Poisson distribution

Poisson distribution gives the probability of a count of independent events in a fixed interval at constant average rate λ. It is fully defined by its single parameter λ, which equals both the mean and variance.

ByHoang TruongUpdated June 30, 2026

What it is

See it move

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A café serves an average of λ = 4 customers per hour, a Poisson process because arrivals are independent and occur at a constant average rate. The probability of exactly 2 customers arriving in a given hour is P(X = 2) = (e⁻⁴ × 4²) ÷ 2! ≈ 0.147, about 15%. Both the mean and the variance of this distribution equal 4.

The formula

LaTeX

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

Variables

$λ$: Average rate of events per interval (must be positive)
$k$: Number of events (0, 1, 2, …)
$e$: Euler's number ≈ 2.71828

Requires: events occur independently, at a constant average rate, and two events cannot occur simultaneously.

LaTeX

\text{Mean} = \text{Variance} = \lambda

Equal mean and variance is a defining property of the Poisson distribution; it can be used to assess whether data follow a Poisson process.

Check yourself

PracticeCORE

A city bus service receives an average of 4 passenger complaints per day. Assuming complaints arrive independently at a constant average rate, the daily count follows a Poisson distribution with λ = 4. Which statement about this distribution is correct?

Select an answer to check your understanding.