npm package: @stdlib/stats-base-dists-negative-binomial-quantile
Description: Negative binomial distribution quantile function.
Author: The Stdlib Authors
Latest version: 0.2.2

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# Quantile Function

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

> [Negative binomial][negative-binomial-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [negative binomial][negative-binomial-distribution] random variable `X` returns for any `k` satisfying `0 <= k <= 1` the value `x` for which

<!-- <equation class="equation" label="eq:negative_binomial_quantile_function" align="center" raw="F(x-1;r,p) < k \le F(x;r,p)" alt="Quantile for a negative binomial distribution."> -->

<div class="equation" align="center" data-raw-text="F(x-1;r,p) &lt; k \le F(x;r,p)" data-equation="eq:negative_binomial_quantile_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/quantile/docs/img/equation_negative_binomial_quantile_function.svg" alt="Quantile for a negative binomial distribution.">
    <br>
</div>

<!-- </equation> -->

holds, where `F` is the cumulative distribution function (CDF) of a negative binomial distribution with parameters `r` and `p`, where `r` is the number of successes until the experiment is stopped and `p` is the success probability. The random variable `X` denotes the number of failures until the `r` success is reached.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### quantile( k, r, p )

Evaluates the [quantile function][quantile-function] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.

```javascript
var y = quantile( 0.9, 20.0, 0.2 );
// returns 106

y = quantile( 0.9, 20.0, 0.8 );
// returns 8

y = quantile( 0.5, 10.0, 0.4 );
// returns 14

y = quantile( 0.0, 10.0, 0.9 );
// returns 0
```

If provided an input value `k` outside of `[0,1]`, the function returns `NaN`.

```javascript
var y = quantile( 1.1, 20.0, 0.5 );
// returns NaN

y = quantile( -0.1, 20.0, 0.5 );
// returns NaN
```

While `r` can be interpreted as the number of successes until the experiment is stopped, the [negative binomial][negative-binomial-distribution] distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the [gamma mixing distribution][negative-binomial-mixture-representation].

```javascript
var y = quantile( 0.5, 15.5, 0.5 );
// returns 15

y = quantile( 0.3, 7.4, 0.4 );
// returns 8
```

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

```javascript
var y = quantile( 0.5, 0.0, 0.5 );
// returns NaN

y = quantile( 0.5, -2.0, 0.5 );
// returns NaN
```

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

```javascript
var y = quantile( NaN, 20.0, 0.5 );
// returns NaN

y = quantile( 0.3, NaN, 0.5 );
// returns NaN

y = quantile( 0.3, 20.0, NaN );
// returns NaN
```

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

```javascript
var y = quantile( 0.3, 20.0, -1.0 );
// returns NaN

y = quantile( 0.3, 20.0, 1.5 );
// returns NaN
```

#### quantile.factory( r, p )

Returns a function for evaluating the [quantile function][quantile-function] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`.

```javascript
var myquantile = quantile.factory( 10.0, 0.5 );
var y = myquantile( 0.1 );
// returns 5

y = myquantile( 0.9 );
// returns 16
```

</section>

<!-- /.usage -->

<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 quantile = require( '@stdlib/stats-base-dists-negative-binomial-quantile' );

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

logEachMap( 'k: %0.4f, r: %0.4f, p: %0.4f, Q(k;r,p): %0.4f', k, r, p, quantile );
```

</section>

<!-- /.examples -->

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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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[negative-binomial-mixture-representation]: https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma.E2.80.93Poisson_mixture

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

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

</section>

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