Poisson Distribution Table

Match lambda to the exposure

X counts events in a specified exposure window. In this table λ is the expected number in that window, not a probability and not a rate without units. At a constant rate of four events per hour, a half-hour has λ = 2.

A homogeneous Poisson-process interpretation requires independent increments and a constant rate, with simultaneous events negligible at sufficiently small time scales. Not every count process satisfies that model. Clustering and varying rates can produce more variability than a Poisson model predicts.

$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}.$

Cells below are P(X ≤ k), rounded to four decimals. The mean and variance of this model both equal λ; sample summaries need not match exactly.

Cumulative probabilities: λ = 0.5 to 2.5

Cumulative probabilities: λ = 3 to 5

Cumulative probabilities: λ = 6 to 10

Cumulative probabilities: λ = 12 to 20

Worked example and tails

Assume a hypothetical constant-rate process has λ = 4 per hour. The probability of at most two arrivals in an hour is 0.2381. The probability of at least five is 1 − P(X ≤ 4) ≈ 1 − 0.6288 = 0.3712.

The exact probability of three is e⁻⁴ × 4³/3! ≈ 0.1954. Subtracting rounded adjacent CDF cells can introduce rounding error. For P(X ≥ k), subtract the CDF at k − 1, not k. A printed 1.0000 can still have a small nonzero upper tail.

A Poisson approximation to Binomial(n,p) uses λ = np when p is small and the approximation suits the required accuracy. Fixed rules such as n ≥ 20 and p ≤ 0.05 do not guarantee every tail is accurate; use the exact binomial when practical.

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Compare discrete distributions and binomial probabilities. NIST provides the Poisson model. This site does not currently offer a dedicated Poisson calculator.