Poisson Distribution Calculator

Frequently Asked Questions

Can λ be non-integer?

Yes. λ is the average rate and can be any non-negative real number. For example, if a store averages 2.3 customer complaints per day, use λ = 2.3.

What is overdispersion and why does it matter?

Overdispersion means the observed variance exceeds the mean, violating Poisson's assumption of mean = variance. It commonly occurs when events cluster together (e.g. flu cases spread through households). A negative binomial distribution models overdispersed count data with an additional dispersion parameter.

How do I scale λ for a different time window?

Multiply by the ratio of windows. If λ = 6 events per hour and you want a 10-minute window, λ<sub>10min</sub> = 6 × (10/60) = 1. Use that scaled λ in the calculator.

When does Poisson approximate the binomial?

When n is large (> 100) and p is small (< 0.01), the binomial with parameters n and p is well approximated by Poisson with λ = np. The approximation degrades as p approaches 0.1 or n drops below 100.

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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 (lambda). Its probability mass function is:

P(X = k) = e^−λ × λ^k ÷ k!

To avoid overflow for large k, this calculator evaluates the PMF in log space: exp(−λ + k·lnλ − ln(k!)). Cumulative probabilities are exact partial sums, and the upper tails follow by complement. The mean and variance are both equal to λ, so the standard deviation is √λ. This equality of mean and variance is a key diagnostic: if your data has variance much larger than the mean, the Poisson model is likely wrong.

Worked Example: λ = 3, k = 2

A call center averages 3 calls per minute. What is the probability of exactly 2 calls in a given minute?

P(X = 2) = e⁻³ × 3² ÷ 2! = 0.049787 × 9 ÷ 2 ≈ 0.2240. Building up: P(0) ≈ 0.0498, P(1) ≈ 0.1494, P(2) ≈ 0.2240, so P(X ≤ 2) ≈ 0.4232 and P(X ≥ 3) ≈ 0.5768. Mean = variance = 3, SD = √3 ≈ 1.732 calls.

When to Use the Poisson Distribution

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

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

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

Binomial limit: when n is large and p is small, set λ = np for a fast approximation of the binomial.

Assumptions and Common Mistakes

The Poisson model requires: events are independent (one event does not trigger or prevent others), the rate λ is constant over the interval, and two events cannot occur at precisely the same instant. When these fail, the data often shows overdispersion (variance > mean) - a negative binomial distribution usually fits better in that case.

A critical error is mismatching the rate and interval. If λ = 5 per hour and you want the probability for a 30-minute window, use λ = 2.5, not 5. Always confirm that λ and the interval use the same time or space unit.

Normal Approximation for Large λ

For λ ≥ 20 or so, the Poisson distribution is well approximated by a normal distribution with mean = λ and variance = λ. With a continuity correction, P(X ≤ k) ≈ Φ((k + 0.5 − λ) ÷ √λ). This is useful for quick interval estimates or when analytical normal-based methods are needed.

The mean = variance = λ property is the fastest check: compute the sample mean and variance from your data. If they are roughly equal, Poisson is plausible. If variance is much larger (overdispersion) or smaller (underdispersion), consider alternative models.

Poisson Distribution Calculator

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