Poisson Distribution Calculator: P(X=k), λ, cumulative, mean

Compute the Poisson distribution P(X=k) = λ^k·e^(-λ)/k!, cumulative probability, interval and mean μ = σ² = λ. Rare events, queues, radioactive decay, photons.

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Poisson distribution

The Poisson distribution (Siméon Poisson, 1837) models the number X of events occurring in a fixed interval (time, space, volume) when: events are independent, the mean rate λ is constant, and no two events occur simultaneously. Probability mass function: P(X = k) = λ^k · e^(−λ) / k!, with λ > 0 mean count in the interval and k = 0, 1, 2, ... observed count. Discrete with infinite support.

When to use Poisson

Poisson is the limit of a binomial for large n and small p with λ = n·p finite. Models rare events over large intervals. Classic examples: calls at a call centre per hour; customer arrivals at a counter; radioactive decay counts; DNA mutations per segment; typos per book; births per hospital; traffic incidents per road; photons detected by a CCD; cosmic rays per telescope.

Probability mass function

P(X = k) = λ^k · e^(−λ) / k!. Examples with λ = 3: P(X = 0) = e^(−3) ≈ 0.0498; P(X = 1) ≈ 0.1494; P(X = 2) ≈ 0.2240 (peak); P(X = 3) ≈ 0.2240; P(X = 5) ≈ 0.1008. Mode is ⌊λ⌋ (or λ and λ − 1 if λ is integer). CDF F(k) = Σ_{i=0}^{k} λ^i · e^(−λ) / i! has no elementary closed form but is computed as partial sum.

Mean and variance

A defining feature: E[X] = Var(X) = λ. Standard deviation σ = √λ. Example: call centre with λ = 100 calls/hour → μ = 100, σ = 10. By the central limit theorem, for large λ (typically ≥ 20), Poisson is approximated by normal: X ≈ N(λ, λ). The dispersion factor Var/μ = 1 is a diagnostic: if data show Var/μ significantly different from 1, Poisson isn't appropriate (over/under-dispersion). Alternative: negative binomial.

Poisson process

A Poisson process is a sequence of events in time at constant rate. Properties: (

  • the number of events in an interval of length t is Poisson with mean λ·t; (
  • inter-event times follow exponential with mean 1/λ; (
  • given n events in [0, T], their positions are uniformly distributed on [0, T]. Basis of queueing theory (M/M/1, M/M/c) for sizing call centres, networks, computer systems

Poisson ↔ binomial approximation

For binomial B(n, p) with large n and small p (typically n ≥ 20, p ≤ 0.05, np ≤ 10): B(n, p) ≈ Poisson(λ = np). Example: exactly 2 winners in 1000 draws with p = 0.005: exact B(1000, 0.005) vs Poisson(5): P(X = 2)_binom ≈ 0.0841; P(X = 2)_poisson = 25·e^(−5)/2 ≈ 0.0842. Error negligible. The approximation avoids computing large binomial coefficients.

Practical applications

Insurance: claim count in a portfolio. Biology: bacterial colonies on agar, mutation events. Physics: radioactive decay (λ = activity), photon detection, Geiger events. Computer science: packet loss rates, HTTP requests per second. Quality control: defects per unit. Finance: jumps in stochastic models for options. Epidemiology: rare-disease case counts. Sports: Poisson model for goals scored in football — underlying most betting systems.

Related distributions

  • Exponential: inter-event times in a Poisson process. PDF: λe^(−λt). Memoryless.
  • Gamma: waiting time for k events. Sum of k exponentials.
  • Negative binomial: generalisation for over-dispersed counts (variance > mean).
  • Zero-inflated Poisson: excess of zeros (e.g. event counts where many zero cases).
  • Compound Poisson: sum of amounts over a random count (e.g. insurance claim sizes).

How to use

Four modes: exact probability (λ, k → P(X = k)); cumulative (λ, k → P(X ≤ k), P(X ≥ k), P(X > k), P(X < k)); interval (λ, a, b → P(a ≤ X ≤ b)); mean and variance (λ → μ, σ², σ, mode). Log-domain computation for large k (avoids factorial overflow), normal approximation for λ ≥ 100.

Updated July 2026

Frequently Asked Questions

What is the Poisson distribution?

It is a probability distribution describing the number of events occurring in a fixed interval of time or space, when the events are independent and happen at a constant average rate λ. The formula is P(X=k) = λ^k·e^(−λ)/k!. It is used to model rare events.

How do you calculate a Poisson probability?

With the formula P(X=k) = λ^k·e^(−λ)/k!, where λ is the mean number of events and k the specific number. For example, if a switchboard receives on average 3 calls per minute (λ=3), the probability of receiving exactly 5 is 3⁵·e⁻³/5! ≈ 0.101, about 10%.

Why are the mean and variance equal in the Poisson?

It is the distinctive property of the Poisson distribution: both the expected value E[X] and the variance Var(X) equal λ. Hence the standard deviation is √λ. This means a single parameter, λ, completely defines the distribution, greatly simplifying its practical use.

When is the Poisson distribution used?

When counting the number of rare, independent events in a fixed interval: calls to a call centre, radioactive decays, typos per page, insurance claims, customer arrivals. The events must be independent and occur at a constant average rate over time or space.

What is the relation to the binomial distribution?

The Poisson is the limit of the binomial when the number of trials n tends to infinity and the probability p tends to zero, keeping the product np = λ constant. Hence it approximates the binomial well when n is large and p small, simplifying calculations of rare events.

What does the parameter λ represent?

λ (lambda) is the mean number of events expected in the interval considered. It is the distribution's only parameter and fully determines its shape: small values give right-skewed distributions, while for large λ the Poisson approaches a normal distribution.

What is the Poisson distribution used for?

It is used in queueing theory (sizing call centres and servers), in physics for the counting statistics of radioactive decays, in insurance, in biology for mutations, in telecommunications for network traffic, and in any analysis of rare events.

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