npm package: @stdlib/stats-base-dists-negative-binomial-cdf
Description: Negative binomial distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Cumulative Distribution 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 [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [negative binomial][negative-binomial-distribution] random variable `X` is

<!-- <equation class="equation" label="eq:negative_binomial_cdf" align="center" raw="F(x;r,p)=1-I_p(x+1,r)" alt="Cumulative distribution function for a negative binomial distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;r,p)=1-I_p(x+1,r)" data-equation="eq:negative_binomial_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/cdf/docs/img/equation_negative_binomial_cdf.svg" alt="Cumulative distribution function for a negative binomial distribution.">
    <br>
</div>

<!-- </equation> -->

where `r` is the number of successes until experiment is stopped, `p` is the success probability in each trial and `I` is the [lower regularized incomplete beta function][incomplete-beta]. 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-cdf
```

</section>

<section class="usage">

## Usage

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

#### cdf( x, r, p )

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

```javascript
var y = cdf( 5.0, 20.0, 0.8 );
// returns ~0.617

y = cdf( 21.0, 20.0, 0.5 );
// returns ~0.622

y = cdf( 5.0, 10.0, 0.4 );
// returns ~0.034

y = cdf( 0.0, 10.0, 0.9 );
// returns ~0.349
```

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 = cdf( 21.0, 15.5, 0.5 );
// returns ~0.859

y = cdf( 5.0, 7.4, 0.4 );
// returns ~0.131
```

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

```javascript
var y = cdf( 2.0, 0.0, 0.5 );
// returns NaN

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

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

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

y = cdf( 0.0, NaN, 0.5 );
// returns NaN

y = cdf( 0.0, 20.0, NaN );
// returns NaN
```

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

```javascript
var y = cdf( 2.0, 20, -1.0 );
// returns NaN

y = cdf( 2.0, 20, 1.5 );
// returns NaN
```

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

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

```javascript
var mycdf = cdf.factory( 10, 0.5 );
var y = mycdf( 3.0 );
// returns ~0.046

y = mycdf( 11.0 );
// returns ~0.668
```

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

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, 0.0, 50.0, opts );
var r = uniform( 10, 0.0, 50.0, opts );
var p = uniform( 10, 0.0, 1.0, opts );

logEachMap( 'x: %0.4f, r: %0.4f, p: %0.4f, F(x;r,p): %0.4f', x, r, p, cdf );
```

</section>

<!-- /.examples -->

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<section class="related">

</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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[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

[incomplete-beta]: https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function

[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

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