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

Standard library statistical functions.

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# Probability Density Function

> [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="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 a [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 randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/stats/base/dists/f/pdf' );

var d1;
var d2;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 4.0;
    d1 = randu() * 10.0;
    d2 = randu() * 10.0;
    y = pdf( x, d1, d2 );
    console.log( 'x: %d, d1: %d, d2: %d, f(x;d1,d2): %d', x.toFixed( 4 ), d1.toFixed( 4 ), d2.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>

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<section class="links">

[f-distribution]: https://en.wikipedia.org/wiki/F_distribution

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

</section>

<!-- /.links -->