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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]

[Chi][chi-distribution] distribution [cumulative distribution function][cdf].

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

math
F(x;\,k) = P\left(k/2,x^{2}/2\right)

where k is the degrees of freedom and P is the lower regularized incomplete gamma function.

## Installation

bash
npm install @stdlib/stats-base-dists-chi-cdf

Alternatively,

- To load the package in a website via a script tag without installation and bundlers, use the [ES Module][es-module] available on the [esm][esm-url] branch (see [README][esm-readme]).
- If you are using Deno, visit the [deno][deno-url] branch (see [README][deno-readme] for usage intructions).
- For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [umd][umd-url] branch (see [README][umd-readme]).

The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

## Usage

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

#### cdf( x, k )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [chi][chi-distribution] distribution with degrees of freedom k.

javascript
var y = cdf( 2.0, 1.0 );
// returns ~0.954

y = cdf( 2.0, 3.0 );
// returns ~0.739

y = cdf( 1.0, 0.5 );
// returns ~0.846

y = cdf( -1.0, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4.0 );
// returns 0.0

y = cdf( +Infinity, 4.0 );
// returns 1.0

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

javascript
var y = cdf( NaN, 1.0 );
// returns NaN

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

If provided k < 0, the function returns NaN.

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

If provided k = 0, the function evaluates the [CDF][cdf] of a [degenerate distribution][degenerate-distribution] centered at 0.

javascript
var y = cdf( 2.0, 0.0 );
// returns 1.0

y = cdf( -2.0, 0.0 );
// returns 0.0

y = cdf( 0.0, 0.0 );
// returns 1.0

#### cdf.factory( k )

Returns a function for evaluating the [cumulative distribution function][cdf] for a [chi][chi-distribution] distribution with degrees of freedom k.

javascript
var mycdf = cdf.factory( 3.0 );

var y = mycdf( 6.0 );
// returns ~1.0

y = mycdf( 1.5 );
// returns ~0.478

## Examples

javascript
var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var cdf = require( '@stdlib/stats-base-dists-chi-cdf' );

var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu()*5.0 );
    y = cdf( x, k );
    console.log( 'x: %d, k: %d, F(x;k): %d', x.toFixed( 4 ), k.toFixed( 4 ), y.toFixed( 4 ) );
}

*

## 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

[![Chat][chat-image]][chat-url]

—-

## License

See [LICENSE][stdlib-license].

## Copyright

Copyright © 2016-2025. The Stdlib [Authors][stdlib-authors].