Poisson Distribution Calculator

Frequently Asked Questions

When is the Poisson distribution used?

For counts of independent events in a fixed interval of time/space at a constant average rate λ - calls per hour, defects per metre, arrivals per minute.

What does it output?

P(X=k), cumulative P(X≤k) and P(X≥k), and the mean and variance, which for Poisson are both equal to λ.

How does it relate to the binomial?

It is the limiting case of the binomial when trials are many and success rare (n large, p small, np = λ). Useful when n is unknown but the rate is.

What assumption matters most?

Events are independent and occur at a constant rate. Clustered or rate-varying events (overdispersion) need a negative binomial instead.

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This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

The Poisson distribution models the number of events in a fixed interval of time or space when events occur independently at a constant average rate λ. Its probability mass function is P(X = k) = e^−λ λ^k ÷ k!. To avoid overflow at large k this calculator evaluates the PMF in log space - −λ + k·lnλ − ln(k!) - then exponentiates. Cumulative probabilities are exact partial sums: P(X ≤ k) = Σ_i=0..k P(X = i), and the upper tails follow by complement, P(X ≥ k) = 1 − P(X ≤ k − 1) and P(X > k) = 1 − P(X ≤ k). A defining feature: for a Poisson the mean and the variance are both equal to λ, so the standard deviation is √λ.

Worked Example: λ = 3, k = 2

A call center averages 3 calls per minute; what is the probability of exactly 2 in a given minute? P(X = 2) = e⁻³ × 3² ÷ 2! = 0.049787 × 9 ÷ 2 ≈ 0.2240. Building up the cumulative: P(0) ≈ 0.0498, P(1) ≈ 0.1494, P(2) ≈ 0.2240, so P(X ≤ 2) ≈ 0.4232 and P(X ≥ 3) ≈ 0.5768. The mean and variance are both 3, so the standard deviation is √3 ≈ 1.732 calls.

When to Use It

Arrivals & queues: customers, calls, web requests, or emails per unit time.

Defects & failures: flaws per square meter, typos per page, breakdowns per month.

Rare events: accidents at an intersection, mutations per genome, decay counts per second.

Binomial limit: large n, small p - set λ = np for a fast approximation.

Common Mistakes & Assumptions

Poisson assumes events are independent, the rate λ is constant over the interval, and two events cannot occur at exactly the same instant. Real data often shows overdispersion - variance exceeding the mean - which breaks the model; a negative binomial fits better there. The rate must match the interval: if λ = 3 per minute but you want a 5-minute window, scale to λ = 15 before computing. Confusing "at least k" (P(X ≥ k), includes k) with "more than k" (P(X > k), excludes k) is a recurring error.

Interpretation & Limitations

The equality of mean and variance is the quickest diagnostic: if sample variance is far from the sample mean, Poisson is the wrong model. For large λ (roughly ≥ 20) the distribution is approximately normal with mean and variance λ, useful for quick interval estimates with a continuity correction. This tool computes the exact Poisson, so it stays accurate across regimes, though summing the PMF for very large k is slower. Displayed probabilities are accurate to six decimals; extreme upper-tail values may carry minor floating-point error.

Poisson Distribution Calculator

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