# Poisson Distribution and Event Counts

`D00-F07-A03` · 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/poisson-distribution-and-event-counts/
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 { poissonDistributionAndEventCounts } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/poisson-distribution-and-event-counts";
```

## Signature

```ts
poissonDistributionAndEventCounts(input)
```

Evaluates the Poisson probability of exactly `k` events, the cumulative probability of at most `k`, and the distribution's mean and variance, for a rate of `lambda` events per interval.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record carrying `lambda` (the expected event count per interval), `k` (the event count to evaluate) and `p`, which the family entry point validates before dispatching. · lambda: zero or greater, k: integer, zero or greater, p: between 0 and 1 inclusive |

## Returns

`D00Output`

An object with `pmf` (probability of exactly `k` events), `cdf` (probability of at most `k`), and `mean` and `variance`, both equal to `lambda`.

## Errors

- When p is missing or outside 0 to 1 — this check runs for every topic in the family — throws RangeError
- When lambda is negative, or k is not an integer at or above zero — throws RangeError

## Complexity

Time `O(k^2)`, space `O(k)`.

## 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
poissonDistributionAndEventCounts(input)
```

### Returns

object with 2 fields: pmf, cdf

```json
{
  "pmf": 0.2706705664732254,
  "cdf": 0.6766764161830635
}
```

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