npm package: @stdlib/stats-base-dists-frechet-pdf
Description: Fréchet distribution probability density function (PDF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Probability Density Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [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="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-frechet-pdf
```

</section>

<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 -->

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<section class="related">

</section>

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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

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

</section>

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