@stdlib/stats/base/dists/gumbel/pdf/README.md

Standard library statistical functions.

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# Probability Density Function

> [Gumbel][gumbel-distribution] distribution [probability density function (PDF)][pdf].

<section class="intro">

The [probability density function][pdf] (PDF) for a [Gumbel][gumbel-distribution] random variable is

<!-- <equation class="equation" label="eq:gumbel_pdf" align="center" raw="f(x;\mu,\beta) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta}+e^{- \frac{x-\mu}{\beta}}\right)}" alt="Probability density function (PDF) for a Gumbel distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\mu,\beta) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta}+e^{- \frac{x-\mu}{\beta}}\right)}" data-equation="eq:gumbel_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/gumbel/pdf/docs/img/equation_gumbel_pdf.svg" alt="Probability density function (PDF) for a Gumbel distribution.">
    <br>
</div>

<!-- </equation> -->

where `mu` is the location parameter and `beta > 0` is the scale parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### pdf( x, mu, beta )

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

```javascript
var y = pdf( 0.0, 0.0, 2.0 );
// returns ~0.184

y = pdf( 0.0, 0.0, 1.0 );
// returns ~0.368

y = pdf( 1.0, 3.0, 2.0 );
// returns ~0.09
```

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

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

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

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

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

```javascript
var y = pdf( 2.0, 0.0, -1.0 );
// returns NaN

y = pdf( 2.0, 8.0, 0.0 );
// returns NaN
```

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

Partially applies `mu` and `beta` to create a reusable `function` for evaluating the PDF.

```javascript
var mypdf = pdf.factory( 10.0, 2.0 );

y = mypdf( 10.0 );
// returns ~0.184

y = mypdf( 12.0 );
// returns ~0.127
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

var beta;
var mu;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    beta = randu() * 10.0;
    y = pdf( x, mu, beta );
    console.log( 'x: %d, µ: %d, β: %d, f(x;µ,β): %d', x, mu, beta, y );
}
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

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<section class="links">

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

[pdf]: https://en.wikipedia.org/wiki/Probability_density_function

</section>

<!-- /.links -->