# Poisson Distribution Calculator

> Poisson Distribution Calculator is a paid API for AI agents from distribution.openverbs.com, paid per call via x402, $0.004/call, status unknown (last checked 2026-09-14).

Computes PMF, CDF, or quantile (inverse CDF) for the Poisson distribution given a rate parameter λ and a count or probability value.

## Facts

- Endpoint: POST https://distribution.openverbs.com/v1/poisson
- Price: $0.004/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-14
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/poisson-distribution-calculator-8778589e
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_E1pb-zepnPMTh5Lq3J2MP

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability poisson-distribution-calculator-8778589e -d '<json body>'
```

Example prompt: What is the probability of exactly 3 events occurring in a Poisson process with a rate of 2.5 events per interval? Use the PMF function.

## When to prefer this

Choose this endpoint when you need fast, on-demand computation of Poisson distribution probabilities or quantiles without setting up a statistics library or runtime environment. Ideal for agents performing probabilistic reasoning about count-based events (arrivals, failures, defects) in real time. Prefer this over a generic math API when you specifically need Poisson PMF, CDF, or inverse CDF evaluated server-side with a simple HTTP call.

## Known failure modes

- Invalid λ ≤ 0 returns a validation error
- Probability p outside (0,1) for quantile function returns an error
- Non-integer or negative k for PMF/CDF may return an error or unexpected result
- Missing required fields (function, x, lambda) returns a schema validation error
- Payment failure via x402 protocol prevents access

## How this service works

Poisson distribution with rate `lambda` (> 0, ≤ 1e9). `function` selects the mass (pmf), cumulative distribution (cdf) or quantile / inverse-CDF.

## Output

Returns a single numeric value: either the probability mass (PMF), the cumulative probability (CDF), or the smallest integer k such that CDF(k) ≥ p (quantile), for the specified Poisson distribution parameters.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "properties": {
    "body": {
     "type": "object",
     "required": [
      "function",
      "x",
      "lambda"
     ],
     "properties": {
      "x": {
       "type": "number",
       "description": "Count k (integer for pmf) for pmf/cdf, or the probability p ∈ (0,1) for quantile."
      },
      "lambda": {
       "type": "number",
       "description": "Rate λ > 0.",
       "exclusiveMinimum": 0
      },
      "function": {
       "enum": [
        "pmf",
        "cdf",
        "quantile"
       ],
       "type": "string"
      }
     },
     "additionalProperties": false
    },
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "POST"
     ],
     "type": "string"
    },
    "bodyType": {
     "enum": [
      "json",
      "form-data",
      "text"
     ],
     "type": "string"
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/poisson-distribution-calculator-8778589e/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from distribution.openverbs.com](https://www.zero.xyz/host/distribution.openverbs.com/llms.txt)