# Normal Distribution Calculator (PDF, CDF, Quantile)

> Normal Distribution Calculator (PDF, CDF, Quantile) 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 the probability density function (PDF), cumulative distribution function (CDF), or quantile (inverse CDF) for a normal (Gaussian) distribution given a mean, standard deviation, and evaluation point.

## Facts

- Endpoint: POST https://distribution.openverbs.com/v1/normal
- 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/normal-distribution-calculator-pdf-cdf-quantile-02b0661a
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_v2vhzWdmsetXYLvpd9ZrJ

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 normal-distribution-calculator-pdf-cdf-quantile-02b0661a -d '<json body>'
```

Example prompt: What is the cumulative probability (CDF) of a normal distribution with mean 0 and standard deviation 1 at x = 1.96?

## When to prefer this

Choose this endpoint when you need fast, on-demand computation of normal distribution statistics (PDF, CDF, or quantile) without setting up a local statistics library. Ideal for AI agents performing probabilistic reasoning, hypothesis testing, or risk analysis that require a single clean API call rather than executing code. Particularly useful in sandboxed environments where running Python or R is not available.

## Known failure modes

- Standard deviation ≤ 0 returns a validation error (schema enforces exclusiveMinimum: 0)
- Quantile called with x outside (0,1) returns an error since p must be a valid probability
- Missing required fields (function or x) returns a 400-level error
- Payment not included or insufficient balance returns a 402 Payment Required response

## How this service works

Normal distribution with optional mean (default 0) and standard deviation sd (default 1). `function` selects the density (pdf), cumulative distribution (cdf) or quantile / inverse-CDF.

## Output

Returns a numeric result corresponding to the requested function: a probability density value (PDF), a cumulative probability between 0 and 1 (CDF), or a variate value (quantile/inverse CDF) for the given normal distribution parameters.

## Request schema (JSON Schema)

```json
{
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "type": "object",
 "properties": {
  "input": {
   "type": "object",
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "type": "string",
     "enum": [
      "POST"
     ]
    },
    "bodyType": {
     "type": "string",
     "enum": [
      "json",
      "form-data",
      "text"
     ]
    },
    "body": {
     "type": "object",
     "properties": {
      "function": {
       "type": "string",
       "enum": [
        "pdf",
        "cdf",
        "quantile"
       ]
      },
      "x": {
       "type": "number",
       "description": "Evaluation point: the variate for pdf/pmf/cdf, or the probability p ∈ (0,1) for quantile."
      },
      "mean": {
       "type": "number",
       "description": "Mean μ. Default 0."
      },
      "sd": {
       "type": "number",
       "exclusiveMinimum": 0,
       "description": "Standard deviation σ > 0. Default 1."
      }
     },
     "required": [
      "function",
      "x"
     ],
     "additionalProperties": false
    }
   },
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "additionalProperties": false
  }
 },
 "required": [
  "input"
 ]
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/normal-distribution-calculator-pdf-cdf-quantile-02b0661a/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)