Discrete probability distribution
Poisson distribution
X ~ Poisson(λ)
Counts of independent events in a fixed interval of time or space when events occur at a constant average rate λ. It is the limit of the Binomial as n → ∞ and p → 0 with np = λ, and its mean equals its variance.
Explore it interactivelyKey facts
- Notation
- X ~ Poisson(λ)
- Type
- discrete
- Parameters
- λ — average rate of events (λ > 0)
- Support
- {0, 1, 2, …}
- PMF
- P(X = k) = e⁻ᵟ · λᵏ / k!
- Mean
- λ
- Variance
- λ
When to use it
- Calls arriving at a call centre per hour
- Typos per page
- Customers arriving per minute
- Rare-event counts
Related distributions
Frequently asked questions
What is the Poisson distribution?
Counts of independent events in a fixed interval of time or space when events occur at a constant average rate λ. It is the limit of the Binomial as n → ∞ and p → 0 with np = λ, and its mean equals its variance.
What are the mean and variance of the Poisson distribution?
For X ~ Poisson(λ), the mean is λ and the variance is λ.
When is the Poisson distribution used?
It is commonly used for: Calls arriving at a call centre per hour; Typos per page; Customers arriving per minute; Rare-event counts.