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

Standard library statistical functions.

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# Kurtosis

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

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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>

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</section>

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<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
```

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</section>

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## 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>

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<section class="links">

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

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

</section>

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