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Poisson Distribution Calculator

Computes the probability of exactly, at most, at least, fewer than or more than k events when events occur independently at a constant average rate λ per interval, plus the mean, variance and standard deviation.

When to use

You count independent events over a fixed interval of time or space with a known average rate (calls per hour, defects per metre, arrivals per minute) and need the probability of a given count.

Do not use when: The number of trials is fixed and each has a success probability (use binomial-distribution), the rate varies over the interval, or events cluster (over-dispersion).

Formula

P(X = k) = e^(−λ) · λ^k / k!; P(X ≤ k) = Σ_{i=0..k} P(X = i); P(X ≥ k) = 1 − P(X ≤ k − 1); mean = variance = λ

Terms are evaluated in log space (log k! accumulated incrementally), so large λ and k do not overflow. Scale λ to the interval you are asking about: 2 calls per minute is λ = 10 for a 5-minute window.

Inputs

Parameter	Type	Unit	Required	Description
`lambda`	number		yes	Expected number of events in the interval (mean of the distribution). Range: > 0, ≤ 10000
`events`	integer		yes	Number of events of interest. Range: ≥ 0, ≤ 100000

Outputs

Output	Type	Description
`probability_exact`	number	Probability of exactly k events.
`probability_at_most`	number	Cumulative probability of k or fewer events.
`probability_at_least`	number	Probability of k or more events = 1 − P(X ≤ k − 1).
`probability_less_than`	number	Probability of fewer than k events.
`probability_more_than`	number	Probability of more than k events = 1 − P(X ≤ k).
`mean`	number	λ.
`variance`	number	λ (equal to the mean for a Poisson distribution).
`std_dev`	number	√λ.

Example

λ = 3, k = 2: {"lambda":3,"events":2} → {"probability_exact":0.224042,"probability_at_most":0.42319,"probability_at_least":0.800852,"probability_less_than":0.199148,"probability_more_than":0.57681,"mean":3,"variance":3,"std_dev":1.7321}

λ = 5, no events: {"lambda":5,"events":0} → {"probability_exact":0.006738,"probability_at_most":0.006738,"probability_at_least":1,"probability_more_than":0.993262,"std_dev":2.2361}

GET https://tttkmbb.com/api/v1/calculate/poisson-distribution?lambda=3&events=2

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/poisson-distribution (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/poisson-distribution · Markdown: https://tttkmbb.com/statistics/poisson-distribution.md · JSON definition: https://tttkmbb.com/statistics/poisson-distribution.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="poisson-distribution"
OpenAPI operationId: calculate_poisson_probability
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.19 Poisson Distribution (government)
Wikipedia – Poisson distribution (reference)

FAQ

How do I get λ from a rate?

Multiply the rate by the length of the interval in the same units: 4 customers per hour over 15 minutes gives λ = 1.

When does Poisson approximate the binomial?

For many trials with a small success probability (n ≥ 20, p ≤ 0.05, or more loosely n·p < 10) the binomial(n, p) is close to Poisson(λ = n·p).

Related calculators

Binomial Distribution Calculator — Exact probabilities for a fixed number of trials.
Probability of Two Events Calculator — Combine probabilities of two events.