What is the Poisson distribution used for?

It models the probability of a given number of events occurring in a fixed interval of time or space, when events happen independently and at a known constant average rate — like phone calls per hour or typos per page.

What does λ (lambda) represent?

λ is the average rate of events per interval — for example, if a call center receives 4 calls per minute on average, λ = 4. It's the single parameter that fully defines a Poisson distribution.

What is the formula for the Poisson probability mass function?

P(X = k) = (λᵏ × e⁻λ) / k!, where k is the exact number of events, λ is the average rate, and e is Euler's number (≈2.71828).

What are the mean and variance of a Poisson distribution?

Both the mean and the variance of a Poisson distribution equal λ — a distinctive property that sets it apart from most other distributions, where the mean and variance are independent parameters.

What does P(X ≤ k) mean?

It's the cumulative probability of observing k or fewer events — the sum of P(X = i) for every i from 0 to k. This calculator computes it automatically alongside the exact P(X = k).

When should I use Poisson instead of Binomial?

Use Poisson when you know the average rate of events over a continuous interval but not a fixed number of trials; use Binomial when you have a fixed number of independent trials, each with the same success probability.

Yes. λ represents an average rate and is very often a decimal, like 2.5 events per hour — the calculator accepts any positive real number for λ.

Where is the Poisson distribution used in real life?

It's used to model call-center traffic, website error rates, radioactive decay counts, network packet arrivals, and manufacturing defect rates — anywhere rare, independent events accumulate at a known average rate.

Poisson Distribution Calculator — Find P(X = k) for Rare Events

Our Poisson Distribution Calculator finds the probability of observing exactly k events in a fixed interval, given the average rate λ at which those events occur, using the Poisson probability mass function P(X = k) = λᵏe⁻λ / k!. It also reports the cumulative probability P(X ≤ k), the complementary P(X ≥ k), and the distribution's mean and variance (both equal to λ) — everything needed to reason about rare, independent events like call-center arrivals, website errors per hour, or typos per page.

Quick Answer

The Poisson distribution gives the probability of exactly k events in a fixed interval given an average rate λ: P(X = k) = λᵏe⁻λ/k!. Enter λ and k below to get the exact probability, cumulative probability, and the distribution's mean and variance (both equal to λ).

Rate (λ)

Number of events (k)

Enter a rate and event count, then click Calculate.

How to Use the Poisson Distribution Calculator — Event Probability Online

1
Enter the average rate (λ) — the expected number of events per interval.
2
Enter the exact number of events (k) you want the probability for.
3
Click 'Calculate' to get P(X = k), the cumulative probability P(X ≤ k), and the mean/variance.
4
Review the step-by-step formula breakdown to see how the probability was derived.

Why Use Poisson Distribution Calculator — Event Probability Online?

The Poisson distribution models the number of times a rare, independent event happens in a fixed window — customer arrivals per minute, defects per batch, earthquakes per year — whenever the average rate is known but the exact timing is random. Computing the formula by hand requires factorials and negative exponentials that are easy to mis-key, and forgetting that both the mean and variance equal λ is a common source of confusion. This calculator computes the exact probability, its cumulative counterpart, and the distribution's summary statistics instantly, with every step shown.

Frequently Asked Questions

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