npm package: @stdlib/stats-base-dists-kumaraswamy-variance
Description: Kumaraswamy's double bounded distribution variance.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Variance

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution [variance][variance].

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

The [variance][variance] for a [Kumaraswamy's double bounded][kumaraswamy-distribution] random variable with first shape parameter `a` and second shape parameter `b` is

<!-- <equation class="equation" label="eq:kumaraswamy_variance" align="center" raw="\operatorname{Var}\left( X \right) = m_2 - m_1^2" alt="Variance for a Kumaraswamy's double bounded distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = m_2 - m_1^2" data-equation="eq:kumaraswamy_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/variance/docs/img/equation_kumaraswamy_variance.svg" alt="Variance for a Kumaraswamy's double bounded distribution.">
    <br>
</div> -->

<!-- </equation> -->

where the raw moments of the distribution are given by

<!-- <equation class="equation" label="eq:kumaraswamy_raw_moments" align="center" raw="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" alt="Raw moments for a Kumaraswamy's double bounded distribution."> -->

<!-- <div class="equation" align="center" data-raw-text="m_n = b \, B\left(1+\tfrac{n}{a}, b \right)" data-equation="eq:kumaraswamy_raw_moments">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/kumaraswamy/variance/docs/img/equation_kumaraswamy_raw_moments.svg" alt="Raw moments for a Kumaraswamy's double bounded distribution.">
    <br>
</div>

<!-- </equation> -->

with `B` denoting the [beta function][beta-function].

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-kumaraswamy-variance
```

</section>

<section class="usage">

## Usage

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

#### variance( a, b )

Returns the [variance][variance] of a [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution with first shape parameter `a` and second shape parameter `b`.

```javascript
var v = variance( 1.0, 1.0 );
// returns ~0.083

v = variance( 4.0, 12.0 );
// returns ~0.017

v = variance( 2.0, 8.0 );
// returns ~0.021
```

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

```javascript
var v = variance( NaN, 2.0 );
// returns NaN

v = variance( 2.0, NaN );
// returns NaN
```

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

```javascript
var y = variance( -1.0, 0.5 );
// returns NaN

y = variance( 0.0, 0.5 );
// returns NaN
```

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

```javascript
var y = variance( 0.5, -1.0 );
// returns NaN

y = variance( 0.5, 0.0 );
// 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 uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var variance = require( '@stdlib/stats-base-dists-kumaraswamy-variance' );

var opts = {
    'dtype': 'float64'
};
var a = uniform( 10, 0.0, 10.0, opts );
var b = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'a: %0.4f, b: %0.4f, Var(X;a,b): %0.4f', a, b, variance );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/kumaraswamy/variance.h"
```

#### stdlib_base_dists_kumaraswamy_variance( a, b )

Returns the [variance][variance] of a [Kumaraswamy's double bounded][kumaraswamy-distribution] distribution with first shape parameter `a` and second shape parameter `b`.

```c
double v = stdlib_base_dists_kumaraswamy_variance( 0.5, 1.0 );
// returns ~0.089
```

The function accepts the following arguments:

-   **a**: `[in] double` first shape parameter.
-   **b**: `[in] double` second shape parameter.

```c
double stdlib_base_dists_kumaraswamy_variance( const double a, const double b );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

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

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/kumaraswamy/variance.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 ) {
    double a;
    double b;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        a = random_uniform( -5.0, 5.0 );
        b = random_uniform( 0.0, 20.0 );
        y = stdlib_base_dists_kumaraswamy_variance( a, b );
        printf( "a: %lf, b: %lf, Var(X;a,b): %lf\n", a, b, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

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<section class="related">

</section>

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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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[beta-function]: https://en.wikipedia.org/wiki/Beta_function

[kumaraswamy-distribution]: https://en.wikipedia.org/wiki/Kumaraswamy_distribution

[variance]: https://en.wikipedia.org/wiki/Variance

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