npm package: @stdlib/stats-base-dists-betaprime-kurtosis
Description: Beta prime distribution excess kurtosis.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Kurtosis

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> [Beta prime][betaprime-distribution] distribution [excess kurtosis][kurtosis].

<!-- 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">

The [excess kurtosis][kurtosis] for a [beta prime][betaprime-distribution] random variable with first shape parameter `α` and second shape parameter `β` is

<!-- <equation class="equation" label="eq:betaprime_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}" alt="Kurtosis for a beta prime distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}" data-equation="eq:betaprime_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/betaprime/kurtosis/docs/img/equation_betaprime_kurtosis.svg" alt="Kurtosis for a beta prime distribution.">
    <br>
</div>

<!-- </equation> -->

when `α > 0` and `β > 4`. Otherwise, the excess kurtosis is not defined.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### kurtosis( alpha, beta )

Returns the [excess kurtosis][kurtosis] of a [beta prime][betaprime-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).

```javascript
var v = kurtosis( 2.0, 5.0 );
// returns 54.0

v = kurtosis( 4.0, 12.0 );
// returns ~5.764

v = kurtosis( 12.0, 6.0 );
// returns ~19.49
```

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

```javascript
var v = kurtosis( NaN, 5.0 );
// returns NaN

v = kurtosis( 2.0, NaN );
// returns NaN
```

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

```javascript
var v = kurtosis( 0.0, 5.0 );
// returns NaN

v = kurtosis( -1.0, 5.0 );
// returns NaN
```

If provided `beta <= 4`, the function returns `NaN`.

```javascript
var v = kurtosis( 1.0, 3.5 );
// returns NaN

v = kurtosis( 1.0, 2.0 );
// returns NaN

v = kurtosis( 1.0, -1.0 );
// returns NaN
```

</section>

<!-- /.usage -->

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

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<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 kurtosis = require( '@stdlib/stats-base-dists-betaprime-kurtosis' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 10.0, opts );
var beta = uniform( 10, 4.0 + EPS, 10.0, opts );

logEachMap( 'α: %0.4f, β: %0.4f, Kurt(X;α,β): %0.4f', alpha, beta, kurtosis );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

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<!-- C usage documentation. -->

<section class="usage">

### Usage

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

#### stdlib_base_dists_betaprime_kurtosis( alpha, beta )

Returns the excess kurtosis of a beta prime distribution.

```c
double out = stdlib_base_dists_betaprime_kurtosis( 2.0, 6.0 );
// returns ~26.143
```

The function accepts the following arguments:

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

```c
double stdlib_base_dists_betaprime_kurtosis( const double alpha, const double beta );
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

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

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/betaprime/kurtosis.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 a;
    double b;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        a = random_uniform( 0.0, 20.0 );
        b = random_uniform( 4.0, 24.0 );
        y = stdlib_base_dists_betaprime_kurtosis( a, b );
        printf( "α: %lf, β: %lf, Kurt(X;α,β): %lf", a, b, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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* * *

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

[kurtosis]: https://en.wikipedia.org/wiki/Kurtosis

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