# 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.

- Calculator id: `poisson-distribution` · Category: Statistics & Probability (`statistics`) · Tool name: `calculate_poisson_probability`
- Canonical page: https://tttkmbb.com/statistics/poisson-distribution · This document: https://tttkmbb.com/statistics/poisson-distribution.md · JSON definition: https://tttkmbb.com/statistics/poisson-distribution.json

## Purpose

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.

**Use when:** 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).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `lambda` | number |  | required | Expected number of events in the interval (mean of the distribution). (> 0, max 10000) |
| `events` | integer |  | required | Number of events of interest. (min 0, max 100000) |

## Output

| Field | Type | Unit | 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. |

## 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.

## Data Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.19 Poisson Distribution — https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm (government, retrieved 2026-09-23)
- Wikipedia – Poisson distribution — https://en.wikipedia.org/wiki/Poisson_distribution (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/poisson-distribution?lambda=…&events=…`
- `POST https://tttkmbb.com/api/v1/calculate/poisson-distribution` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/poisson-distribution · OpenAPI operationId `calculate_poisson_probability` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "poisson-distribution", "inputs": {…}}`

## Example

- λ = 3, k = 2: inputs `{"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: inputs `{"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
```

## Limitations

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). 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. All values are computed from the formula above; no measurement or live data is involved.

## 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

- [Binomial Distribution Calculator](https://tttkmbb.com/statistics/binomial-distribution.md) — Exact probabilities for a fixed number of trials.
- [Probability of Two Events Calculator](https://tttkmbb.com/statistics/probability-of-events.md) — Combine probabilities of two events.