npm package: @stdlib/stats-base-dists-betaprime-pdf
Description: Beta prime distribution probability density function (PDF).
Author: The Stdlib Authors
Latest version: 0.3.1

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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] -->

> [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.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/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="installation">

## Installation

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

</section>

<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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var pdf = require( '@stdlib/stats-base-dists-betaprime-pdf' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 5.0, opts );
var beta = uniform( 10, EPS, 5.0, opts );
var x = uniform( 10, 0.0, 1.0, opts );

logEachMap( 'x: %0.4f, α: %0.4f, β: %0.4f, f(x;α,β): %0.4f', x, alpha, beta, pdf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text.  Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/betaprime/pdf.h"
```

#### stdlib_base_dists_betaprime_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).

```c
double y = stdlib_base_dists_betaprime_pdf( 0.5, 1.0, 1.0 );
// returns ~0.444
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **alpha**: `[in] double` first shape parameter.
-   **beta**: `[in] double` second shape parameter.

```c
double stdlib_base_dists_betaprime_pdf( const double x, const double alpha, const double beta );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section`
element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/betaprime/pdf.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double alpha;
    double beta;
    double x;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        x = random_uniform( 0.0, 1.0 );
        alpha = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 5.0 );
        beta = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 5.0 );
        y = stdlib_base_dists_betaprime_pdf( x, alpha, beta );
        printf( "x: %lf, α: %lf, β: %lf, f(x;α,β): %lf\n", x, alpha, beta, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- 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

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---

## License

See [LICENSE][stdlib-license].


## 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-betaprime-pdf/tree/deno
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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-betaprime-pdf/main/LICENSE

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

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

</section>

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