# Probability Distributions – t-Distribution

> Probability Distributions – t-Distribution 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 PDF, CDF, or quantile (inverse CDF) for the Student's t-distribution given a value and degrees of freedom

## Facts

- Endpoint: POST https://distribution.openverbs.com/v1/t
- 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/probability-distributions-t-distribution-97aedd46
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_4_U1U4uqWue9c1ACaVboj

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 probability-distributions-t-distribution-97aedd46 -d '<json body>'
```

Example prompt: What is the cumulative probability (CDF) of the t-distribution at x = 1.96 with 30 degrees of freedom?

## When to prefer this

Choose this endpoint when you need precise numerical evaluation of Student's t-distribution functions (PDF, CDF, or quantile/inverse CDF) for statistical computations such as hypothesis testing, confidence interval construction, or Bayesian inference, especially when you need a lightweight, pay-per-call API rather than running a local stats library.

## Known failure modes

- Invalid degrees of freedom (df ≤ 0) returns a validation error
- For the quantile function, x must be in (0,1); values outside this range return an error
- Missing required fields (function, x, df) return a schema validation error
- Extremely large or small x values may produce numerical precision issues
- Unsupported function enum values return a bad request error

## How this service works

Student's t distribution with df degrees of freedom (> 0). `function` selects the density (pdf), cumulative distribution (cdf) or quantile / inverse-CDF.

## Output

Returns a numerical result representing either the probability density (PDF), cumulative probability (CDF), or the quantile value (inverse CDF) of the Student's t-distribution for the specified evaluation point and degrees of freedom.

## 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."
      },
      "df": {
       "type": "number",
       "exclusiveMinimum": 0,
       "description": "Degrees of freedom ν > 0."
      }
     },
     "required": [
      "function",
      "x",
      "df"
     ],
     "additionalProperties": false
    }
   },
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "additionalProperties": false
  }
 },
 "required": [
  "input"
 ]
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/probability-distributions-t-distribution-97aedd46/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)