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

Standard library statistical functions.

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# Probability Density Function

> [Fréchet][frechet-distribution] distribution [probability density function][pdf].

<section class="intro">

The [probability density function][pdf] for a [Fréchet][frechet-distribution] random variable is

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

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

<!-- </equation> -->

where `alpha > 0` is the shape, `s > 0` the scale and `m` the location parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### pdf( x, alpha, s, m )

Evaluates the [probability density function][pdf] (PDF) for a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m` at a value `x`.

```javascript
var y = pdf( 10.0, 2.0, 3.0, 5.0 );
// returns ~0.1

y = pdf( -3.0, 1.0, 2.0, -4.0 );
// returns ~0.271

y = pdf( 0.0, 2.0, 1.0, -1.0 );
// returns ~0.736
```

If provided `x <= m`, the function returns `0`.

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

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

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

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

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

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

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

```javascript
var y = pdf( 2.0, -0.1, 1.0, 1.0 );
// returns NaN

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

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

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

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

#### pdf.factory( alpha, s, m )

Returns a function for evaluating the [probability density function][pdf] of a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m`.

```javascript
var mypdf = pdf.factory( 3.0, 3.0, 5.0 );

var y = mypdf( 10.0 );
// returns ~0.104

y = mypdf( 7.0 );
// returns ~0.173
```

</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/frechet/pdf' );

var alpha;
var m;
var s;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
    alpha = randu() * 10.0;
    x = randu() * 10.0;
    s = randu() * 10.0;
    m = randu() * 10.0;
    y = pdf( x, alpha, s, m );
    console.log( 'x: %d, α: %d, s: %d, m: %d, f(x;α,s,m): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

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

<section class="related">

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[frechet-distribution]: https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution

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

</section>

<!-- /.links -->