npm package: @stdlib/stats-base-dists-discrete-uniform-kurtosis
Description: Discrete uniform distribution excess kurtosis.
Author: The Stdlib Authors
Latest version: 0.2.2

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# Kurtosis

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> [Discrete uniform][discrete-uniform-distribution] distribution [excess kurtosis][kurtosis].

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<section class="intro">

The [excess kurtosis][kurtosis] for a [discrete uniform][discrete-uniform-distribution] random variable with minimum support `a` and maximum support `b` is

<!-- <equation class="equation" label="eq:discrete_uniform_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = -\frac{6(n^{2}+1)}{5(n^{2}-1)}" alt="Excess kurtosis for a discrete uniform distribution."> -->

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

<!-- </equation> -->

where `n = b - a + 1`.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### kurtosis( a, b )

Returns the [excess kurtosis][kurtosis] of a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var v = kurtosis( 0, 1 );
// returns -2.0

v = kurtosis( 4, 12 );
// returns ~-1.23

v = kurtosis( 2, 8 );
// returns ~-1.25
```

If `a` or `b` is not an integer value, the function returns `NaN`.

```javascript
var v = kurtosis( 0.1, 2 );
// returns NaN

v = kurtosis( 0, 2.2 );
// returns NaN

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

v = kurtosis( 2, NaN );
// returns NaN
```

If provided `a > b`, the function returns `NaN`.

```javascript
var v = kurtosis( 3, 2 );
// returns NaN

v = kurtosis( -1, -2 );
// 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 randint = require( '@stdlib/random-base-discrete-uniform' );
var kurtosis = require( '@stdlib/stats-base-dists-discrete-uniform-kurtosis' );

var randa = randint.factory( 0, 10 );
var randb = randint.factory();
var a;
var b;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    a = randa();
    b = randb( a, a+randa() );
    v = kurtosis( a, b );
    console.log( 'a: %d, b: %d, Kurt(X;a,b): %d', a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/discrete-uniform/kurtosis.h"
```

#### stdlib_base_dists_discrete_uniform_kurtosis( a, b )

Returns the [excess kurtosis][kurtosis] of a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```c
double out = stdlib_base_dists_discrete_uniform_kurtosis( 0, 1 );
// returns -2.0
```

The function accepts the following arguments:

-   **a**: `[in] int32_t` minimum support.
-   **b**: `[in] int32_t` maximum support.

```c
double stdlib_base_dists_discrete_uniform_kurtosis( const int32_t a, const int32_t b );
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

<!-- /.notes -->

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

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/discrete-uniform/kurtosis.h"
#include "stdlib/math/base/special/round.h"
#include <stdint.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 ) {
    int32_t a;
    int32_t b;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        a = stdlib_base_round( random_uniform( 0.0, 10.0 ) );
        b = stdlib_base_round( random_uniform( 0.0, 10.0 ) ) + a;
        y = stdlib_base_dists_discrete_uniform_kurtosis( a, b );
        printf( "a: %d, b: %d, Kurt(X;a,b): %lf\n", 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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[kurtosis]: https://en.wikipedia.org/wiki/Kurtosis

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