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Variance

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

[Beta][beta-distribution] distribution [variance][variance].

The [variance][variance] for a [beta][beta-distribution] random variable is

math
\mathop{\mathrm{Var}}\left( X \right) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

where α > 0 is the first shape parameter and β > 0 is the second shape parameter.

## Installation

bash
npm install @stdlib/stats-base-dists-beta-variance

Alternatively,

- To load the package in a website via a script tag without installation and bundlers, use the [ES Module][es-module] available on the [esm][esm-url] branch (see [README][esm-readme]).
- If you are using Deno, visit the [deno][deno-url] branch (see [README][deno-readme] for usage intructions).
- For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [umd][umd-url] branch (see [README][umd-readme]).

The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

## Usage

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

#### variance( alpha, beta )

Returns the [variance][variance] of a [beta][beta-distribution] distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

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

v = variance( 4.0, 12.0 );
// returns ~0.011

v = variance( 8.0, 2.0 );
// returns ~0.015

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 alpha <= 0, the function returns NaN.

javascript
var v = variance( 0.0, 1.0 );
// returns NaN

v = variance( -1.0, 1.0 );
// returns NaN

If provided beta <= 0, the function returns NaN.

javascript
var v = variance( 1.0, 0.0 );
// returns NaN

v = variance( 1.0, -1.0 );
// returns NaN

## Examples

javascript
var randu = require( '@stdlib/random-base-randu' );
var EPS = require( '@stdlib/constants-float64-eps' );
var variance = require( '@stdlib/stats-base-dists-beta-variance' );

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + EPS;
    v = variance( alpha, beta );
    console.log( 'α: %d, β: %d, Var(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}

## C APIs

### Usage

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

#### stdlib_base_dists_beta_variance( alpha, beta )

Returns the [variance][variance] of a [beta][beta-distribution] distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

c
double out = stdlib_base_dists_beta_variance( 1.0, 1.0 );
// returns ~0.083

The function accepts the following arguments:

- alpha: [in] double first shape parameter.
- beta: [in] double second shape parameter.

c
double stdlib_base_dists_beta_variance( const double alpha, const double beta );

### Examples

c
#include "stdlib/stats/base/dists/beta/variance.h"
#include "stdlib/constants/float64/eps.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 alpha;
    double beta;
    double y;
    int i;
    for ( i = 0; i < 25; i++ ) {
        alpha = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        beta = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        y = stdlib_base_dists_beta_variance( alpha, beta );
        printf( "α: %lf, β: %lf, Var(X;α,β): %lf\n", alpha, beta, y );
    }
}

*

## 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].

#### Community

[![Chat][chat-image]][chat-url]

—-

## License

See [LICENSE][stdlib-license].

## Copyright

Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].