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# Probability Density Function

> [Beta prime][betaprime-distribution] distribution probability density function (PDF).

<section class="intro">

The [probability density function][pdf] (PDF) for a [beta prime][betaprime-distribution] random variable is

<!-- <equation class="equation" label="eq:betaprime_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta prime distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} &amp; \text{ for } x &gt; 0 \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:betaprime_pdf">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/docs/img/equation_betaprime_pdf.svg" alt="Probability density function (PDF) for a beta prime distribution.">
    <br>
</div>

<!-- </equation> -->

where `α > 0` is the first shape parameter and `β > 0` is the second shape parameter.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### pdf( x, alpha, beta )

Evaluates the [probability density function][pdf] (PDF) for a [beta prime][betaprime-distribution]  distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var y = pdf( 0.5, 0.5, 1.0 );
// returns ~0.385

y = pdf( 0.1, 1.0, 1.0 );
// returns ~0.826

y = pdf( 0.8, 4.0, 2.0 );
// returns ~0.301
```

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

```javascript
var y = pdf( -0.1, 1.0, 1.0 );
// returns 0.0

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

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

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

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

```javascript
var y = pdf( 0.5, 0.0, 1.0 );
// returns NaN

y = pdf( 0.5, -1.0, 1.0 );
// returns NaN
```

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

```javascript
var y = pdf( 0.5, 1.0, 0.0 );
// returns NaN

y = pdf( 0.5, 1.0, -1.0 );
// returns NaN
```

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

Returns a `function` for evaluating the [PDF][pdf] for a [beta prime][betaprime-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var mypdf = pdf.factory( 0.5, 0.5 );

var y = mypdf( 0.8 );
// returns ~0.198

y = mypdf( 0.3 );
// returns ~0.447
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/float64/eps' );
var pdf = require( '@stdlib/stats/base/dists/betaprime/pdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    y = pdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<section class="links">

[betaprime-distribution]: https://en.wikipedia.org/wiki/Beta_prime_distribution

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

</section>

<!-- /.links -->
