# Statistical Distribution Calculator (PDF, CDF, Quantile)

> Statistical Distribution Calculator (PDF, CDF, Quantile) is a paid API for AI agents from math-stats.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-10-29).

Computes probability density/mass, cumulative probability, survival probability, and quantiles for normal, binomial, Poisson, and Student t distributions.

## Facts

- Endpoint: GET https://math-stats.x402tookit.com/distribution?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-29
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/statistical-distribution-calculator-pdf-cdf-quantile-6ad1dbf3
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_rNs7inRHhv_5veEdMr7he

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 statistical-distribution-calculator-pdf-cdf-quantile-6ad1dbf3
```

Example prompt: What's the cumulative probability and survival probability for a normal distribution with mean 0 and standard deviation 1 at x = 1.96, and also give me the two-sided tail probability?

## When to prefer this

Choose this endpoint when you need precise statistical distribution values — PDF, CDF, quantile, or survival probability — for normal, binomial, Poisson, or Student t distributions without setting up a local math library. Ideal for hypothesis testing support, confidence interval critical values, or computing tail probabilities in automated statistical workflows. Prefer over general-purpose code execution when a lightweight, pay-per-call API call is sufficient.

## Known failure modes

- Missing required 'dist' parameter returns an error
- Specifying 'binomial' without both 'n' and 'prob' causes a validation failure
- Specifying 't' without 'df' causes a missing-parameter error
- Requesting a quantile (using 'p') on binomial or Poisson (unsupported) may return null or error
- Providing out-of-range values (e.g., p outside [0,1], negative lambda) returns an error
- Omitting both 'x' and 'p' when one is required results in incomplete or null output

## How this service works

Probability density or mass, cumulative probability and quantiles for normal, binomial, Poisson and Student t distributions.

## Output

Returns a JSON object containing the input x value, PDF (probability density or mass), CDF (cumulative probability), survival probability (1 - CDF), z-score (for normal), two-sided tail probability, distribution name, parameters used, and a plain-English summary. Some fields may be null depending on the distribution and query type.

## 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"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [
      "dist"
     ],
     "properties": {
      "n": {
       "type": "number",
       "description": "Binomial trials"
      },
      "p": {
       "type": "number",
       "description": "Probability for a quantile (normal and t only), 0 to 1"
      },
      "x": {
       "type": "number",
       "description": "Value (k for binomial and poisson)"
      },
      "df": {
       "type": "number",
       "description": "t degrees of freedom"
      },
      "sd": {
       "type": "number",
       "description": "Normal standard deviation (default 1)"
      },
      "dist": {
       "type": "string",
       "description": "normal, binomial, poisson or t"
      },
      "mean": {
       "type": "number",
       "description": "Normal mean (default 0)"
      },
      "prob": {
       "type": "number",
       "description": "Binomial success probability"
      },
      "lambda": {
       "type": "number",
       "description": "Poisson rate"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string"
    },
    "example": {
     "type": "object",
     "properties": {
      "x": {
       "type": [
        "null",
        "number"
       ]
      },
      "cdf": {
       "type": [
        "null",
        "number"
       ]
      },
      "pdf": {
       "type": [
        "null",
        "number"
       ]
      },
      "asOf": {
       "type": [
        "null",
        "string"
       ]
      },
      "source": {
       "type": [
        "null",
        "string"
       ]
      },
      "zScore": {
       "type": [
        "null",
        "number"
       ]
      },
      "reasons": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "summary": {
       "type": [
        "null",
        "string"
       ]
      },
      "verdict": {
       "type": [
        "null",
        "string"
       ]
      },
      "degraded": {
       "type": [
    
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/statistical-distribution-calculator-pdf-cdf-quantile-6ad1dbf3/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from math-stats.x402tookit.com](https://www.zero.xyz/host/math-stats.x402tookit.com/llms.txt)