# Lognormal Distribution and Positive Quantities

`D00-F07-A06` · 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/lognormal-distribution-and-positive-quantities/
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 { lognormalDistributionAndPositiveQuantities } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/lognormal-distribution-and-positive-quantities";
```

## Signature

```ts
lognormalDistributionAndPositiveQuantities(input)
```

Evaluates the lognormal density and cumulative probability at `input.x` for a variable whose logarithm is normal with mean `mu` and standard deviation `sigma`, and returns the distribution's median.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record carrying `x` (the point to evaluate, on the original positive scale), `mu` and `sigma` (the mean and standard deviation of the underlying log scale), 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`, `cdf` and `median`. The median is always `exp(mu)`. For `x` at or below zero the function returns `pdf` and `cdf` of zero rather than throwing, since the lognormal puts no mass there.

## 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
lognormalDistributionAndPositiveQuantities(input)
```

### Returns

object with 2 fields: pdf, cdf

```json
{
  "pdf": 0.3989422804014327,
  "cdf": 0.5
}
```

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