# PMF, PDF, CDF, Survival, and Quantile Functions

`D00-F07-A01` · 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/pmf-pdf-cdf-survival-and-quantile-functions/
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 { pmfPdfCdfSurvivalAndQuantileFunctions } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/pmf-pdf-cdf-survival-and-quantile-functions";
```

## Signature

```ts
pmfPdfCdfSurvivalAndQuantileFunctions(input)
```

Reads an empirical sample from `input.values` and reports where the threshold `input.x` sits inside it, plus the sample quantile at probability `input.p`.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record. `values` is the observed sample, `x` is the threshold to evaluate the empirical CDF and survival at, and `p` is the quantile probability. Other keys in the record are ignored. · values: non-empty list of finite numbers, p: between 0 and 1 inclusive |

## Returns

`D00Output`

An object with `cdf` (share of `values` at or below `x`), `survival` (share strictly above `x`) and `quantile` (the linearly interpolated sample quantile at `p`).

## Errors

- When p is missing or outside 0 to 1 — this check runs for every topic in the family — throws RangeError
- When values is missing, empty, or contains a non-finite number — throws RangeError

## Complexity

Time `O(n log n)`, space `O(n)`.

## 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
pmfPdfCdfSurvivalAndQuantileFunctions(input)
```

### Returns

object with 2 fields: cdf, survival

```json
{
  "cdf": 0.8,
  "survival": 0.2
}
```

## 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/pmf-pdf-cdf-survival-and-quantile-functions/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/probability-distributions-and-simulation-basics/pmf-pdf-cdf-survival-and-quantile-functions/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt