npm package: @stdlib/stats-base-dists-f-pdf
Description: F 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] -->

> [F][f-distribution] distribution probability density function (PDF).

<section class="intro">

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

<!-- <equation class="equation" label="eq:f_pdf" align="center" raw="f(x; d_1,d_2) = \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}" alt="Probability density function (PDF) for an F distribution."> -->

<div class="equation" align="center" data-raw-text="f(x; d_1,d_2) = \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}" data-equation="eq:f_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/f/pdf/docs/img/equation_f_pdf.svg" alt="Probability density function (PDF) for an F distribution.">
    <br>
</div>

<!-- </equation> -->

where `d1` is the numerator degrees of freedom and `d2` is the denominator degrees of freedom and `B` is the `Beta` function.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### pdf( x, d1, d2 )

Evaluates the [probability density function][pdf] (PDF) for a [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```javascript
var y = pdf( 2.0, 0.5, 1.0 );
// returns ~0.057

y = pdf( 0.1, 1.0, 1.0 );
// returns ~0.915

y = pdf( -1.0, 4.0, 2.0 );
// returns 0.0
```

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

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

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

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

If provided `d1 <= 0`, the function returns `NaN`.

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

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

If provided `d2 <= 0`, the function returns `NaN`.

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

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

#### pdf.factory( d1, d2 )

Returns a `function` for evaluating the [PDF][pdf] of an [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```javascript
var mypdf = pdf.factory( 6.0, 7.0 );
var y = mypdf( 7.0 );
// returns ~0.004

y = mypdf( 2.0 );
// returns ~0.166
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var pdf = require( '@stdlib/stats-base-dists-f-pdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, 0.0, 4.0, opts );
var d1 = uniform( 10, 0.0, 10.0, opts );
var d2 = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, d1: %0.4f, d2: %0.4f, f(x;d1,d2): %0.4f', x, d1, d2, pdf );
```

</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="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].

#### Community

[![Chat][chat-image]][chat-url]

---

## Copyright

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

</section>

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[deno-url]: https://github.com/stdlib-js/stats-base-dists-f-pdf/tree/deno
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[f-distribution]: https://en.wikipedia.org/wiki/F_distribution

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

</section>

<!-- /.links -->