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

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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.

Where it fits

TopicProbability & DistributionsAdvancedSubjectData Analysis & StatisticsAdvanced

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.