npm package: @stdlib/stats-base-dists-binomial-cdf
Description: 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] -->

> [Binomial][binomial-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

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

<!-- <equation class="equation" label="eq:binomial_cdf" align="center" raw="F(x;n,p) = \sum_{i=0}^{\lfloor x \rfloor} {n\choose i}p^i(1-p)^{n-i}" alt="Cumulative distribution function for a Binomial distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;n,p) = \sum_{i=0}^{\lfloor x \rfloor} {n\choose i}p^i(1-p)^{n-i}" data-equation="eq:binomial_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/binomial/cdf/docs/img/equation_binomial_cdf.svg" alt="Cumulative distribution function for a Binomial distribution.">
    <br>
</div> -->

<!-- </equation> -->

where `n` is the number of trials and `p` is the success probability. The CDF can be equivalently expressed as

<!-- <equation class="equation" label="eq:binomial_cdf_incomplete_beta" align="center" raw="F(x;n,p) = I_{1-p}( n-x, x+1 )" alt="Cumulative distribution function for a Binomial distribution expressed using the incomplete beta function."> -->

<!-- <div class="equation" align="center" data-raw-text="F(x;n,p) = I_{1-p}( n-x, x+1 )" data-equation="eq:binomial_cdf_incomplete_beta">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/binomial/cdf/docs/img/equation_binomial_cdf_incomplete_beta.svg" alt="Cumulative distribution function for a Binomial distribution expressed using the incomplete beta function.">
    <br>
</div>

<!-- </equation> -->

where `I` is the [lower regularized incomplete beta function][incomplete-beta].

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### cdf( x, n, p )

Evaluates the [cumulative distribution function][cdf] for a [binomial][binomial-distribution] distribution with number of trials `n` and success probability `p`.

```javascript
var y = cdf( 3.0, 20, 0.2 );
// returns ~0.411

y = cdf( 21.0, 20, 0.2 );
// returns 1.0

y = cdf( 5.0, 10, 0.4 );
// returns ~0.834

y = cdf( 0.0, 10, 0.4 );
// returns ~0.006
```

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

```javascript
var y = cdf( NaN, 20, 0.5 );
// returns NaN

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

y = cdf( 0.0, 20, NaN );
// returns NaN
```

If provided a number of trials `n` which is not a nonnegative integer, the function returns `NaN`.

```javascript
var y = cdf( 2.0, 1.5, 0.5 );
// returns NaN

y = cdf( 2.0, -2.0, 0.5 );
// 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( n, p )

Returns a function for evaluating the [cumulative distribution function][cdf] of a [binomial][binomial-distribution] distribution with number of trials `n` and success probability `p`.

```javascript
var mycdf = cdf.factory( 10, 0.5 );

var y = mycdf( 3.0 );
// returns ~0.172

y = mycdf( 1.0 );
// returns ~0.011
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var cdf = require( '@stdlib/stats-base-dists-binomial-cdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, 0.0, 20.0, opts );
var n = discreteUniform( 10, 0, 100, opts );
var p = uniform( 10, 0.0, 1.0, opts );

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

</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].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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

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

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

</section>

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