npm package: @stdlib/stats-base-dists-negative-binomial-variance
Description: Negative binomial distribution variance.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Variance

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> [Negative binomial][negative-binomial-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 [negative binomial][negative-binomial-distribution] random variable is

<!-- <equation class="equation" label="eq:negative_binomial_variance" align="center" raw="\operatorname{Var}\left( X \right) = \frac{pr}{(1-p)^{2}}" alt="Variance for a negative binomial distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = \frac{pr}{(1-p)^{2}}" data-equation="eq:negative_binomial_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/variance/docs/img/equation_negative_binomial_variance.svg" alt="Variance for a negative binomial distribution.">
    <br>
</div>

<!-- </equation> -->

where `r` is the number of successes until experiment is stopped and `p` is the success probability in each trial. The random variable `X` denotes the number of failures until the `r` success is reached.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### variance( r, p )

Returns the [variance][variance] of a [negative binomial][negative-binomial-distribution] distribution with parameters `r` (number of successes until experiment is stopped) and `p` (success probability).

```javascript
var v = variance( 100, 0.2 );
// returns ~2000.0

v = variance( 50, 0.5 );
// returns 100.0
```

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

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

v = variance( 20, NaN );
// returns NaN
```

If provided a `r` which is not a positive number, the function returns `NaN`.

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

If provided a success probability `p` outside of `[0,1]`, the function returns `NaN`.

```javascript
var v = variance( 20, -1.0 );
// returns NaN

v = variance( 20, 1.5 );
// 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-negative-binomial-variance' );

var opts = {
    'dtype': 'float64'
};
var r = uniform( 10, 0.0, 100.0, opts );
var p = uniform( 10, 0.0, 1.0, opts );

logEachMap( 'r: %0.4f, p: %0.4f, Var(X;r,p): %0.4f', r, p, 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/negative-binomial/variance.h"
```

#### stdlib_base_dists_negative_binomial_variance( r, p )

Returns the variance of a negative binomial distribution with parameters `r` (number of successes until experiment is stopped) and `p` (success probability).

```c
double v = stdlib_base_dists_negative_binomial_variance( 100.0, 0.2 );
// returns ~2000.0
```

The function accepts the following arguments:

-   **r**: `[in] double` number of successes until experiment is stopped.
-   **p**: `[in] double` success probability.

```c
double stdlib_base_dists_negative_binomial_variance( const double r, const double p );
```

</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/negative-binomial/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 r;
    double p;
    double v;
    int i;


    for ( i = 0; i < 10; i++ ) {
        r = random_uniform( -1.0, 100.0 );
        p = random_uniform( 0.0, 1.0 );
        v = stdlib_base_dists_negative_binomial_variance( r, p );
        printf( "r: %.4f, p: %.4f, Var(X;r,p): %.4f\n", r, p, v );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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</section>

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<section class="main-repo" >

* * *

## 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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[negative-binomial-distribution]: https://en.wikipedia.org/wiki/Negative_binomial_distribution

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

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