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# Quantile Function

> [Gumbel][gumbel-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [Gumbel][gumbel-distribution] random variable is

<!-- <equation class="equation" label="eq:gumbel_quantile_function" align="center" raw="Q(p)=\mu-\beta\ln(-\ln(p))" alt="Quantile function for a Gumbel distribution."> -->

<div class="equation" align="center" data-raw-text="Q(p)=\mu-\beta\ln(-\ln(p))" data-equation="eq:gumbel_quantile_function">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/gumbel/quantile/docs/img/equation_gumbel_quantile_function.svg" alt="Quantile function for a Gumbel distribution.">
    <br>
</div>

<!-- </equation> -->

for `0 <= p < 1`, where `mu` is the location parameter and `beta > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### quantile( p, mu, beta )

Evaluates the [quantile function][quantile-function] for a [Gumbel][gumbel-distribution] distribution with parameters `mu` (location parameter) and `beta` (scale parameter).

```javascript
var y = quantile( 0.8, 0.0, 1.0 );
// returns ~1.5

y = quantile( 0.5, 4.0, 2.0 );
// returns ~4.733

y = quantile( 0.5, 4.0, 4.0 );
// returns ~5.466
```

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

```javascript
var y = quantile( 1.9, 0.0, 1.0 );
// returns NaN

y = quantile( -0.1, 0.0, 1.0 );
// returns NaN
```

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

```javascript
var y = quantile( NaN, 0.0, 1.0 );
// returns NaN

y = quantile( 0.0, NaN, 1.0 );
// returns NaN

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

If provided `beta <= 0`, the function returns `NaN`.

```javascript
var y = quantile( 0.4, 0.0, -1.0 );
// returns NaN

y = quantile( 0.4, 0.0, 0.0 );
// returns NaN
```

#### quantile.factory( mu, beta )

Returns a function for evaluating the [quantile function][quantile-function] of a [Gumbel][gumbel-distribution] distribution with parameters `mu` and `beta`.

```javascript
var myquantile = quantile.factory( 10.0, 2.0 );

var y = myquantile( 0.2 );
// returns ~9.048

y = myquantile( 0.8 );
// returns ~13.00
```

</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/gumbel/quantile' );

var beta;
var mu;
var p;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
    p = randu();
    mu = randu() * 10.0;
    beta = randu() * 10.0;
    y = quantile( p, mu, beta );
    console.log( 'p: %d, µ: %d, β: %d, Q(p;µ,β): %d', p.toFixed( 4 ), mu.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[gumbel-distribution]: https://en.wikipedia.org/wiki/Gumbel_distribution

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

</section>

<!-- /.links -->
