# Joint, Marginal, and Conditional Distributions

`D00-F06-A09` · Financial Mathematics, Statistics, and Data Foundations → Probability and Random Variables · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/probability-and-random-variables/joint-marginal-and-conditional-distributions/
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 { jointMarginalAndConditionalDistributions } from "fintech-algorithms/foundations/probability-and-random-variables/joint-marginal-and-conditional-distributions";
```

## Signature

```ts
jointMarginalAndConditionalDistributions(input)
```

Collapses a joint probability table over each axis in turn to give the two marginal distributions, then renormalises one slice of it to give the conditional distribution of X at a chosen value of Y.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `{ pA: number; pB: number; pAB: number; randomValues: number[]; probabilities: number[]; joint: { x: number; y: number; p: number }[]; conditionY: number \| string }` | yes | `joint` is the probability table as a flat list of cells, each carrying an `x` level, a `y` level, and its probability `p`. `conditionY` selects the Y level to condition on and is matched by string form, so `1` and `"1"` select the same slice. The family-wide `pA`, `pB`, `pAB`, `randomValues`, and `probabilities` are validated on every call. · pA: 0 <= pA <= 1, pB: 0 <= pB <= 1, pAB: 0 <= pAB <= min(pA, pB), joint: non-empty, every `p` nonnegative, all `p` summing to 1 within 1e-12, conditionY: must name a Y level whose marginal probability is not zero |

## Returns

`{ marginalX: Record<string, number>; marginalY: Record<string, number>; conditionalXGivenY: Record<string, number> }`

`marginalX` and `marginalY` sum the cell probabilities by X level and by Y level, keyed by the level rendered as a string. `conditionalXGivenY` divides the cells at the selected Y level by that level's marginal, so its entries sum to 1.

## Errors

- When `input` is null, an array, or not an object — throws TypeError
- When `pA`, `pB`, or `pAB` falls outside [0, 1], or `pAB` exceeds `min(pA, pB)` — throws RangeError
- When `randomValues` or `probabilities` is missing, empty, or holds a non-finite entry — throws RangeError
- When `randomValues` and `probabilities` differ in length, a probability is negative, or the probabilities do not sum to one within 1e-12 — throws RangeError
- When `joint` is empty, a cell probability is negative, or the cell probabilities do not sum to one within 1e-12 — throws RangeError
- When `conditionY` names a Y level with zero or missing marginal probability — throws RangeError

## Complexity

Time `O(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
{
  "pA": 0.6,
  "pB": 0.5,
  "pAB": 0.3,
  "outcomes": ["up", "flat", "down"],
  "event": ["up", "flat"],
  "prior": 0.01,
  "sensitivity": 0.9,
  "falsePositiveRate": 0.05,
  "randomValues": [0, 1, 2],
  "probabilities": [0.2, 0.5, 0.3],
  "randomVariableKind": "discrete",
  "joint": [
    {
      "x": 0,
      "y": 0,
      "p": 0.3
    },
    {
      "x": 0,
      "y": 1,
      "p": 0.2
    },
    {
      "x": 1,
      "y": 0,
      "p": 0.1
    }
  ],
  "conditionY": 1
}
```

### Call

```ts
jointMarginalAndConditionalDistributions(input)
```

### Returns

object with 2 fields: marginalX, marginalY

```json
{
  "marginalX": {
    "0": 0.5,
    "1": 0.5
  },
  "marginalY": {
    "0": 0.4,
    "1": 0.6000000000000001
  }
}
```

## 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-and-random-variables/joint-marginal-and-conditional-distributions/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/probability-and-random-variables/joint-marginal-and-conditional-distributions/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt