the expected count over the exact interval in the question. If a counter averages 3 arrivals per hour, a 30-minute window has λ = 1.5 and a 2-hour window has λ = 6. Using the wrong interval is the most common error. For λ = 6, the chance of zero events is 0.002479.

When is the Poisson model a poor fit?

When events cluster or influence one another, when the rate changes during the interval (rush hours), or when the variance of observed counts is much larger than their mean. The Poisson distribution forces mean and variance to be equal to λ. Overdispersed counts need another model, and counts that cannot be zero need a zero-truncated version.

How is it related to the binomial distribution?

Poisson is the limit of a binomial with many trials and a small success chance, with λ = np. It suits rare events counted over time or space. If you have a fixed number of trials with a known probability and the count is bounded by that number, the binomial is the exact model and Poisson is only an approximation.