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# Quantile Function

> [Rayleigh][rayleigh-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [Rayleigh][rayleigh-distribution] random variable is

<!-- <equation class="equation" label="eq:rayleigh_quantile_function" align="center" raw="Q(p;\sigma)=\sigma \sqrt{-\ln[(1 - p)^2]}" alt="Quantile function for a Rayleigh distribution."> -->

<div class="equation" align="center" data-raw-text="Q(p;\sigma)=\sigma \sqrt{-\ln[(1 - p)^2]}" data-equation="eq:rayleigh_quantile_function">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/rayleigh/quantile/docs/img/equation_rayleigh_quantile_function.svg" alt="Quantile function for a Rayleigh distribution.">
    <br>
</div>

<!-- </equation> -->

for `0 <= p < 1`, where `sigma > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var quantile = require( '@stdlib/stats/base/dists/rayleigh/quantile' );
```

#### quantile( p, sigma )

Evaluates the [quantile function][quantile-function] for a [Rayleigh][rayleigh-distribution] distribution with parameter `sigma` (scale parameter).

```javascript
var y = quantile( 0.8, 1.0 );
// returns ~1.794

y = quantile( 0.5, 4.0 );
// returns ~4.71
```

If provided a probability `p` outside the interval `[0,1]`, the function returns `NaN`.

```javascript
var y = quantile( 1.9, 1.0 );
// returns NaN

y = quantile( -0.1, 1.0 );
// returns NaN
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = quantile( NaN, 1.0 );
// returns NaN

y = quantile( 0.0, NaN);
// returns NaN
```

If provided `sigma < 0`, the function returns `NaN`.

```javascript
var y = quantile( 0.4, -1.0 );
// returns NaN
```

If provided `sigma = 0`, the function evaluates the [quantile function][quantile-function] of a [degenerate distribution][degenerate-distribution] centered at `0`.

```javascript
var y = quantile( 0.3, 0.0 );
// returns 0.0

y = quantile( 0.9, 0.0 );
// returns 0.0
```

#### quantile.factory( sigma )

Returns a function for evaluating the [quantile function][quantile-function] of a [Rayleigh][rayleigh-distribution] distribution with scale parameter `sigma`.

```javascript
var myQuantile = quantile.factory( 0.4 );

y = myQuantile( 0.4 );
// returns ~0.404

y = myQuantile( 1.0 );
// returns Infinity
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var randu = require( '@stdlib/random/base/randu' );
var quantile = require( '@stdlib/stats/base/dists/rayleigh/quantile' );

var sigma;
var p;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    p = randu();
    sigma = randu() * 10.0;
    y = quantile( p, sigma );
    console.log( 'p: %d, σ: %d, Q(p;σ): %d', p.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

[rayleigh-distribution]: https://en.wikipedia.org/wiki/Rayleigh_distribution

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

</section>

<!-- /.links -->
