npm package: @stdlib/stats-base-dists-pareto-type1-variance
Description: Pareto (Type I) 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] -->

> [Pareto (Type I)][pareto-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 [Pareto (Type I)][pareto-distribution] random variable is

<!-- <equation class="equation" label="eq:pareto_type1_variance" align="center" raw="\operatorname{Var}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 \end{cases}" alt="Variance for a Pareto (Type I) distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = \begin{cases} \infty &amp; \text{for }\alpha\in(0,2] \\ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} &amp; \text{for }\alpha&gt;2 \end{cases}" data-equation="eq:pareto_type1_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/pareto-type1/variance/docs/img/equation_pareto_type1_variance.svg" alt="Variance for a Pareto (Type I) distribution.">
    <br>
</div>

<!-- </equation> -->

where `α > 0` is the shape parameter and `β > 0` is the scale parameter.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-pareto-type1-variance
```

</section>

<section class="usage">

## Usage

```javascript
var variance = require( '@stdlib/stats-base-dists-pareto-type1-variance' );
```

#### variance( alpha, beta )

Returns the [variance][variance] of a [Pareto (Type I)][pareto-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (scale parameter).

```javascript
var v = variance( 2.5, 1.0 );
// returns ~2.222

v = variance( 4.0, 12.0 );
// returns 32.0

v = variance( 8.0, 2.0 );
// returns ~0.109
```

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 `0 < alpha <= 2`, the function returns `+Infinity`.

```javascript
var v = variance( 0.5, 2.0 );
// returns Infinity

v = variance( 1.5, 1.0 );
// returns Infinity
```

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
```

</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 EPS = require( '@stdlib/constants-float64-eps' );
var variance = require( '@stdlib/stats-base-dists-pareto-type1-variance' );

var opts = {
    'dtype': 'float64'
};
var alpha = uniform( 10, EPS, 10.0, opts );
var beta = uniform( 10, EPS, 10.0, opts );

logEachMap( 'α: %0.4f, β: %0.4f, Var(X;α,β): %0.4f', alpha, beta, variance );
```

</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/pareto-type1/variance.h"
```

#### stdlib_base_dists_pareto_type1_variance( alpha, beta )

Returns the [variance][variance] of a [Pareto (Type I)][pareto-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (scale parameter).

```c
double out = stdlib_base_dists_pareto_type1_variance( 4.0, 12.0 );
// returns 32.0
```

The function accepts the following arguments:

-   **alpha**: `[in] double` shape parameter.
-   **beta**: `[in] double` scale parameter.

```c
double stdlib_base_dists_pareto_type1_variance( const double alpha, const double beta );
```

</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/pareto-type1/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 alpha;
    double beta;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        alpha = random_uniform( 0.0, 10.0 );
        beta = random_uniform( 0.0, 10.0 );
        y = stdlib_base_dists_pareto_type1_variance( alpha, beta );
        printf( "α: %lf, β: %lf, Var(X;α,β): %lf\n", alpha, beta, 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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[pareto-distribution]: https://en.wikipedia.org/wiki/Pareto_distribution

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

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