# Normal Distribution and Standard Normal

`D00-F07-A05` · 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/normal-distribution-and-standard-normal/
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 { normalDistributionAndStandardNormal } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/normal-distribution-and-standard-normal";
```

## Signature

```ts
normalDistributionAndStandardNormal(input)
```

Evaluates the normal density and cumulative probability at `input.x` for a distribution with mean `mu` and standard deviation `sigma`, and returns the standardised score that maps `x` onto the standard normal.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record carrying `x` (the point to evaluate), `mu` and `sigma` (the distribution's mean and standard deviation), plus `p`, which the family entry point validates before dispatching. · sigma: strictly positive, p: between 0 and 1 inclusive |

## Returns

`D00Output`

An object with `pdf` (the density at `x`), `cdf` (the cumulative probability at `x`, computed from an Abramowitz-Stegun error function approximation rather than an exact integral) and `z` (the standardised score).

## 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

## 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
normalDistributionAndStandardNormal(input)
```

### Returns

object with 2 fields: pdf, cdf

```json
{
  "pdf": 0.24197072451914337,
  "cdf": 0.8413447460685428
}
```

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