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

Standard library statistical functions.

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# Kurtosis

> [F][f-distribution] distribution [excess kurtosis][kurtosis].

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The [excess kurtosis][kurtosis] for a [F][f-distribution] random variable with numerator degrees of freedom `d1` and denominator degrees of freedom `d2` is

<!-- <equation class="equation" label="eq:f_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}." alt="Excess kurtosis for an F distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}." data-equation="eq:f_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/f/kurtosis/docs/img/equation_f_kurtosis.svg" alt="Excess kurtosis for an F distribution.">
    <br>
</div>

<!-- </equation> -->

for `d1 > 0` and `d2 > 8`. Otherwise, the kurtosis is not defined.

</section>

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<section class="usage">

## Usage

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

#### kurtosis( d1, d2 )

Returns the [excess kurtosis][kurtosis] of a [F][f-distribution] distribution with parameters `d1` (numerator degrees of freedom) and `d2` (denominator degrees of freedom).

```javascript
var v = kurtosis( 4.0, 9.0 );
// returns ~114.273

v = kurtosis( 4.0, 12.0 );
// returns ~26.143

v = kurtosis( 8.0, 9.0 );
// returns ~100.167
```

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

```javascript
var v = kurtosis( NaN, 7.0 );
// returns NaN

v = kurtosis( 3.0, NaN );
// returns NaN
```

If provided `d1 <= 0`, the function returns `NaN`.

```javascript
var v = kurtosis( 0.0, 9.0 );
// returns NaN

v = kurtosis( -1.0, 9.0 );
// returns NaN
```

If provided `d2 <= 8`, the function returns `NaN`.

```javascript
var v = kurtosis( 1.0, 8.0 );
// returns NaN

v = kurtosis( 2.2, 5.5 );
// returns NaN

v = kurtosis( 3.0, -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/f/kurtosis' );

var d1;
var d2;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    d1 = ( randu()*10.0 ) + EPS;
    d2 = ( randu()*20.0 ) + EPS;
    v = kurtosis( d1, d2 );
    console.log( 'd1: %d, d2: %d, skew(X;d1,d2): %d', d1.toFixed( 4 ), d2.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

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</section>

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<section class="links">

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

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

</section>

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