# distributions/continuous

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## Purpose

The `continuous` module implements the nine continuous probability distributions most commonly used in simulations, analytics, and game mechanics: Normal (Gaussian), LogNormal, Exponential, Gamma, Beta, Pareto, Weibull, Cauchy and Logistic. Each is a stateless function taking parameters and an optional `rng?`; together they cover the overwhelming majority of non-uniform sampling needs.

## Features

- `normal(mean, stddev, rng?)` — Box-Muller transform
- `logNormal(mu, sigma, rng?)` — exponential of a Normal
- `exponential(lambda, rng?)` — inverse-CDF method
- `gamma(shape, scale, rng?)` — Marsaglia–Tsang for α ≥ 1, Ahrens-Dieter boost for α < 1
- `beta(alpha, beta, rng?)` — two Gamma draws
- `pareto(xm, alpha, rng?)` — inverse-CDF
- `weibull(k, lambda, rng?)` — inverse-CDF
- `cauchy(x0, gamma, rng?)` — tangent-transform
- `logistic(mu, s, rng?)` — inverse-CDF
- Every function validates positivity of scale parameters and throws `RandomError.DISTRIBUTION.INVALID_PARAM`

## Usage: Basic Example

```ts
import { normal, exponential } from "@suaveplan/random/distributions";

const z = normal(0, 1);
const wait = exponential(2);
```

## Usage: Advanced Example

```ts
import { Sfc32Rng } from "@suaveplan/pseudo-random";
import { gamma, beta } from "@suaveplan/random/distributions";

const rng = new Sfc32Rng("mc-sim-42");
const serviceTime = gamma(2, 3, rng);
const conversionRate = beta(8, 92, rng); // ~8% with informative prior
```

## API Reference

All functions: `(...params, rng?): number`. Parameter positivity is validated; violations raise `RandomError.DISTRIBUTION.INVALID_PARAM`. Default parameter values match the standard form of each distribution (e.g. `normal()` returns a standard Normal).

## Implementation Notes

- Box-Muller discards the paired draw; this is the simpler variant of the transform and has no measurable performance cost at JS call scales.
- The Marsaglia–Tsang Gamma sampler is the fastest acceptance-rejection method known for α ≥ 1. For α < 1 we use the boost trick `gamma(α) = gamma(α+1) · U^(1/α)`.
- Beta uses two Gamma draws from the same source, which leaves the mean and variance exactly right without requiring a specialised sampler.