@stdlib/stats/base/dists/betaprime/kurtosis/README.md

Standard library statistical functions.

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# Kurtosis

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

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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>

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

</section>

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## 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>

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</section>

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## Examples

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

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

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + 4.0 + EPS;
    v = kurtosis( alpha, beta );
    console.log( 'α: %d, β: %d, Kurt(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

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

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

</section>

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