# Student-t Distribution and Heavy Tails

`D00-F07-A07` · Financial Mathematics, Statistics, and Data Foundations → Probability Distributions and Simulation Basics · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/probability-distributions-and-simulation-basics/student-t-distribution-and-heavy-tails/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { studentTDistributionAndHeavyTails } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/student-t-distribution-and-heavy-tails";
```

## Signature

```ts
studentTDistributionAndHeavyTails(input)
```

Evaluates the Student t density at `input.x` for `df` degrees of freedom and reports the distribution's variance, which only exists once `df` passes two.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record carrying `x` (the point to evaluate) and `df` (degrees of freedom), plus `p` and `sigma`, which the family entry point validates before dispatching. · df: strictly positive, sigma: strictly positive, p: between 0 and 1 inclusive |

## Returns

`D00Output`

An object with `pdf` (the density at `x`, using a Lanczos gamma approximation for the normalising constant), `degreesOfFreedom` (echoes `df`) and `variance`, which is `df / (df - 2)` when `df` exceeds two and `null` otherwise.

## Errors

- When p is missing or outside 0 to 1 — this check runs for every topic in the family — throws RangeError
- When sigma is zero or negative — this check runs for every topic from A05 onward — throws RangeError
- When df is zero or negative — throws RangeError

## Complexity

Time `O(1)`, space `O(1)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "values": [0.2, 0.5, 0.7, 1, 1.4],
  "x": 1,
  "p": 0.3,
  "n": 5,
  "k": 2,
  "lambda": 2,
  "seed": 42,
  "sampleCount": 8,
  "mu": 0,
  "sigma": 1,
  "df": 5,
  "shape": 2,
  "scale": 1.5,
  "components": [
    {
      "weight": 0.7,
      "mean": 0,
      "sd": 1
    },
    {
      "weight": 0.3,
      "mean": 3,
      "sd": 0.8
    }
  ]
}
```

### Call

```ts
studentTDistributionAndHeavyTails(input)
```

### Returns

object with 2 fields: pdf, degreesOfFreedom

```json
{
  "pdf": 0.21967979735098053,
  "degreesOfFreedom": 5
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/probability-distributions-and-simulation-basics/student-t-distribution-and-heavy-tails/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/probability-distributions-and-simulation-basics/student-t-distribution-and-heavy-tails/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt