npm package: @stdlib/stats-base-dists-hypergeometric-kurtosis
Description: Hypergeometric distribution excess kurtosis.
Author: The Stdlib Authors
Latest version: 0.2.1

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# Kurtosis

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> [Hypergeometric][hypergeometric-distribution] distribution [excess kurtosis][kurtosis].

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<section class="intro">

Imagine a scenario with a population of size `N`, of which a subpopulation of size `K` can be considered successes. We draw `n` observations from the total population. Defining the random variable `X` as the number of successes in the `n` draws, `X` is said to follow a [hypergeometric distribution][hypergeometric-distribution]. The [excess kurtosis][kurtosis] for a [hypergeometric][hypergeometric-distribution] random variable is

<!-- <equation class="equation" label="eq:hypergeometric_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = \frac{(N-1) N^{2} \left[ N(N+1)-6K(N-K)-6n(N-n) \right]+6nK(N-K)(N-n)(5N-6)}{nK(N-K)(N-n)(N-2)(N-3)}" alt="Excess kurtosis for a hypergeometric distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = \frac{(N-1) N^{2} \left[ N(N+1)-6K(N-K)-6n(N-n) \right]+6nK(N-K)(N-n)(5N-6)}{nK(N-K)(N-n)(N-2)(N-3)}" data-equation="eq:hypergeometric_kurtosis">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/hypergeometric/kurtosis/docs/img/equation_hypergeometric_kurtosis.svg" alt="Excess kurtosis for a hypergeometric distribution.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### kurtosis( N, K, n )

Returns the [excess kurtosis][kurtosis] of a [hypergeometric][hypergeometric-distribution] distribution with parameters `N` (population size), `K` (subpopulation size), and `n` (number of draws).

```javascript
var v = kurtosis( 16, 11, 4 );
// returns ~-0.326

v = kurtosis( 4, 2, 2 );
// returns 0.0
```

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

```javascript
var v = kurtosis( NaN, 10, 4 );
// returns NaN

v = kurtosis( 20, NaN, 4 );
// returns NaN

v = kurtosis( 20, 10, NaN );
// returns NaN
```

If provided a population size `N`, subpopulation size `K`, or draws `n` which is not a nonnegative integer, the function returns `NaN`.

```javascript
var v = kurtosis( 10.5, 5, 2 );
// returns NaN

v = kurtosis( 10, 1.5, 2 );
// returns NaN

v = kurtosis( 10, 5, -2.0 );
// returns NaN
```

If the number of draws `n` or the subpopulation size `K` exceed population size `N`, the function returns `NaN`.

```javascript
var v = kurtosis( 10, 5, 12 );
// returns NaN

v = kurtosis( 10, 12, 5 );
// returns NaN
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

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<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var kurtosis = require( '@stdlib/stats-base-dists-hypergeometric-kurtosis' );

var v;
var i;
var N;
var K;
var n;

for ( i = 0; i < 10; i++ ) {
    N = round( randu() * 20 );
    K = round( randu() * N );
    n = round( randu() * K );
    v = kurtosis( N, K, n );
    console.log( 'N: %d, K: %d, n: %d, Kurt(X;N,K,n): %d', N, K, n, v.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

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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; 2016-2024. The Stdlib [Authors][stdlib-authors].

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[hypergeometric-distribution]: https://en.wikipedia.org/wiki/Hypergeometric_distribution

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

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