# Covariance and Correlation of Random Variables

`D00-F06-A10` · 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/covariance-and-correlation-of-random-variables/
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 { covarianceAndCorrelationOfRandomVariables } from "fintech-algorithms/foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables";
```

## Signature

```ts
covarianceAndCorrelationOfRandomVariables(input)
```

Computes the probability-weighted covariance of the two variables in a joint distribution table and divides it by the product of their standard deviations to give the correlation.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `{ pA: number; pB: number; pAB: number; randomValues: number[]; probabilities: number[]; joint: { x: number; y: number; p: number }[] }` | yes | `joint` is the probability table as a flat list of cells, each carrying an `x` level, a `y` level, and its probability `p`. These are distributional moments taken over the table, not sample moments over observed data. 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, and neither variable degenerate |

## Returns

`{ covariance: number; correlation: number }`

`covariance` is the probability-weighted sum of the cross-deviations from the two expected values, and `correlation` is that covariance divided by the square root of the product of the two variances.

## 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 either variable has zero variance under the joint table, leaving the correlation denominator at zero — 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
covarianceAndCorrelationOfRandomVariables(input)
```

### Returns

object with 2 fields: covariance, correlation

```json
{
  "covariance": 0.1,
  "correlation": 0.4082482904638631
}
```

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